CONTRIBUTIONS TO GAME THEORY AND

ST. PETERSBURG STATE UNIVERSITY THE INTERNATIONAL SOCIETY OF DYNAMIC GAMES (Russian Chapter)

CONTRIBUTIONS TO GAME THEORY AND MANAGEMENT Volume VII

The Seventh International Conference Game Theory and Management June 26–28, 2013, St. Petersburg, Russia

Collected papers Edited by Leon A. Petrosyan and Nikolay A. Zenkevich

St. Petersburg State University St. Petersburg 2014

УДК 518.9, 517.9, 681.3.07 Contributions to game theory and management, vol. VII. Collected papers presented on the Seventh International Conference Game Theory and Management / Editors Leon A. Petrosyan, Nikolay A. Zenkevich. – SPb.: Graduate School of Management SPbSU, 2014. – 438 p. The collection contains papers accepted for the Seventh International Conference Game Theory and Management (June 26–28, 2013, St. Petersburg State University, St. Petersburg, Russia). The presented papers belong to the field of game theory and its applications to management. The volume may be recommended for researches and post-graduate students of management, economic and applied mathematics departments. Sited and reviewed in: Math-Net.Ru and RSCI. Abstracted and indexed in: Mathematical Reviews, Zentralblatt MATH and VINITI.

c Copyright of the authors, 2014

c St. Petersburg State University, 2014

ISSN 2310-2608

Успехи теории игр и менеджмента. Вып. 7. Сб. статей седьмой международной конференции по теории игр и менеджменту / Под ред. Л.А. Петросяна и Н.А. Зенкевича. – СПб.: Высшая школа менеджмента СПбГУ, 2014. – 438 с. Сборник статей содержит работы участников седьмой международной конференции «Теория игр и менеджмент» (26–28 июня 2013 года, Высшая школа менеджмента, Санкт-Петербургский государственный университет, СанктПетербург, Россия). Представленные статьи относятся к теории игр и ее приложениям в менеджменте. Издание представляет интерес для научных работников, аспирантов и студентов старших курсов университетов, специализирующихся по менеджменту, экономике и прикладной математике. Электронные версии серии «Теория игр и менеджмент» размещены в: Math-Net.Ru и РИНЦ. Аннотации и ссылки на статьи цитируются в следующих базах данных: Mathematical Reviews, Zentralblatt MATH и ВИНИТИ.

c Коллектив авторов, 2014

c Санкт-Петербургский государственный университет, 2014

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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On Monotonicity of the SM-nucleolus and the α-nucleolus . . . . . . . . . . Sergei V. Britvin, Svetlana I. Tarashnina

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Efficient Myerson Value for Union Stable Structures . . . . . . . . . . . . . . . . 17 Hua Dong, Hao Sun, Genjiu Xu On the Inverse Problem and the Coalitional Rationality for Binomial Semivalues of Cooperative TU Games . . . . . . . . . . . . . . . . . 24 Irinel Dragan Stackelberg Oligopoly Games: the Model and the 1-concavity of its Dual Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Theo Driessen, Aymeric Lardon, Dongshuang Hou On Uniqueness of Coalitional Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Michael Finus, Pierre von Mouche and Bianca Rundshagen Quality Level Choice Model under Oligopoly Competition on a Fitness Service Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Margarita A. Gladkova, Maria A. Kazantseva, Nikolay A. Zenkevich A Problem of Purpose Resource Use in Two-Level Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Olga I. Gorbaneva, Guennady A. Ougolnitsky Multicriteria Coalitional Model of Decision-making over the Set of Projects with Constant Payoff Matrix in the Noncooperative Game Xeniya Grigorieva

93

Differential Games with Random Duration: A Hybrid Systems Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Dmitry Gromov, Ekaterina Gromova Simulations of Evolutionary Models of a Stock Market . . . . . . . . . . . . . . 120 Gubar Elena Equilibrium in Secure Strategies in the Bertrand-Edgeworth Duopoly Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Mikhail Iskakov, Alexey Iskakov Equilibrium Strategies in Two-Sided Mate Choice Problem with Age Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 Anna A. Ivashko, Elena N. Konovalchikova Stationary State in a Multistage Auction Model . . . . . . . . . . . . . . . . . . . 151 Aleksei Y. Kondratev

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Phenomenon of Narrow Throats of Level Sets of Value Function in Differential Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Sergey S. Kumkov, Valerii S. Patsko Strictly Strong (n − 1)-equilibrium in n-person Multicriteria Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Denis V. Kuzyutin, Mariya V. Nikitina, Yaroslavna B. Pankratova The Nash Equilibrium in Multy-Product Inventory Model . . . . . . . . . . 191 Elena A. Lezhnina, Victor V. Zakharov Nash Equilibria Conditions for Stochastic Positional Games . . . . . . . . 201 Dmitrii Lozovanu, Stefan Pickl Pricing in Queueing Systems M/M/m with Delays . . . . . . . . . . . . . . . . . 214 Anna V. Melnik How to arrange a Singles’ Party: Coalition Formation in Matching Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Joseph E. Mullat Evolution of Agents Behavior in the Labor Market . . . . . . . . . . . . . . . . . 239 Maria A. Nastych, Nikolai D. Balashov An Axiomatization of the Proportional Prenucleolus . . . . . . . . . . . . . . . . 246 Natalia Naumova Competition Form of Bargaining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 Tatyana E. Nosalskaya Interval Obligation Rules and Related Results . . . . . . . . . . . . . . . . . . . . . . 262 Osman Palancı, Sırma Zeynep Alparslan Gök, Gerhald Wilhelm Weber Stable Cooperation in Graph-Restricted Games . . . . . . . . . . . . . . . . . . . . 271 Elena Parilina, Artem Sedakov Power in Game Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 Rodolfo Coelho Prates Completions for Space of Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 Victor V. Rozen Bridging the Gap between the Nash and Kalai-Smorodinsky Bargaining Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 Shiran Rachmilevitch Unravelling Conditions for Successful Change Management Through Evolutionary Games of Deterrence . . . . . . . . . . . . . . . . . . . . . . . . 313 Michel Rudnianski, Cerasela Tanasescu Applying Game Theory in Procurement. An Approach for Coping with Dynamic Conditions in Supply Chains . . . . . . . . . . . . 326 Günther Schuh, Simone Runge

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An Axiomatization of the Myerson Value . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 Özer Selçuk, Takamasa Suzuki Multi-period Cooperative Vehicle Routing Games . . . . . . . . . . . . . . . . . . 349 Alexander Shchegryaev, Victor V. Zakharov Mechanisms of Endogenous Allocation of Firms and Workers in Urban Area: from Monocentric to Polycentric City . . . . . . . . . . . . . . 360 Alexandr P. Sidorov The Irrational Behavior Proof Condition for Linear-Quadratic Discrete-time Dynamic Games with Nontransferable Payoffs . . . . . . . . 384 Anna V. Tur Von Neumann-Morgernstern Modified Generalized Raiffa Solution and its Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 Radim Valenčík, Ondřej Černík Subgame Consistent Cooperative Solution of Stochastic Dynamic Game of Public Goods Provision . . . . . . . . . . . . . . . . . . . . . . . . . . 404 David W.K. Yeung, Leon A. Petrosyan Joint Venture’s Dynamic Stability with Application to the Renault-Nissan Alliance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 Nikolay A. Zenkevich, Anastasia F. Koroleva Symmetric Core of Cooperative Side Payments Game . . . . . . . . . . . . . 428 Alexandra B. Zinchenko

Preface

This edited volume contains a selection of papers that are an outgrowth of the Seventh International Conference on Game Theory and Management with a few additional contributed papers. These papers present an outlook of the current development of the theory of games and its applications to management and various domains, in particular, finance, mechanism design, environment and economics. The International Conference on Game Theory and Management, a three day conference, was held in St. Petersburg, Russia in June 26-28, 2013. The conference was organized by Graduate School of Management St. Petersburg State University in collaboration with The International Society of Dynamic Games (Russian Chapter) and Faculty of Applied Mathematics and Control Processes (SPbSU). More than 100 participants from 26 countries had an opportunity to hear state-of-the-art presentations on a wide range of game-theoretic models, both theory and management applications. Plenary lectures covered different areas of games and management applications. They had been delivered by Professor Finn Kydland, Nobel Prize in Economic Sciences, 2004, University of California, Santa Barbara (USA); Professor Burkhard Monien, Paderborn University (Germany); Professor Bernard De Meyer, Université Paris 1, Panthéon-Sorbonne (France); Professor Leon Petrosyan, St. Petersburg State University (Russia). The importance of strategic behavior in the human and social world is increasingly recognized in theory and practice. As a result, game theory has emerged as a fundamental instrument in pure and applied research. The discipline of game theory studies decision making in an interactive environment. It draws on mathematics, statistics, operations research, engineering, biology, economics, political science and other subjects. In canonical form, a game takes place when an individual pursues an objective(s) in a situation in which other individuals concurrently pursue other (possibly conflicting, possibly overlapping) objectives and in the same time the objectives cannot be reached by individual actions of one decision maker. The problem is then to determine each individual’s optimal decision, how these decisions interact to produce equilibrium, and the properties of such outcomes. The foundations of game theory were laid more than sixty years ago by von Neumann and Morgenstern (1944). Theoretical research and applications in games are proceeding apace, in areas ranging from aircraft and missile control to inventory management, market development, natural resources extraction, competition policy, negotiation techniques, macroeconomic and environmental planning, capital accumulation and investment. In all these areas, game theory is perhaps the most sophisticated and fertile paradigm applied mathematics can offer to study and analyze decision making under real world conditions. The papers presented at this Seventh International Conference on Game Theory and Management certainly reflect both the maturity and the vitality of modern day game theory and management science in general, and of

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dynamic games, in particular. The maturity can be seen from the sophistication of the theorems, proofs, methods and numerical algorithms contained in the most of the papers in these contributions. The vitality is manifested by the range of new ideas, new applications, the growing number of young researchers and the expanding world wide coverage of research centers and institutes from whence the contributions originated. The contributions demonstrate that GTM2013 offers an interactive program on wide range of latest developments in game theory and management. It includes recent advances in topics with high future potential and exiting developments in classical fields. We thank Anna Tur from the Faculty of Applied Mathematics (SPbSU) for displaying extreme patience typesetting the manuscript.

Editors, Leon A. Petrosyan and Nikolay A. Zenkevich

On Monotonicity of the SM-nucleolus and the α-nucleolus Sergei V. Britvin and Svetlana I. Tarashnina St.Petersburg State University, Faculty of Applied Mathematics and Control Processes, Bibliotechnaya pl. 2, St.Petersburg, 198504, Russia E-mail: [email protected] E-mail: [email protected] Abstract In this paper two single-valued solution concepts of a TU-game with a finite set of players, the SM-nucleolus and the α-nucleolus, are considered. Based on the procedure of finding lexicographical minimum, there was proposed an algorithm allowing to calculate the SM-nucleolus as well as the prenucleolus. This algorithm is modified to calculate the α-nucleolus for any fixed α ∈ [0, 1]. Using this algorithm the monotonicity properties of the SM-nucleolus and the α-nucleolus are studied by means of counterexamples. Keywords: cooperative TU-game, solution concept, aggregate and coalitional monotonicity, the SM-nucleolus, the α-nucleolus.

1.

Introduction

In this paper we examine two single-valued solution concepts of a transferable utility game (TU-game) with a finite set of players — the SM-nucleolus (Tarashnina, 2011) and the α-nucleolus (Smirnova and Tarashnina, 2011). Both of these solution concepts take into account "the blocking power" of a coalition, the amount which the coalition cannot be prevented from by the complement coalition. Based on the procedure (Maschler et al., 1979) of finding lexicographical minimum, there was proposed an algorithm (Britvin and Tarashnina, 2013) allowing to calculate the SM-nucleolus as well as the prenucleolus (Schmeidler, 1969). By introducing the special numbering of coalitions the problem of finding the SM-nucleolus of a cooperative n-person game is reduced to solving a single linear program with 2n rows and (n + 1) columns. The initial values of the problem coefficients are 0, 1, −1. This algorithm is modified to calculate the α-nucleolus for any fixed α ∈ [0, 1]. In this work we consider two properties of single-valued solution concepts of TU-games: aggregate and coalitional monotonicity. Aggregate monotonicity means that if the worth of the grand coalition icreases while the worths of all other coalitions remain the same, then the players payoffs should not decrease. Coalitional monotonicity applies this rule to any coalition S ⊂ N in a game. The Shapley value (Shapley, 1953) satisfies aggregate and coalitional monotonicity. N. Megiddo (Megiddo, 1974) presented an example of nine person cooperative TU-game that shows that another well-known single-valued solution, the nucleolus (Schmeidler, 1969), violates aggregate monotonicity. It is known that the nucleolus does not satisfy coalitional monotonicity too (Young, 1985). In this paper we verify that the SM-nucleolus does not satisfy aggregate and coalitional monotonicity. The paper is organized as follows. In section 2 basic definitions and notations are given. For any fixed α we describe the algorithm of finding the α-nucleolus (including the SM-nucleolus) in section 3. Using this algorithm we study the monotonicity properties of the considered solution concepts by means of counterexamples in section 4.

On Monotonicity of the SM-nucleolus and the α-nucleolus 2.

9

Basic definitions and notations

In this paper we consider cooperative games with transferable utilities (TU-games). A TU-game is a pair (N, v), where N = {1, ..., n} is the set of players and v : 2N → R is a characteristic function with v(∅) = 0. More information about cooperative game theory can be found in (Petrosjan et al., 2012) and (Pecherskiy and Yanovskaya, 2004). The set of all TU-games with the fixed set of players N is denoted by GN . Consider a game (N, v) from GN . Assume that the players have formed the maximal coalition N and consider the distribution of v(N ) among all the players. We define the set of feasible payoff vectors as follows: X X(N, v) = {x ∈ Rn | xi ≤ v(N )}. i∈N

The set X (N, v) ⊂ X(N, v) such that 0

X 0 (N, v) = {x ∈ Rn |

X

xi = v(N )}

(1)

i∈N

is a set of group rational payoff vectors of the game (N, v). It follows from (1) that x ∈ X 0 (N, v) if and only if for all S ⊂ N it holds x(S) + x(N \S) = v(N ), where x(S) =

X

xi .

i∈S

Definition 1. A solution of a TU-game on GN is a mapping f that matches for every game (N, v) ∈ GN the subset f (N, v) of X(N, v). In the paper we study two single-valued solution concepts: the SM-nucleolus and the α-nucleolus. To introduce the definitions of these solution concepts, we should define the excess of a coalition.

Definition 2. The excess e(x, S, v) of a coalition S at x ∈ X 0 (N, v) is calculated as e(x, S, v) = v(S) − x(S). (2) Let (N, v) be a TU-game. The dual game (N, v ∗ ) of (N, v) is defined by v ∗ (S) = v(N ) − v(N \S) for all coalitions S. Let us clarify the notion of the constructive and the blocking power of S. The constructive power of S is the worth of the coalition, or exactly what S can reach by cooperation. By the blocking power of coalition S we understand the amount v ∗ (S) that this coalition brings to N if the last will be formed — its contribution to the grand coalition. The difference between v(N ) and v(N \S) is a subject which should be taken into account in a solution of a game. In our opinion, the blocking power can be judged as a measure of necessity of S for N — how much S contributes to N . So, each coalition S is estimated by N in this spirit. In order to introduce the SM-nucleolus we define the sum-excess of the coalition.

10

Sergei V. Britvin, Svetlana I. Tarashnina

Definition 3. The sum-excess e¯(x, S, v) of a coalition S at x ∈ X 0 (N, v) in the game (N, v) is 1 1 e¯(x, S, v) = e(x, S, v) + e(x, S, v ∗ ). 2 2 We define for some ϕ ∈ Rn the mapping θ : Rn → Rn such that ψ = θ(ϕ) ∈ Rn means that ψ is obtained from ϕ by ordering its components in non-increasing order. After calculating the sum-excess for each S ⊆ N we obtain the sum-excess vector e¯(x, v) = {¯ e(x, S, v)}S⊆N of dimension 2n . Definition 4. The SM-nucleolus of the game (N, v) is the set XSM ⊂ X 0 (N, v) such that for every x ∈ XSM vector θ({¯ e(x, S, v)}S⊆N ) is lexicographically the smallest: XSM (N, v) = {x ∈ X 0 |θ({¯ e(x, S, v)}S⊆N ) lex θ({¯ e(y, S, v)}S⊆N ), ∀y ∈ X 0 (N, v)}. To introduce the α-nucleolus we define the α-excess of a coalition. Definition 5. The α-excess eα (x, S, v) of a coalition S at x ∈ X 0 (N, v) is eα (x, S, v) = αe(x, S, v) + (1 − α)e(x, S, v ∗ ), α ∈ [0, 1].

(3)

After calculating the α-excess for each S ⊆ N we obtain the α-excess vector eα (x, v) = {eα (x, S, v)}S⊆N of dimension 2n . Definition 6. The α-nucleolus of the game (N, v) is the set Xα ⊂ X 0 (N, v) such that for every x ∈ Xα vector θ({eα (x, S, v)}S⊆N ) is lexicographically the smallest: Xα (N, v) = {x ∈ X 0 (N, v)|θ({eα (x, S, v)}S⊆N ) lex θ({eα (y, S, v)}S⊆N ),

∀y ∈ X 0 (N, v)}.

It is important to note that both solution concepts represent a unique point in X 0 , so they are single-valued solutions (Smirnova and Tarashnina, 2011). Obviously, if α = 21 , then Xα (N, v) = XSM (N, v). This means that the SM-nucleolus is a special case of the α-nucleolus. 3.

Algorithm

In the literature there was presented an algorithm of finding the SM-nucleolus of any TU-game (Britvin and Tarashnina, 2013). Here we modify this algorithm for calculation the α-nucleolus. First, we should replace the excess in the procedure of finding the lexicographical minimum (Maschler et al., 1979) to the α-excess. We obtain the following procedure.

On Monotonicity of the SM-nucleolus and the α-nucleolus

11

1. Consider a pair (X 0 , J 0 ), where J 0 consists of all possible coalitions except the empty one. 2. Recursively find ut = min max eα (x, S, v), (4) x∈X t−1 S⊆J t−1

X t = {x ∈ X t−1 | eα (x, S, v) ≤ ut , ∀S ⊆ J t−1 }, Jt = {S ⊆ J t−1 | eα (x, S, v) = ut , ∀x ∈ X t }, J t = J t−1 \Jt .

3. If J t = ∅, then we stop, otherwise we go to step 2 with t = t + 1. In the game there may be formed 2n coalitions (including the empty one). We will not consider the empty coalition. Let us enumerate all the other coalitions in the following way. Suppose that n-person game has been built dynamically by adding one player at each step. In a game with one player the single coalition {1} is number 1. When the second player enters the game he brings there two additional coalitions. The sequence of coalitions in ascending order for two-person game is as follows: {1}, {2}, {1, 2}. Further, for a three-person game we have: {1}, {2}, {1,2}, {3}, {1,3}, {2,3}, {1,2,3}. And so on. Assume that the sequence of coalitions has been formed in ascending order for a k-person game. Adding to this game the (k + 1)-th player entails forming 2k coalitions. We determine the order of the added coalitions. Let coalition {k + 1} be the first of the added coalitions. Among the remaining coalitions we do not pay attention to the (k + 1)-th player, then we obtain a set of coalitions for a k-person game, which is already built in ascending order. Finally, we extend this numbering to the additional coalitions. As a result, each coalition in a (k + 1)-person game will be numbered. For example, the coalitions in 4-person game in ascending order look like {1}, {2}, {1, 2}, {3}, {1, 3}, {2, 3}, {1, 2, 3}, {4}, {1, 4}, {2, 4}, {1, 2, 4}, {3, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4}.

Problem (4) for t = 1 is equivalent to the following task  min u1 ,    u1 ≥ eα (x, S, v),  S ⊆ J 0,    x ∈ X 0.

By using formulas (1), (2) and (3) we transform it to the following form  1  min uP, 1 u + i∈S xi ≥ αv(S) + (1 − α)(v(N ) − v(N \S)), S ⊆ J 0 ,  P i∈N xi = v(N ).

The resulting problem is a linear programming. Given the suggested order of the coalitions in J 0 we obtain the matrix form

12

Sergei V. Britvin, Svetlana I. Tarashnina  T  min c z, Az ≥ b,   Aeq z = beq ,

with

z=

1 u . x

(5)

The parameters of this linear programming are the following:      1 1 100 0 1 0 1 0           c = 0 , A = I A∗ , I = 1 , A∗ = 1 1 0  ..   ..   .. . . . 0 1 011 

v(1) v(2) v(1, 2) .. .





 v(N ) − v(2, 3, ..., n)  v(N ) − v(1, 3, ..., n)        + (1 − α) v(N ) − v(3, 4, ..., n) ,    ..    .

   b = α   v(2, 3, ..., n) Aeq = 1 1 1 ... 1 ,

 ... 0 0 ... 0 0  ... 0 0 , ..  .. . . ... 1 1

v(N ) − v(1)

beq = v(N ).

Based on the theorem from (Britvin and Tarashnina, 2013), it is easy to prove that there exists a unique solution z ∗ of this linear programming. So, the calculation procedure is stopped and we obtain the α-nucleolus in the form   ∗ z2 (α)  z3∗ (α)    Xα =   , α ∈ [0, 1]. ..   . ∗ zn+1 (α)

4.

The monotonicity of the SM-nucleolus and the α-nucleolus

In this paper we investigate the monotonicity properties of single-valued solution concepts of TU-games: aggregate monotonicity and coalitional monotonicity. First, let us define these properties. Definition 7. A single-valued solution concept f satisfies aggregate monotonicity if for every pair of games (N, v) and (N, w) such that v(N ) < w(N ),

(6)

v(S) = w(S) for all S ⊂ N,

(7)

fi (N, v) ≤ fi (N, w) for all i ∈ N.

(8)

it follows that


13

Definition 8. A single-valued solution concept f satisfies coalitional monotonicity if for every pair of games (N, v) and (N, w) such that

it follows that

v(T ) < w(T ) for any T ⊂ N,

(9)

v(S) = w(S) for all S ⊂ N, S 6= T,

(10)

fi (N, v) ≤ fi (N, w) for all i ∈ T.

(11)

Let us give the following example (Megiddo, 1974) that contains of two cooperative games and illustrates the absence of aggregate monotonicity of the SMnucleolus. Example 1. Let N = {1, 2, ..., 9} and ϕ = (1, 1, 1, 2, 2, 2, 1, 1, 1). Consider two groups of coalitions: A = {(1, 2, 3), (1, 4), (2, 4), (3, 4), (1, 5), (2, 5), (3, 5), (7, 8, 9)}, B = {(1, 2, 3, 6, 7), (1, 2, 3, 6, 8), (1, 2, 3, 6, 9), (4, 5, 6)}.

Define the characteristic function v as  6, if S ∈ A,    9, if S ∈ B, v=  12, if S = N,   P i∈S ϕi − 1, otherwise. The characteristic function w has the following form

 6, if S ∈ A,    9, if S ∈ B, w=  13, if S = N,   P i∈S ϕi − 1, otherwise.

It is obvious that conditions (6) and (7) hold for characteristic functions v and w. For some fixed α ∈ [0, 1] we calculate the α-nucleolus for games (N, v) and (N, w) using the algorithm, presented in section 3. Assuming that the parameter α is moving along the interval [0, 1] with the step of 0.1, we have the following payoff vectors presented in Table 1. The special case α = 12 with XSM = Xα is shown in bold. Consider the payoffs that player 6 gets according to the α-nucleolus for α ≥ 0.1. We can see that Xα6 (N, v) > Xα6 (N, w). Therefore, inequality (8) is not satisfied. So, by means of the counterexample we can verify that aggregate monotonicity does not hold for the α-nucleolus with 0.1 ≤ α ≤ 1. By using the dichotomy method for this pair of games we can approximately calculate the maximum α∗ such that 0 < α∗ < 0.1 for which aggregate monotonicity is satisfied and for some α > α∗ aggregate monotonicity is not satisfied. In the current example α∗ ≈ 0.075.

14

Sergei V. Britvin, Svetlana I. Tarashnina Table 1: The α-nucleolus for (N, v) and (N, w). α 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Xα (N, v) Xα (N, w) (1, 1, 1, 2, 2, 2, 1, 1, 1) (1.11, 1.11, 1.11, 2.11, 2.11, 2.11, 1.11, 1.11, 1.11) (1, 1, 1, 2, 2, 2, 1, 1, 1) (1.11, 1.11, 1.11, 2.22, 2.22, 1.92, 1.11, 1.11, 1.11) (1, 1, 1, 2, 2, 2, 1, 1, 1) (1.11, 1.11, 1.11, 2.22, 2.22, 1.89, 1.11, 1.11, 1.11) (1, 1, 1, 2, 2, 2, 1, 1, 1) (1.11, 1.11, 1.11, 2.22, 2.22, 1.89, 1.11, 1.11, 1.11) (1, 1, 1, 2, 2, 2, 1, 1, 1) (1.11, 1.11, 1.11, 2.22, 2.22, 1.89, 1.11, 1.11, 1.11) (1,1,1,2,2,2,1,1,1) (1.11,1.11,1.11,2.22,2.22,1.89,1.11,1.11,1.11) (1, 1, 1, 2, 2, 2, 1, 1, 1) (1.11, 1.11, 1.11, 2.22, 2.22, 1.89, 1.11, 1.11, 1.11) (1, 1, 1, 2, 2, 2, 1, 1, 1) (1.11, 1.11, 1.11, 2.22, 2.22, 1.89, 1.11, 1.11, 1.11) (1, 1, 1, 2, 2, 2, 1, 1, 1) (1.11, 1.11, 1.11, 2.22, 2.22, 1.89, 1.11, 1.11, 1.11) (1, 1, 1, 2, 2, 2, 1, 1, 1) (1.11, 1.11, 1.11, 2.22, 2.22, 1.89, 1.11, 1.11, 1.11) (1, 1, 1, 2, 2, 2, 1, 1, 1) (1.11, 1.11, 1.11, 2.22, 2.22, 1.89, 1.11, 1.11, 1.11)

Let us give one more example. Example 2. Consider the characteristic function v ′ that coincides with v from Example 1: v ′ (S) = v(S) for all S ⊆ N. The characteristic function w′ is constructed in the following way ( 7, if S = (4, 7), w′ (S) = v(S), otherwise. It is obvious that conditions (9) and (10) hold for characteristic functions v ′ and w . For some fixed α ∈ [0, 1] we calculate the α-nucleolus for games (N, v ′ ) and (N, w′ ) using the algorithm, presented in section 3. Assuming that the parameter α is moving along the interval [0, 1] with the step of 0.1, we have the payoff vectors presented in Table 2. The special case α = 21 with XSM = Xα is shown in bold. ′

Table 2: The α-nucleolus for (N, v ′ ) and (N, w′ ). α 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Xα (N, v ′ ) Xα (N, w′ ) (1, 1, 1, 2, 2, 2, 1, 1, 1) (1, 1, 1, 2, 2, 2, 1, 1, 1) (1, 1, 1, 2, 2, 2, 1, 1, 1) (1.01, 1.01, 1.01, 1.98, 1.98, 2.03, 1.11, 0.94, 0.94) (1, 1, 1, 2, 2, 2, 1, 1, 1) (1.01, 1.01, 1.01, 1.96, 1.96, 2.05, 1.21, 0.88, 0.88) (1, 1, 1, 2, 2, 2, 1, 1, 1) (1.02, 1.02, 1.02, 1.94, 1.94, 2.08, 1.32, 0.82, 0.82) (1, 1, 1, 2, 2, 2, 1, 1, 1) (1.03, 1.03, 1.03, 1.93, 1.93, 2.10, 1.43, 0.76, 0.76) (1,1,1,2,2,2,1,1,1) (1.04,1.04,1.04,1.91,1.91,2.13,1.54,0.70,0.70) (1, 1, 1, 2, 2, 2, 1, 1, 1) (1.04, 1.04, 1.04, 1.89, 1.89, 2.16, 1.64, 0.64, 0.64) (1, 1, 1, 2, 2, 2, 1, 1, 1) (1.05, 1.05, 1.05, 1.87, 1.87, 2.18, 1.75, 0.59, 0.59) (1, 1, 1, 2, 2, 2, 1, 1, 1) (1.06, 1.06, 1.06, 1.85, 1.85, 2.21, 1.86, 0.53, 0.53) (1, 1, 1, 2, 2, 2, 1, 1, 1) (1.07, 1.07, 1.07, 1.83, 1.83, 2.23, 1.97, 0.47, 0.47) (1, 1, 1, 2, 2, 2, 1, 1, 1) (1.07, 1.07, 1.07, 1.81, 1.81, 2.26, 2.07, 0.41, 0.41)

Consider the payoffs that player 4 gets according to the α-nucleolus for α ≥ 0.1. We can see that Xα4 (N, v) > Xα4 (N, w). Therefore, inequality (11) is not satisfied.


15

So, by means of counterexample we can verify that coalitional monotonicity does not hold for the α-nucleolus with 0.1 ≤ α ≤ 1. By using the dichotomy method for this pair of games we can approximately calculate the maximum α∗ such that 0 < α∗ < 0.1 for which coalitional monotonicity is satisfied and for some α > α∗ coalitional monotonicity is not satisfied. In the current example α∗ ≈ 0. 5.

Conclusion

The result of this work is not surprising. Although aggregate and coalitional monotonicity are considered to be desirable and natural properties of a solution in a TU-game (Maschler, 1992). There are very few solution concepts satisfying even aggregate monotonicity, the weakest form of it. In the paper we have investigated the monotonicity of the SM-nucleolus and come across to some negative conclusion. In a general game it violates the both aggregate and coalitional monotonicity. At the same time, the α-nucleolus due to the arbitrary choice of a real parameter α demonstrates for some α better properties than the SM-nucleolus. The intervals for which the α-nucleolus satisfies aggregate and coalitional monotonicity are approximately calculated for the given examples. The investigation may be extended and deepened in the direction of getting analytical formulas for this interval. In (Tauman and Zapechelnyuk, 2010), the authors argue that monotonicity may not be a proper requirement for some economic context from which a cooperative game arises. They provide an example of a simple 4-person game that marks out a class of economic problems where the monotonicity property of a solution concept is not as attractive as it may seem at the beginning. So, sometimes there is a competition between monotonicity and other attractive properties of a solution in a TU-game. References Britvin, S., Tarashnina, S. (2013). Algorithms of finding the prenucleolus and the SMnucleolus in cooperative TU-games. Mathematical Game Theory and its Applications, 5(4), 14–32 (in Russian). Maschler, M., Peleg, B., Shapley, L. S. (1979). Geometric properties of the kernel, nucleolus and related solution concepts. Mathematics of Operations Research, 4(4), 303–338. Maschler, M. (1992). The Bargaining set, Kernel and Nucleolus. Handbook of Game Theory (R. Aumann, S. Hart, eds), Elsevier Science Publishers BV, 591–665. Megiddo, N. (1974). On nonmonotonicity of the bargaining set, the kernel and the nucleolus of a game. SIAM Journal on Applied Mathemetics, 27(2), 355–358. Pecherskiy, S. L., Yanovskaya, E. B. (2004). Cooperative games: soutions and axioms. European University press: St. Petersburg, 443 p. (in Russian). Petrosjan, L. A., Zenkevich, N. A., Shevkoplyas, E. V. (2012). Game theory. SaintPetersburg: BHV-Petersburg, 432 p. (in Russian). Schmeidler, D. (1969). The nucleolus of a characteristic function game. SIAM Journal on Applied Mathematics, 17(6), 1163-1170. Shapley, L. S. (1953). A value for n-person games. In: Kuhn and Tucker (eds.) Contributions to the theory of games II. Princeton University press, pp. 307–311. Smirnova, N., Tarashnina, S. (2011). On generalisation of the nucleolus in cooperative games. Journal of Applied and Industrial Mathematics, 18(4), 77–93 (in Russian). Tarashnina, S. (2011). The simplified modified nucleolus of a cooperative TU-game. Operations Research and Decision Theory, 19(1), 150–166.

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Sergei V. Britvin, Svetlana I. Tarashnina

Tauman, Y., Zapechelnyuk, A. (2010). On (non-) monotonicity of cooperative solutions. International Journal of Game Theory, 39(1), 171–175. Young, H. P. (1985). Monotonic solution of cooperative games. International Journal of Game Theory, 14, 65–72.

Efficient Myerson Value for Union Stable Structures

⋆

Hua Dong, Hao Sun and Genjiu Xu Department of Applied Mathematics Northwestern Polytechnical University, Xi’an, 710072,China E-mail: [email protected],[email protected],[email protected]

Abstract In this work, an axiomatization of a new value for union stable structures, efficient Myerson value, is shown by average equity, redundant fairness, superfluous component property and other three properties. And the independence of the axioms is illustrated. Besides, the difference of three values, efficient Myerson value, the two-step Shapley value and collective value, is shown. Keywords: Union stable structure; average equity; redundant fairness.

1.

Introduction

A situation in which a finite set of players can obtain payoffs by cooperation can be described by a cooperative game with transferable utility, shortly TU-game, being a pair consisting of a finite set of players and a characteristic function on the set of coalitions of players assigning a worth to each coalition of players. In practice, since the cooperation restrictions exist, only some subgroup of players can form a coalition. One way to describe the structure of partial cooperation in the context of cooperative games is to specify sets of feasible coalitions. Algaba, et al (Algaba, 2000) considered union stable systems as such sets. A union stable system of two intersecting feasible coalitions is also feasible, which can be interpreted as follows: players who are common members of two feasible coalitions are able to act as intermediaries to elicit cooperation among all the players in either of these coalitions, and so their union should be a feasible coalition. And a TU game with a union stable system is called a union stable structure. Besides, the union stable structure is a generalization of games with communication structure and games with permission structure, which are respectively proposed by Myerson (Myerson, 1977) and Gilles (Gilles,1992). Hamiache (Hamiache, 2012) presented a matrix approach to construct extensions of the Shapley value on the games with coalition structures and communication structures. This paper aims to generalize this matrix approach to union stable structures, a generalized communication structures. 2.

Preliminaries

2.1. Matrix Approach To Shapley Value A cooperative game with transferable utility, or simply a TU-game, being a pair (N, v), where N is the finite set of all players, and v : 2N → R is a characteristic function satisfying v(∅) = 0. The collection of all games with player set N is denoted by G. A game (N, v) is called an inessential game if for any two disjoint coalitions ⋆

This work was supported by the NSF of China under grants Nos. 71171163, 71271171 and 71311120091.

18

Hua Dong, Hao Sun, Genjiu Xu

S, T ⊆ N , v(S ∪ T ) = v(S) + v(T ). Here the cardinality of any coalition S ⊆ N is denoted by |S| or the lower case letter s. A payoff vector for a game is a vector x ∈P RN assigning a payoff xi to player i ∈ N . In the sequel, for all S ⊆ N , x(S) = i∈S xi . A single-valued solution is a function that assigns to any game (N, v) ∈ G a unique payoff vector. The most well-known single-valued efficient solution is the Shapley value (1953) given by Shi (N, v) =

X s!(n − s − 1)! (v(S ∪ i) − v(S)). n!

i∈N \S

In fact, the explicit expression of Shapley value can be presented as Shi (N, v) = (M Sh · v)[{i}], where the matrix M Sh = [M Sh ]i∈N,S⊆N \∅ is defined by [M

Sh

]i,S =

(

(s−1)!(n−s)! , n! s!(n−s−1)! − , n!

if i ∈ S , otherwise.

(1)

Next, we recall the axiomatic characterization of Shapley value (Shapley, 1953) illustrated in Hamiache (Hamiache, 2001)and the matrix approach in Xu, et al (Xu, 2008) and Hamiache (Hamiache, 2010) for the analysis of the associated consistency. For all games (N, v) ∈ G, the associated game (N, vλSh ) defined in Hamiache (Hamiache, 2001)for all parameters λ(0 < λ < n2 ) as follows, vλSh (S) = v(S) + λ

X

i∈N \S

[v(S ∪ i) − v(S) − v({i})] for all S ⊆ N

Definition 1. A matrix A is called a row (column)-coalition matrix if its rows (column) are indexed by coalitions S ⊆ N in the lexicographic order. A is called square-coalitional if it is both row-coalitional and column-coalitional. And a rowcoalition matrix A = [a]S,T is called row-inessential or inessential, if A = [a]S,T = P i∈S ai,T for all S ⊆ N .

Since the associated game is a linear transformation of the original game, the associated game can be expressed as vλSh = Mλ · v, where Mλ is a square-coalitional matrix of order 2n−1 , for detailed information, please refer to Xu, et al (Xu, 2008) and Hamiache (Hamiache, 2010). The sequence of associated games illustrated in Hamiache (2001) can be expressed by matrix approach in Xu, et al (Xu, 2008) and Hamiache(Hamiache, 2010) as follows, k vkλ = (v(k−1)λ )Sh λ = Mλ · v(k−1)λ = ... = (Mλ ) · v, for all k ≥ 2 .

And the sequence of games {(N, vkλ )}∞ k=1 converges to an inessential game (N, vL ), denote the corresponding coefficient matrix as ML , then limk→∞ (Mλ )k = ML , and ML is inessential. 2.2.

Union Stable Structures

Definition 2. A union stable system is a pair (N, F ) with F ⊆ 2N verifying that {i} ∈ F for all i ∈ N and for all S, T ∈ F with S ∩ T 6= ∅, S ∪ T ∈ F .

Efficient Myerson Value for Union Stable Structures

19

Given a union stable system (N, F ), B(F ) is called the basis of F , it is denoted by the set of all feasible coalitions which cannot be expressed as a union of feasible coalitions with nonempty intersection, the elements of the basis B(F ) are called supports of F . Especially, the set of non-singleton supports is denoted by C(F ) = {B ∈ B(F ) : |B| ≥ 2}. A union stable structure is a triple (N, v, F ), i.e., a TU game (N, v) with union stable system (N, F ). The set of such union stable structure with player set N is denoted by U S N . Definition 3. Let E ⊆ 2N be a set system and S ⊆ N . A set T ⊆ S is called a ′ ′ E-component of S if T ∈ E and there exists no T ∈ E such that T ( T ⊆ S. Especially, the collection of F -component of N is denoted by β = CF (N ) = {B1 , B2 , ..., Br } with 1 ≤ r ≤ |N | and ∪B∈β B = N , Bi ∩ Bj 6= ∅ for any Bi , Bj ∈ β. Given (N, v, F ) ∈ U S N , define the intermediate game (β, v βP ) by v β (R) = F F v(∪B∈R B) for all R ⊆ β and the quotient game (N, v ) by v (S) = T ∈CF (S) v(T ) for all S ⊆ N . For coalition S ⊆ N \ ∅, define coalitions S and S respectively by the following, S = ∪{K ∈ β|K ⊆ S}, i.e., the maximal union of components of N which belongs to coalition S. S = ∪{K ∈ β|K ∩ S 6= ∅}, i.e., the minimal union of components of N covering coalition S. 3. 3.1.

Efficient Myerson Value For Union Stable Structures Definition

In order to give the formal definition of the efficient Myerson value, two matrices closely related to union stable structures are constructed. Let us define a {0, 1}-squared matrix P of order 2n − 1, which is closely related to union stable structure (N, v, F ). So that for all S, T ⊆ 2N \ ∅, P [S, T ] =

(

1, if T ∈ CF (S) , 0, otherwise.

(2)

Note that for all coalitions S ⊆ N , v F (S) = (P ·v)[S], thus ϕi (N, v, F ) = Shi (N, v F ) = (ML ·P ·v)[i], where ϕ(N, v, F ) is the Myerson value for union stable structure(N, v, F ). Next, we shall make a modification of the matrix, and define the matrix Q as follows,   1, if T = S , (3) Q[S, T ] = 1, if T ∈ CF (S \ S),   0, otherwise. Lemma 1. Given (N, v, F ) ∈ U S N and the intermediate game (β, v β ) defined before, the vector of weights are w = (b1 , b2 , ..., br ), bl = |Bl | for all l ∈ {1, 2, ..., r}. Then for all Bl ∈ β and all players i ∈ Bl , (ML · (Q − P ) · v)[{i}] = Shi (N, (Q − P ) · v) 1 β = (Shw Bl (β, v ) − v(Bl )). bl

20


The proof is similar to the computation in Hamiache (Hamiache, 2012),we omit here. Next we give the definition of the efficient Myerson value and some axioms that will be used to axiomatize the value. Definition 4. For all union stable structures (N, v, F ), define the the efficient Myerson value η as ηi (N, v, F ) = (ML · Q · v)[{i}] = Shi (N, Q · v) for all i ∈ N. From Lemma 1, it is obvious that for solution η and all i ∈ N , ηi (N, v, F ) = ϕi (N, v, F ) +

1 β (Shw Bl (β, v ) − v(Bl )). bl

We can interpret efficient Myerson value in the following sense. In the first step, every player obtains the payoff of Myerson value. In the second step, since the allocation rule satisfies efficiency, define a quotient game, and every component Bl obtains the payoff of weighted Shapley value, following the principle of fairness β among the members of component Bl , the surplus Shw Bl (β, v ) − v(Bl ) is split equally. It can be seen that the efficient Myerson value and the collective value, which was proposed by Kamijo(Kamijo, 2011), is similar, and the difference lies in the allocation rule of first step. Compared with a prior coalition structure, given a component of union stable system, some subset of the component may be not feasible. And the formation of the component of union stable system lies on the contribution of common players, while collective value cannot illustrate the contribution of the intermediate members make during the cooperation, so the collective value is not suitable for union stable structures. So they are irreplaceable for each other. Let Ci (F ) denote the collection given by {C ∈ C(F ) : i ∈ C}, to provide axiomatic characterizations of the efficient Myerson value, the following definitions and properties are introduced. 3.2.

Axiomatization

Definition 5. A union stable structure (N, v, F ) is called point anonymous if there exists a function f : {0, 1, ..., |D|} → R with f (0) = 0 such that v F (S) = f (|S ∩ D|) for all S ⊆ N , where D = {i ∈ N : Ci (F ) 6= ∅}. Definition 6. For any (N, v, F ) ∈ U S N , a player i ∈ N is called superfluous for (N, v, F ) if v F (S ∪ i) = v F (S) for all S ⊆ N \ {i}. Let ψ : U S N → Rn be a solution, then we call P it satisfies the above properties, if Efficiency (EFF) For all (N, v, F ) ∈ U S N , i∈N ψi (N, v, F ) = v(N ). Additivity (ADD) For any (N, u, F ), (N, v, F ) ∈ U S N , ψ(u + v) = ψ(u) + ψ(v), where (u + v)(S) = u(S) + v(S) for all S ⊆ N . Average equity (AE) For all unanimity games uT with T ⊆ N \ ∅, if there exists two components Bl , Bk ∈ β with Bl ∩ T 6= ∅, Bk ∩ T 6= ∅, then X X |Bl |−1 ψi (N, uT , F ) = |Bk |−1 ψj (N, uT , F ). i∈Bl

j∈Bk

Point anonymity (PA) For all point anonymous union stable structures (N, v, F ),


21

there exists b ∈ R such that ψi (N, v, F ) = b for all i ∈ D, ψi (N, v, F ) = 0 otherwise. Redundant fairness (RF) If there exists two superfluous players i, j ∈ Bk with Bk ∈ β, then ψi = ψj . Superfluous component property (SCP) Given component Bk ∈ β, if v(R ∪ Bk ) = P v(R) for all R ⊆ β, then i∈Bk ψi (N, v, F ) = 0. Theorem 1. The efficient Myerson value is the unique value on U S N that satisfies efficiency, additivity, average fairness, point anonymity, redundant fairness and superfluous component property.

Proof. It is straightforward to verify that the efficient Myerson value satisfies EFF, ADD, AE, RF and SCP. In the following, we will only verify the property of point anonymity. Let (N, v, F ) ∈ U S N be point anonymous. If D = ∅, then the restricted game F v (S) = f (|S ∩ ∅|) = f (0) = 0 for all S ⊆ N . Hence, the efficient Myerson value ηi (N, v, F ) = 0 for all i ∈ N . Let D 6= ∅, we will show that there exists a unique component Bk ∈ β such that D ⊆ Bk . Otherwise, assume there are two components Bi , Bj ∈ β such that D = Bi ∪ Bj , let S = D, we have v F (S) = v(Bi ) + v(Bj ) = f (|Bi ∩D|)+f (|Bj ∩D|), which contradicts with v F (S) = f (|S∩D|) = f (|D|). Hence, let us suppose Bk ∈ β is the unique component such that D ⊆ Bk , then for any β R ⊆ β, v(|(Bl ∪R)∩D|) = f (|R ∩D|) = v(R) for all Bl 6= Bk , Shw Bk (β, v ) = v(Bk ). β Consequently, for any Bl ∈ β, Shw Bk (β, v ) − v(Bk ) = 0, the efficient Myerson value is equal to the Myerson value, i.e., ηi = ϕi = f (|D|)/|D| for all i ∈ D, otherwise, ηi = 0. Thus the efficient Myerson value verifies point anonymity. Next, we will show the converse part. Let ψ ∈ Rn be a solution on U S N satisfying the above six properties. Given T ⊆ N \ ∅, let (N, uT , F ) be a unanimity game with union stable system. Given c ∈ R, let cuT be a unanimity game uT multiplied by a scalar c, Then by additivity, it suffices to show that ψ(N, v, F ) is uniquely determined by the above six properties. For all T ⊆ N \ ∅, , let us consider the following two cases: T ∈ / F and T ∈ F . Case 1 T ∈ / F , define T ⊆ β by {B ∈ β, B ∩ T 6= ∅}. Then the unanimity game (β, (cuT )β ) is a T -unanimity game multiplied by c, i.e., (β, cuT ). It is obvious that any component Bl ∈ β \ T is superfluous. From superfluous component property, P we have i∈Bl ψi (N, cuT , F ) = 0 for all Bl ∈ β \ T . Together with average equity P P together and efficiency, we have that i∈Bl ψi (N, cuT , F ) = c( Bl ∈T |Bl |)−1 |Bl | P for all Bl ∈ T , i∈Bl ψi (N, cuT , F ) = 0 otherwise. Furthermore, we assert that any player i ∈ N is superfluous for (N, cuT , F ) ∈ F U S N , i.e., given any i ∈ N , uF T (S) = uT (S ∪ i) for all S ⊆ N . Consequently, given any Bk ∈ β, due to the redundancy fairness of ψ(N, cuT , F ), then ψi = ψjPfor all i, j ∈ Bk . From the above arguments, we have that ψi (N, cuT , F ) = c( Bl ∈T |Bl |)−1 for all i ∈ Bk and Bk ∈ T , ψi (N, cuT , F ) = 0 otherwise. Hence, for any unanimity game with union stable structure (N, cuT , F ) ∈ U S N with T ∈ / F, ψ(N, cuT , F ) is uniquely determined. The remaining task is to show all players are superfluous. In the following, we show that any player i ∈ N is superfluous for (N, cuT , F ) ∈ U S N . If there exists a unique component Bk ∈ β such that T ⊆ Bk , then uF T (S) = F uF T (S ∪ i) = 1, uT (S) = 0 for all S ⊆ N \ Bk . Otherwise, there exists no such component, then uF T (S) = 0 for all S ⊆ N . This completes the proof for case 1. Case 2 T ∈ F , we show that (N, cuT , F ) ∈ U S N is point anonymous. First we show

22


that for T ∈ F , (cuT )F (S) = c if and only if T ⊆ S. Due to whether the coalition S is feasible or not, we distinguish the following two cases: (1)If S ∈ F , then (cuT )F (S) = cuT (S) = c if and only if T ⊆ S, i.e., T ∩ S = T . (2)If S ∈ / F and T ⊆ S, we will show that there exists a unique feasible coalition K ∈ F and K ⊆ S such that T ⊆ K. If T ∈ CF (S), let K = T , (cuT )F (S) = cuT (T ) = c. Otherwise, there exists a series of feasible coalitions A1 , A2 , ..., Al ∈ F with Ai ∩ Aj = ∅ for any i, j = 1, 2, ..., l(l ≥ 2) and i 6= j such that S = ∪lk=1 Ak , since S ∈ / F and |S| ≥ 2. Hence there exists a unique feasible coalition Aj (1 ≤ j ≤ l) such that T ⊆ Aj , let K = Aj , consequently, (cuT )F (S) = cuT (Aj ) = c. If S ∈ /F and T * S, it is easy to verify that (cuT )F (S) = 0. From the arguments above, we have that if T ∈ F , (cuT )F (S) = c if and only if T ⊆ S. Therefore, there exists a function f : {0, 1, 2, ..., |T |} → R such that cuF T (S) = f (|S ∩ T |) for all S ⊆ N where f (0) = f (1) = ... = f (|T | − 1) = 0 and f (|T |) = c. Hence,(N, cuT , F ) is point anonymous, applying the point anonymity to the solution ψ, there exists b ∈ R such P that ψi = b if i ∈ T and ψi = 0 otherwise. By efficiency, we have that cuT (N ) = i∈T ψi = b|T | = c, let b = c/|T |, thus the solution ψ(N, v, F ) is uniquely determined by ψi = b for all i ∈ T , ψi = 0 otherwise. So, ψ(N, v, F ) is unique determined in both cases. Since the efficient Myerson value verifies the six properties, ψ(N, v, F ) = η(N, v, F ). Also the axioms of theorem 1 are logically independent as shown by the following alternative solutions. Example 1. The zero solution given by ψi (N, v, F ) = 0 for all i ∈ N satisfies ADD, AE, PA, RF and SCP. It does not satisfy efficiency. Example 2. The equal division given by ( Shw (β,vβ )−v(B ) ψi (N, v, F ) =

k

Bk

0,

|Bk \SU|

+ ψi (N, v, F ), if i ∈ Bk \ SU ,

if i ∈ Bk ∩ SU .

(4)

for all i ∈ Bk , Bk ∈ β, where SU denotes the set of all superfluous players in (N, v, F ) and the weight system is the same with the definition of Lemma 1. This solution satisfies all properties except additivity. (β,v β )−v(B )

Sh

k Example 3. The solution given by ψi (N, v, F ) = ϕi (N, v, F ) + Bk |Bk | for all i ∈ Bk , Bk ∈ β, satisfies EFF, ADD, PA, RF and SCP. It does not satisfy average equity.

Example 4. The solution given by ψi (N, v, F ) = all properties except point anonymity.

β Shw Bk (β,v ) |Bk |

for all i ∈ Bk satisfies

Example 5. Define the solution ψ(N, v, F ) by ψi (N, v, F ) [ShBk (β,v β )−v(Bk )]wi P for j∈Bk wj n w ∈ R with wi 6= wj

=

ϕi (N, v, F )+

all i ∈ Bk ,Bk ∈ β, for some exogenous weight system

for any two players i 6= j in the same component, and there exists a constant number a ∈ R such that for any component Bk ∈ β, P w(Bk ) = w = |B | · a. It is straightforward to verify that this solution k i∈Bk i satisfies EFF, ADD, AE, PA and SCP, except redundant fairness.


23

Example 6. The solution given by  v(N )−α(v) , if i ∈ D,   |D| ψi (N, v, F ) = 0, if i ∈ Bk \ D,   α(v) , if i ∈ N \ Bk . |N \Bk |

(5)

for all i ∈ Bk , Bk ∈ β, where α : v → R is a linear operator, i.e., satisfying α(v + w) = α(v) + α(w), and α(v) = 0 when the union stable structure(N, v, F ) is point anonymous, otherwise 0 < α(v) < v(N )/|D|. Since there exists only one component Bk such that D ⊆ Bk . It is straightforward to verify that this solution satisfies EFF, ADD, AE, PA and RF. It does not verify superfluous component property. 4.

Conclusion

This paper mainly focus on the axiomatization of efficient Myerson value for union stable structures. And three new axioms:average equity, redundant fairness, superfluous component property and other three properties. And the independence of the axioms is illustrated. Besides, the difference between the value and collective value is remarked. References Algaba, E., Bilbao, J. M., P., R. and J. J. Lopez. (2000). The position value for union stable systems. Mathematical Methods of Operations Research., 52, 221–236. Princeton University Press: Princeton, NJ. Algaba, E., Bilbao, J. M., P., R. and J. J. Lopez. (2001). The Myerson value for union stable structures. Mathematical Methods of Operations Research., 54, 359–371. Driessen, T. S. H. (2010). Associated consistency and values for TU games. International Journal of Game Theory., 39, 467–482. Gilles RP, Owen G, Brink R. (1992). Games with permission structures: the conjunctive approach. International Journal of Game Theory., 20, 277–293. Hamiache, G. (2001). Associated consistency and Shapley value. International Journal of Game Theory., 30, 279–289. Hamiache, G. (2010). A matrix approach to Shapley value. International Game Theory Review., 12 , 1–13 Hamiache, G. (2012). A matrix approach to TU games with coalition and communication structures. Social Choice and Welfare., 38, 85–100. Myerson, R. B. (1977). Graphs and cooperation in games. Mathematics of operations research., 2, 225–229. Shapley, L. S. (1953). A value for n-person games. In:contributions to the Theory of Games (Tucker, A.W.and Kuhn, H.W.,ends), Princeton University Press. Xu, G., Driessen T.and Sun, H. (2008). Matrix analysis for associated consistency in cooperative game theory. Linear Algebra and its Applications., 428, 1571–1586. van den Brink R, Khmelnitskaya, A and Van der Laan G. (2012). An efficient and fair solution for communication graph games. Economics Letters, 117, 786–789. Kamijo, Y. (2011). The collective value: a new solution for games with coalition structures. TOP., 1–18.

On the Inverse Problem and the Coalitional Rationality for Binomial Semivalues of Cooperative TU Games Irinel Dragan University of Texas, Mathematics, Arlington, Texas 76019-0408, USA E-mail: [email protected]

Abstract In an earlier work (Dragan, 1991), we introduced the Inverse Problem for the Shapley Values and Weighted Shapley Values of cooperative transferable utilities games (TU-games). A more recent work (Dragan, 2004) is solving the Inverse Problem for Semivalues, a more general class of values of TU games. The Binomial Semivalues have been introduced recently (Puerte, 2000), and they are particular Semivalues, including among other values the Banzhaf Value. The Inverse problem for Binomial Semivalues was considered in another paper (Dragan, 2013). As these are, in general, not efficient values, the main tools in evaluating the fairness of such solutions are the Power Game and the coalitional rationality, as introduced in the earlier joint work (Dragan/Martinez-Legaz, 2001). In the present paper, we are looking for the existence of games belonging to the Inverse Set, and for which the a priori given Binomial Semivalue is coalitional rational, that is belongs to the Core of the Power Game. It is shown that there are games in the Inverse Set for which the Binomial Semivalue is coalitional rational, and also games for which it is not coalitional rational. An example is illustrating the procedure of finding games in the Inverse Set belonging to both classes of games just mentioned. Keywords: Inverse Problem, Inverse Set, Semivalues, Binomial Semivalues, Power Game, Coalitional rationality.

Introduction In a cooperative transferable utilities game (TU game), (N, v), defined by a finite set of players N, n = |N | , and the characteristic function v : P (N ) → R, with v(∅) = 0, where P (N ) is the set of nonempty subsets of N, called coalitions, the main classical problem is to divide fairly the win of the grand coalition v(N ). An early solution was the Shapley Value (1953), defined axiomatically, to satisfy some fairness conditions (the axioms), and proved to be given by the formula SHi (N, v) =

X

S:i∈S⊆N

(s − 1)!(n − s)! .[v(S) − v(S − {i})], ∀i ∈ N, n!

where s = |S| , S ⊆ N. It is easy to prove that SH is always efficient, that is we have the sum of components equal v(N ). The Shapley Value may belong to the Core of the game and in this case it is coalitional rational. The Semivalues, introduced by Dubey, Neyman and Weber (1981), who tried to avoid the efficiency axiom, in general are not efficient, so that they do not belong to the Core, and coalitional rationality is a problem in evaluating the fairness. The Binomial Semivalues were introduced by Puerte (2000), as extensions of the most known Semivalue, the Banzhaf Value (1965). To evaluate the fairness of such a solution, an algebraic structure is needed,

On the Inverse Problem and the Coalitional Rationality

25

and let us denote by G(N ) the set of all games with the set of players N. Two operations are defined, addition and scalar multiplication by v = v1 + v2 ⇔ v(S) = v1 (S) + v2 (S), ∀S ⊆ N,

where (N, v1 ) and (N, v2 ) are any two TU games in G(N ), and v = γv1 ⇔ v(S) = γv1 (S), ∀S ⊆ N, ∀γ ∈ R,

where (N, v1 ) is any TU game. It is easy to check that G(N ) is a linear vector space and its dimension is 2n − 1. Now, for every coalition S ⊆ N , the restriction of (N, v) to S is the game denoted by (S, v). Obviously, this finite set and the operations shown above define on S again a linear vector space, G(S), and the union of all spaces G(S), ∀S ⊆ N, is denoted by GN . A value Φ defined on GN is any functional defined on each G(S) with values in Rs . The Shapley Value is defined on GN by a formula similar to the first formula above, where N has been changed into S ⊆ N, and n into s. In the first section we introduce the Semivalues and the Binomial Semivalues; the solution of the Inverse Problem for Semivalues is shown in the second section. The Power Game and the coalitional rationality, together with the main result of the paper about Binomial Semivalues are discussed in the last section. 1.

Semivalues and Binomial Semivalues

To give the definition of a Semivalue, we need a weight vector pn ∈ Rn , satisfying a normalization condition n X n−1 s=1

s−1

pns = 1,

(1.1)

together with the interpretation: pns is the common weight of all coalitions of size s. This would be enough for the games in G(N ), but for all the games in GN we need a sequence of weight vectors defined recursively as follows: pn−1 = pns + pns+1 , s

s = 1, 2, ..., n − 1,

(1.2)

will give the weight vector for the space G(S) with |S| = n − 1. Then, the sequence of weight vectors pn−2 , pn−3 , ..., p2 , p1 is defined by formulas similar to (1.2), going up to p11 = 1. From (1.2) it is easy to show that these vectors satisfy a normalization condition like (1.1). It has been said earlier that (1.2) are the inverse Pascal triangle conditions, as any sequence shows triangles similar to those present when the Pascal triangle conditions were defined. Now, we can define the Semivalue associated with any sequence of weight vectors, p1 , p2 , ..., pn subject to (1.1) and connected by (1.2), as the value defined on GN by SEi (T, v, pt ) =

X

S:i∈S⊆T

pts [v(S) − v(S − {i})], ∀i ∈ T, ∀T ⊆ N, T 6= ∅.

(1.3)

Recall that here (T, v) is the TU game, a subgame of (N, v), obtained as a restriction of the characteristic function to T, so that (T, v) ∈ GT . Notice that for t 6 n the weight vectors

26

Irinel Dragan (s − 1)!(t − s)! , s = 1, 2, ..., t, pts = 21−s , s = 1, 2, ..., t, t! give the Shapley Value, and the Banzhaf Value, respectively. pts =

(1.4)

Example 1. Consider the weight vector p3 = ( 18 , 14 , 83 ), and p2 = ( 38 , 85 ), p1 = (1), derived via (1.2). Consider the game v({1}) = v({2}) = v({3}) = 0, v({1, 2}) = v({1, 3}) = v({2, 3}) = v(1, 2, 3}) = 1, (1.5) a constant sum game. We may compute the Semivalue of this game, by the formula (3), to get for the game (1.5) the outcome SE({1, 2, 3}, v, p3) = ( 12 , 12 , 12 ), which is not efficient, because the sum of components makes a number different of 1. Then, the Semivalue does not belong to the Core, as the efficiency is missing. As this is the case in most situations, we have to define the coalitional rationality in some other way, following the ideas from the earlier work (Dragan/Martinez-Legaz, 2001), namely to consider the Power Game of the given game, in which the Semivalue is efficient and may be coalitional rational. Thus we have to define the Power Game. We may compute also the Semivalues of the subgames 5 5 SE({1, 2}, v, p2) = SE({1, 3}, v, p2) = SE({2, 3}, v, p2) = ( , ). (1.6) 8 8 They are looking all similar, due to the symmetry in (1.5) of the worth of the characteristic function of the players. Of course, the Semivalues of singletons are all zero. Then, we got a new game

w({1}) = w({2}) = w({3}) = 0, w({1, 2}) = w({1, 3}) = w({2, 3}) =

5 , 4

w({1, 2, 3}) =

3 . 2

were we used (6) to satisfy the definition which follows. Definition 1. For a TU game (N, v), the Power Game, relative to a Semivalue associated with a weight vector pn , is the game (N, π, pn ) defined by formula X π(T, v, pt ) = SEi (T, v), ∀T ⊆ N, (1.7) i∈T

where the components of the Semivalue were given by formula (1.3).

As seen above in example 1, it is not easy to compute the Power Game by means of (7). However, this may be done by using the following result: Theorem (Dragan, 2000). Let a Semivalue SE(N, v) be associated with the weight vector pn , and the Power Game (N, π, pn ), relative to the Semivalue, given by formula (7). Then, we have X π(T, v, pt ) = [spts − (t − s)pts+1 ]v(S), ∀T ⊆ N, (1.8) S⊆T

where

ptt+1

is an arbitrary number.

27


Example 2. Return to the game of Example 1 and recall that the computation of the Power Game, relative to the Semivalue, by using the definition (7), led to π({1, 2}, v, p2) = π({1, 3}, v, p2 ) = π({2, 3}, v, p2) =

5 , 4

π({1, 2, 3}, v, p3) =

3 . 2 (1.9)

Now, by using the theorem, as we have the bracket in (8) given by 5 1 9 , and 2p32 − p33 = , 3p33 = , (1.10) 4 8 8 with the second equality used three times for coalitions of size two, from (1.10) and formula (1.8) we get the same worth for the characteristic function as in (1.9). Beside definition 1, we illustrated the usefulness of (1.8), relative to (1.7). Now, that we have the Power Game (1.9) for our given game (1.5), and the Semivalue is efficient in this game, which obviously is always true, we may check whether or not, the Semivalue is in the Core of the Power Game, and conclude that this is not true; hence according to the ideas from Dragan/Martinez-Legaz (2001), the Semivalue of (1.5) is not coalitional rational. Of course, it may be possible that the Semivalue does belong to the Core of the Power Game, and in this last case it will be coalitional rational. For example, if we consider the Banzhaf Value, the most popular Semivalue, defined by p3 = ( 14 , 41 , 14 ), and compute the Power Game, then we find out that the value is coalitional rational. Looking at our Example 2 we think that it is justified to introduce the following: 2p22 =

Definition 2. The Semivalue of a given game is coalitional rational if it belongs to the Core of the Power Game relative to the Semivalue (or, the Power Core of the game). Now, let us consider the Binomial Semivalues, introduced by Puerte (2000) and discussed also in the work by Puerte/Freixas (2002), where definition 2 applies. Definition 3. The Semivalue SE associated with the sequence of normalized weight vectors p1 , p2 , ..., pn connected by the inverse Pascal triangle relationships, is a Binomial Semivalue, if the weight vectors satisfy also for some number r ∈ (0, 1], the equalities pn2 pn pn = 3n = ... = nn = r. n p1 p2 pn−1

(1.11)

Now, that the concepts of Power Game and coalitional rationality have been explained and the computation of the Power Game has been given, we notice: Lemma 1. In GN , the weights of a Binomial Semivalue are given by the equalities rs−1 , s = 1, 2, ..., t, t 6 n, (1 + r)t−1 so that the Binomial Semivalue is given by the formula pts =

SEi (T, v, pt ) =

X

S:i∈S⊆T

(1.12)

rs−1 [v(S) − v(S − {i})], ∀i ∈ T, ∀T ⊆ N, T 6= ∅. (1 + r)t−1 (1.13)

28

Irinel Dragan

Proof. Follows from (1.11) and the normalization condition (1.1), as well as formula (1.3). ⊔ ⊓ ę Remarks: (a) From the inverse Pascal triangle relationships it follows that all weight vectors of the sequence are given by formulas similar to (1.12), obtained for different values of t = 1, 2, ..., n, and while the game is replaced by the Value, a fact which justifies the study of the Binomial Semivalues, that should have properties similar to those of the Banzhaf Value. Now, the problem to be considered in this paper is: for a given vector L, and a given TU game (N, v), such that the Binomial Semivalue corresponding to a parameter r is not coalitional rational, find out in the Inverse Set of L a TU game (N, w), for which the Binomial Semivalue with parameter r is the same, but belongs to the Core of the Power Game. Clearly, we have to explain the procedure in two steps: • How do we find the Inverse Set of a Binomial Semivalue associated with a parameter r, and an a priori given value L? From our previous work on general Semivalues we know that this should be determined by an explicit formula, hence the Inverse Set will be available. This will be discussed in the second section. • In the Inverse Set, how do we get a TU game for which the Binomial Semivalue of the original game is in the Core of its Power Game? This will be discussed in the last section. What about games for which the Binomial Semivalue of the original game does not belong to the Power Core? like the one from example 1. 2.

The Null Space and the Inverse Set

In a recent work (Dragan, 2004), it has been shown that the Semivalue, associated with a sequence of weight vectors derived from pn by means of formulas of type (1.2), has a potential function and for a game (N, v), it is given by the formula X P (N, v, pn ) = pns v(S). (2.1) S⊆N

Thus, for a Binomial Semivalue (2.1) becomes P (N, v, pn ) =

X 1 rs−1 v(S). n−1 (1 + r)

(2.2)

S⊆N

Obviously, to make (2.2) computationally better, the sum may be written as P (N, v, pn ) =

n X 1 rs−1 ds (N, v), (1 + r)n−1 s=1

(2.3)

where ds (N, v) is the sum of worth of the characteristic function for all subcoalitions of size s in the set of players N. Example 3. Returning to the game (1.5) of Example 1, and the weight vector p3 = ( 18 , 41 , 38 ), we see that d1 (N, v) = 0, d2 (N, v) = 3, d3 (N, v) = 1, so that from (2.3) we 1 2 get P (N, v, p3 ) = (1+r) 2 (3r + r ), where the expressions of weights in terms of the

29


ratio r were used. Now, for any coalition of size two, again from (2.3), we get the r potential P (N − {i}, v, p2) = 1+r , i = 1, 2, 3, Hence, the Binomial Semivalue is SEi (N, v, p3 ) = P (N, v, p3 ) − P (N − {i}, v, p2 ) =

3r + r2 r 2r − = , 2 (1 + r) 1+r (1 + r)2 i = 1, 2, 3, (2.4)

which for the Banzhaf Value (r = 1) becomes as above B(N, v) = ( 21 , 12 , 21 ). In the more recent work (Dragan, 2013), we considered a basis for the space G(N), consisting of the linearly independent vectors in the set

defined by the formulas

l

wT (T ) =

1 , ptt

(2.5)

W = {wT ∈ Rn : T ⊆ N, T 6= ∅},

wT (S) =

s−t (−1) X l=0

s−t l

pt+l t+l

, ∀S ⊃ T,

wT (S) = 0,

otherwise.

(2.6) For all these vectors were computed the Binomial Semivalues associated with the weight vectors in the sequence generated by the vector pn , via (1.2), by using the formulas for the Binomial Semivalues (1.3) obtained in Lemma 1. Here, we try to compute the same Binomial Semivalues by means of the potential, using SEi (N, wT , pn ) = P (N, wT , pn ) − P (N − {i}, wT , pn−1 ), ∀i ∈ N,

(2.7)

like in example 3 above. To get the experience needed in this computation let us consider first an example. Example 4. From (2.6), the general three person game shows the basis W, that taking into account (1.2) becomes 1 1 2 1 1 1 1 , 1 − 2 , 0, 1 − 2 + 3 ) = (1, 0, 0, − , − , 0, 2 ), 2 p2 p2 p2 p3 r r r 1 1 2 1 1 1 1 w{2} = (0, 1, 0, 1 − 2 , 0, 1 − 2 , 1 − 2 + 3 ) = (0, 1, 0, − , 0, − , 2 ), p2 p2 p2 p3 r r r 1 1 2 1 1 1 1 w{3} = (0, 0, 1, 0, 1 − 2 , 1 − 2 , 1 − 2 + 3 ) = (0, 0, 1, 0, − , − , 2 ), p2 p2 p2 p3 r r r 1 1 1 1+r 1+r w{1,2} = (0, 0, 0, 2 , 0, 0, 2 − 3 ) = (0, 0, 0, , 0, 0, − 2 ), p2 p2 p3 r r 1 1 1 1+r 1+r w{1,3} = (0, 0, 0, 0, 2 , 0, 2 − 3 ) = (0, 0, 0, 0, , 0, − 2 ), p2 p2 p3 r r 1 1 1 1+r 1+r w{2,3} = (0, 0, 0, 0, 0, 2 , 2 − 3 ) = (0, 0, 0, 0, 0, , − 2 ), p2 p2 p3 r r 2 1 (1 + r) w{1,2,3} = (0, 0, 0, 0, 0, 0, 3 ) = (0, 0, 0, 0, 0, 0, ). p3 r2 w{1} = (1, 0, 0, 1 −

(2.8)

30

Irinel Dragan Obviously, these are linearly independent vectors and their number equals the dimension of the space, hence they form a basis. Let us compute the potentials of all basic vectors, by using formula (2.3). For i, j, k = 1, 2, 3, we have the sums 2 1 d1 (N, w{i} ) = 1, d2 (N, w{i} ) = − , d3 (N, w{i} ) = 2 , r r 1+r 1+r d1 (N, w{i,j} ) = 0, d2 (N, w{i,j} ) = , d3 (N, w{i,j} ) = − 2 , r r (1 + r)2 d1 (N, w{i,j,k} ) = d2 (N, w{i,j,k} ) = 0, d3 (N, w{i,j,k} ) = , r2

(2.9)

and from (2.3) we obtain p3s =

rs−1 , (1 + r)2

s = 1, 2, 3.

(2.10)

Now, by (2.9) and (2.10), for every basic vector (N, w) shown above in (2.8), formula (2.1) written as P (N, w, p3 ) = will give P (N, ws , p3 ) =

3 X 1 rs−1 ds (N, w), (1 + r)2 s=1

1 [1 + (−1)]s−1 = 0, ∀S ⊂ N. (1 + r)s−1

(2.11)

(2.12)

while the potential of the last game equals 1. Now, the potentials of the subgames would be computed, by using a formula similar to (2.11), namely

P (N − {i}, wS , p2 ) =

2 1 X s−1 r dS (N, w), ∀S ⊂ N − {i}, i ∈ N. 1 + r s=0

(2.13)

In the same way, we get zero, and the potential of the last game equals 1. Then, the Semivalues computed by the formula (2.7) are SE(N, w{i} , p1 ) = (0, 0, 0), ∀i ∈ N, SEj (N, wN −{i} , p2 ) = −δji ,

i = 1, 2, 3;

(2.14)

SE(N, wN , p3 ) = (1, 1, 1).

A similar approach is hepful in proving, by computing the potentials, the result: Theorem (Thm.3, Dragan, 2013). Let a Binomial Semivalue be defined by a parameter r, and let W be the basis of the space provided by formulas (2.5), (2.6). Then, we have SE(N, wT , pt ) = 0, ∀T ⊂ N, |T | 6 n − 2, T 6= ∅,

31


SEi (N, wN −{i} pn ) = −1, ∀i ∈ N, SEj (N, wN −{i} , pn ) = 0, j 6= i, ∀i ∈ N, (2.15) SEi (N, wN , pn ) = 1, ∀i ∈ N.

As a Corollary, by the linearity of the Semivalue, we get the Inverse Set, where r enters only the basic vectors: Theorem (Thm.6, Dragan, 2013). Let a Binomial Semivalue for a game (N, w), defined by a parameter r, be SE(N, v, pn ) = L. Let W given by (2.5), (2.6) be a basis for the space G(N ). Then, the solution of the Inverse Problem is expressed by the formula w=

X

aS wS + aN (wN +

X

i∈N

S⊂N,|S|6n−2

X

wN −{i} ) −

i∈N

Li wN −{i} ,

(2.16)

where the constants multiplying the basic games are arbitrary. 3.

The Power Game and the coalitional rational inverse

Consider in the Inverse Set the family of games, to be called the “almost null games”, obtained for aS = 0, ∀S ⊂ N, |S| 6 n − 2, S 6= ∅. This family, as seen in the formula (2.16), is given by X X w = aN (wN + wN −{i} ) − Li wN −{i} , (3.1) i∈N

i∈N

where aN is the parameter of the family; of course, the parameter r of the Binomial Semivalue occurs in the basic vectors. Now, by using the weight vectors (2.6), written in terms of r, as shown in (Dragan, 2013), we have

wT (T ) =

(1 + r)t−1 , ∀T ⊆ N, rt−1

s−t

wT (S) =

(−1)

(1 + r)t−1 , ∀S ⊃ T, rs−1

(3.2)

and wT (S) = 0, otherwise, so that from (3.2) we obtain

wN −{i} (N − {i}) =

(1 + r)n−2 , ∀i ∈ N, rn−2

wN −{i} (N ) = −

(1 + r)n−2 , rn−1

(3.3)

(1 + r)n−1 . (3.4) rn−1 In this way, from (3.3), (3.4), the components different of zero in (3.1) are: wN −{i} (N − {j}) = 0, ∀j 6= i,

(1 + r)n−2 , ∀i ∈ N, rn−2

(3.5)

X (1 + r)n−2 [aN (r − n + 1) + Li ]. n−1 r

(3.6)

w(N − (i}) = (aN − Li ) w(N ) =

wN (N ) =

i∈N

32

Irinel Dragan

Now, we compute the Power Game of an arbitrary game in the almost null family set, given in (3.5), (3.6), where the null values of the characteristic function are omitted, and we get: (3.7)

π(N − {i}, v, pn−1 ) = (n − 1)(aN − Li ), ∀i ∈ N, π(N, v, pn ) =

X

(3.8)

Li .

i∈N

As stated in definition 2 before, a Semivalue of a given game is coalitional rational if it belongs to the Core of the Power Game, or Power Core. If the Binomial Semivalue is L > 0, and, as seen in (3.8), this is efficient in the Power Game, then the only Core conditions are those obtained from the coalitions of size n − 1, that have the worth shown in (3.7), namely X Lj > (n − 1)(aN − Li ), ∀i ∈ N, (3.9) or

j∈N −{i}

aN 6

X 1 [ n−1

j∈N −{i}

Lj + (n − 1)Li ], ∀i ∈ N.

(3.10)

We proved Theorem 2. A Binomial Semivalue associated with the parameter r, and given by a nonnegative vector L ∈ Rn , is coalitional rational, in the Power Game of the game X X w = aN (wN + wN −{i} ) − Li wN −{i} , (3.11) i∈N

i∈N

If and only if aN satisfies the inequality aN 6

X 1 M in{ n−1

j∈N −{i}

Lj + (n − 1)Li },

(3.12)

Notice that there is also an infinite set of games in the almost null Inverse Set for which the Binomial Semivalue is not coalitional rational. Example 5. Return to the game considered in Example 1, for which the Banzhaf Value is B(N, v) = ( 12 , 12 , 21 ), so that the inequality (3.12) is aN 6 1. We compute the almost null game, relative to the Banzhaf Value by (3.5) and (3.6) , and we obtain w({1}) = w({2}) = w({3}) = 1, w({1, 2}) = w({1, 3}) = w({2, 3}) = w({1, 2, 3}) = 1, (3.13) that incidentally coincides with the given game (1.5). Obviously, the Banzhaf Value is the same, that is B(N, w) = ( 12 , 12 , 21 ). Then, we compute the Power Game, of (3.13) by (3.7) and (3.8) and we get 3 , 2 (3.14)

π({1, 2}, w, p2 ) = π({1, 3}, w, p2 ) = π({2, 3}, w, p2 ) = 1, π({1, 2, 3}, w, p3) =

33


while we have null values for the singletons. Now, in the new game (3.14) the old Banzhaf Value is efficient and we may see that it is also in the Power Core. This happened because for our game we have aN = 1, which satisfies (3.12) and we have to modify only w({1, 2, 3}) = 1, into π({1, 2, 3}, w, p3) = 32 , to get the Banzhaf Value in the Power Core. Notice that the Power Game does not have the same Banzhaf Value as the original one, or the almost null game in the Inverse Set; for example, in our case the Banzhaf Value of (3.14) is B(N, π) = ( 58 , 85 , 58 ). Obviously, this is not efficient again and the coalitional rationality conditions do not hold. Consider the same game, but take aN = 23 , and compute again the almost null game, relative to the Banzhaf Value by (3.5) and (3.6), and we obtain

w({1}) = w({2}) = w({3}) = 0, w({1, 2}) = w({1, 3}) = w({2, 3}) = 2, w({1, 2, 3}) = 0, (3.15) which gives the same old Banzhaf Value. Further, we compute the Power Game and we get 3 , 2 (3.16) in which the old Banzhaf Value is efficient, but it is not coalitional rational, because (3.12) does not hold. These two examples illustrate theorem 2 and the technique to build the game in the almost null inverse family, for which the given Banzhaf Value is coalitional rational. π({1, 2}, w, p2 ) = π({1, 3}, w, p2 ) = π({2, 3}, w, p2 ) = 2, π({1, 2, 3}, w, p3) =

References Banzhaf, J.F. (1965). Weighted voting doesn’t work; a mathematical analysis. Rutgers Law Review, 19, 317–343. Dragan, I. (1991). The potential basis and the weighted Shapley value. Libertas Mathematica, 11, 139–150. Dragan, I. (1996). New mathematical properties of the Banzhaf value. E.J.O.R., 95, 451– 463. Dragan, I. and Martinez-Legaz, J. E. (2001). On the Semivalues and the Power Core of cooperative TU games. IGTR, 3, 2&3, 127–139. Dragan, I. (2004). On the inverse problem for Semivalues of cooperative TU Games. IJPAM, 4, 545–561. Dragan, I. (2013). On the Inverse Problem for Binomial Semivalue. Proc. GDN2013 Conference, Stockholm, 191–198. Dubey, P., Neyman, A. and Weber, R. J. (1981). Value theory without efficiency. Math.O.R.,6, 122–128. Puente, M. A. (2000). Contributions to the representability of simple games and to the calculus of solutions for this class of games. Ph.D.Thesis, University of Catalonya, Barcelona, Spain. Freixas, J., and Puente, M. A., (2002). Reliability importance measures of the components in a system based upon semivalues and probabilistic values. Ann.,O.R., 109, 331–342. Shapley, L. S., (1953). A value for n=person games. Annals of Math.Studies, 28, 307–317.

Stackelberg Oligopoly Games: the Model and the 1-concavity of its Dual Game Theo Driessen1 , Aymeric Lardon2 and Dongshuang Hou3 University of Twente, Department of Applied Mathematics, P.O. Box 217, 7500 AE Enschede, The Netherlands E-mail: [email protected] 2 Université Jean Monnet, Saint-Etienne, France, E-mail: [email protected] 3 North-Western Politechnical University, Faculty of Applied Mathematics, Xi’an, China, E-mail: [email protected] 1

Abstract This paper highlights the role of a significant property for the core of Stackelberg Oligopoly cooperative games arising from the noncooperative Stackelberg Oligopoly situation with linearly decreasing demand functions. Generally speaking, it is shown that the so-called 1-concavity property for the dual of a cooperative game is a sufficient and necessary condition for the core of the game to coincide with its imputation set. Particularly, the nucleolus of such dual 1-concave TU-games agree with the center of the imputation set. Based on the explicit description of the characteristic function for the Stackelberg Oligopoly game, the aim is to establish, under certain circumstances, the 1-concavity of the dual game of Stackelberg Oligopoly games. These circumstances require the intercept of the inverse demand function to be bounded below by a particular critical number arising from the various cost figures. Keywords: Stackelberg oligopoly game; imputation set; core; efficiency; 1concavity

1.

Introduction of game theoretic notions

A cooperative savings game (with transferable utility) is given by a pair hN, wi, where its characteristic function w : P(N ) → R is defined on the power set P(N ) = {S | S ⊆ N } of the finite set N , of which the elements are called players, while the elements of the power set are called coalitions. The so-called real-valued worth w(S) of coalition S ⊆ N in the game hN, wi represents the maximal amount of monetary benefits due to the mutual cooperation among the members of the coalition, on the understanding that there are no benefits by absence of players, that is w(∅) = 0. In the framework of the division problem of the benefits w(N ) of the grand coalition N among the potential players, any allocation scheme of the P form x = (xi )i∈N ∈ RN is supposed to meet, besides the efficiency principle i∈N xi = w(N ), the so-called individual rationality condition in that each player is allocated at least the individual worth, i.e., xi > w({i}) for all i ∈ N . Concerning the development of the solution part, a (multi- or single-valued) solution concept σ assigns to any cooperative game hN, wi a (possibly empty) subset of its imputation set I(N, w), that is σ(N, w) ⊆ I(N, w), where

35

Stackelberg Oligopoly Games

I(N, w) = {(xi )i∈N ∈ RN |

X

xi = w(N )

i∈N

and xi > w({i}) for all i ∈ N }.

The best known multi-valued solution concept called core requires the group rationality condition in that the aggregate allocation to the members of any coalition is at least its coalitional worth, that is X → CORE(N, w) = {− x ∈ I(N, w) | xi > w(S) i∈S

for all S ⊆ N , S 6= N , S 6= ∅} (1.1)

Of significant importance is the upper core bound composed of the marginal contributions mw i = w(N ) − w(N \{i}), i ∈ N , with respect to the formation of the grand coalition N in the game hN, wi. Obviously, xi 6 mw i for all i ∈ N and all → − x ∈ CORE(N, w). In this context, we focus on the following core catcher called CoreCover X CC(N, w) = {(xi )i∈N ∈ RN | xi = w(N ) and xi 6 mw i i∈N

for all i ∈ N } (1.2)

In the framework of the core, a helpful Ptool appears to be the so-called gap function g w : P(N ) → R defined by g w (S) = i∈S mw i − w(S) for all S ⊆ N , S 6= ∅, where g w (∅) = 0. So, the gap g w (S) of any coalition S measures how much the coalitional worth w(S) differs from the aggregate allocation based on the individually marginal contributions. The interrelationship between the gap function and the general inclusion CORE(N, w) ⊆ CC(N, w) is the following equivalence (Driessen, 1988): CORE(N, w) = CC(N, w)

⇐⇒

0 6 g w (N ) 6 g w (S) for all S ⊆ N , S 6= ∅ (1.3)

In words, the core catcher CC(N, w) coincides with the core CORE(N, w) only if the non-negative gap function g w attains its minimum at the grand coalition N . If the latter property (1.3) holds, the savings game hN, wi is said to be 1-convex. With every cooperative savings game hN, wi there is associated its dual game hN, w∗ i defined by w∗ (S) = w(N ) − w(N \S) for all S ⊆ N . That is, the worth of any coalition in the dual game is given by the coalitionally marginal contribution with respect to the formation of the grand coalition N in the original game. Partic(w ∗ ) ularly, w∗ (∅) = 0, w∗ (N ) = w(N ), and so, mi = w∗ (N ) − w∗ (N \{i}) = w({i}) for all i ∈ N . We arrive at the first main result. Proposition 1.1. Three equivalent statements for any cooperative savings game hN, wi. X ∗ I(N, w) 6= ∅ ⇐⇒ w(N ) > w({i}) ⇐⇒ g (w ) (N ) 6 0 (1.4) i∈N

36

Theo Driessen, Aymeric Lardon, Dongshuang Hou

In fact, the dual game hN, w∗ i of any cooperative savings game hN, wi is treated as a cost game such that the core equality CORE(N, w∗ ) = CORE(N, w) holds, on the understanding that the core of any cost game is defined through the reversed → → inequalities of (1.1). Thus, − x ∈ CORE(N, w∗ ) iff − x ∈ CORE(N, w). As the counterpart to 1-convex savings games (with non-negative gap functions), we deal with so-called 1-concave cost games (with non-positive gap functions). Definition 1.2. A cooperative cost game hN, wi is said to be 1-concave if its nonpositive gap function attains its maximum at the grand coalition N , i.e., g w (S) 6 g w (N ) 6 0 for all S ⊆ N , S 6= ∅.

(1.5)

Theorem 1.3. Three equivalent statements for any cooperative savings game hN, wi. (i) The dual game hN, w∗ i is 1-concave, that is (1.5) applied to hN, w∗ i holds (ii) w(N ) >

X

w({i})

i∈N

(iii) I(N, w) 6= ∅

and

w(S) 6

X

w({i})

i∈S

and

for all S ⊆ N , S 6= N , S 6= ∅

(1.6)

CORE(N, w) = I(N, w)

Proof. In view of Proposition 1.1, together with CORE(N, w) ⊆ I(N, w), it remains to prove the implication (iii) =⇒ (ii). By contra-position, suppose (ii) does not hold P in that there exists S ⊆ N , S 6= N , S 6= ∅ with w(S) > i∈S w({i}). Define the → 1 allocation − x = (xi )i∈N ∈ RN by xi = w({i}) for all i ∈ S and xi = w({i}) + n−s · P → w(N ) − w({j}) for all i ∈ N \S. Obviously, − x ∈ I(N, w)\CORE(N, w). ✷ j∈N

2.

The Stackelberg oligopoly game

The normal form game of the non-cooperative Stackelberg oligopoly situation1 is modeled as a cooperative TU-game as follows. Throughout the paper we fix the set N of firms with (possibly identical) strategy sets Xi = [0, wi ), i ∈ N , with reference to (possibly unlimited) capacities wi ∈ [0, ∞], i ∈ N , (possibly distinct) marginal costs ci > 0, i ∈ N , and the inverse demand function p(x) = a − x for all x 6 a and p(x) = 0 for all x > a. In this framework, the corresponding individual profit functions πi : Πk∈N Xk → R, i ∈ N , and coalitional profit functions πT : Πk∈N Xk → R, T ⊆ N , T 6= ∅, are defined by X πi ((xk )k∈N ) = (a − X(N ) − ci ) · xi and πT ((xk )k∈N ) = πj ((xk )k∈N ) (2.7) j∈T

P

where X(N ) = k∈N xk ∈ R represents the aggregate production and a > 2 · n · maxi∈N ci . The Stackelberg oligopoly model is based on a two-stage procedure. Given that the 1

For a description of the oligopoly situation, we refer to the PhD thesis of Aymeric Lardon (Lardon, 2011).

37


members of the coalition S are supposed to perform their leadership in the first stage maximizing its coalitional profit by taking into account the best responses of individual followers j ∈ N \S (i.e., the non-members of S) during the second stage. So, the second stage is devoted to the maximization problems maxxj ∈Xj πj (xj , (xk )k∈N \{j} ) for all j ∈ N \S. Theorem 2.1. Let S ⊆ N , S 6= N . Write cT =

P

k∈T

ck for all T ⊆ N , T 6= ∅.

(i) The best response of any individual i ∈ N \S during the second stage is given by yi =

cN \S + X(N \S) a − X(S) + cN \S − ci = − ci n−s n+1−s

(ii) The worth v(S) of coalition S is determined by (x∗iS )2 ∗ 1 v(S) = where xiS = 2 · a + cN \S − (n + 1 − s) · ciS n+1−s

is the maximizer of the profit function of player iS ∈ S with the smallest marginal contribution among members of S, supposing other members of S produce nothing. Note that x∗i,S > 0 because of a > n · maxi∈N ci .

Proof. Fix coalition S 6= N . For all i ∈ N \S the maximization problem of the player’s profit function πi ((xk )k∈N ) = (a − X(N ) − ci ) · xi = −(xi )2 + xi · (a − ci − i X(N \{i})) is solved through its first order condition ∂π ∂xi = 0, yielding 1 or equivalently, xi = a − ci − X(N ) xi = 2 · a − ci − X(N \{i}) Summing up the latter equations over all i ∈ N \S yields X(N \S) = (n − s) · a − cN \S − (n − s) · X(N ) and so, a − X(N ) = Hence, by substitution, it holds for all i ∈ N \S yi = (a − X(N )) − ci =

cN \S + X(N \S) n−s

cN \S + X(N \S) a − X(S) + cN \S − ci = − ci n−s n+1−s

This proves part (i). Given these best responses by players in N \S, the maximization problem of the coalitional profit function πS is, due to unlimited capacities, equivalent to the maximization problem of the profit function of the firm iS ∈ S with smallest marginal cost among members of S, (i.e., ciS 6 ci for all i ∈ S), supposing that the other members of S produce nothing. In this framework, a − xiS + cN \S − ci for all i ∈ N \S and thus, n+1−s n−s y(N \S) = · a − xiS + cN \S − cN \S n+1−s 1 = a − xis − · a − xiS + cN \S n+1−s yi =

38


Hence, we focus on the player’s profit function of the form → − πiS ((yi )i∈N \S , xiS , ( 0 )i∈S\{iS } ) = a − X(N ) − ciS · xiS = a − xiS − y(N \S) − ciS · xiS

1 · a − xiS + cN \S − ciS · xiS n+1−s 1 = · a + cN \S − (n + 1 − s) · ciS − xiS · xiS n+1−s =

The first order condition yields that the maximizer of this quadratic profit function is given by ∗ 1 xiS = 2 · a + cN \S − (n + 1 − s) · ciS and finally, → − v(S) = πiS ((yi )i∈N \S , x∗iS , ( 0 )i∈S\{iS } ) 1 = · a + cN \S − (n + 1 − s) · ciS − x∗iS · x∗iS n+1−s (x∗iS )2 1 ∗ ∗ = · 2 · xiS − xiS · x∗iS = n+1−s n+1−s

It remains to check the non-negativity constraint for the maximizer x∗iS (since production levels are supposed to be non-negative). Of course, the non-negativity constraint also applies to any player j ∈ N \S. For that purpose, choose a sufficiently large in that a > 2 · n · maxi∈N ci (or to be exact, a > 2 · n · maxi∈N ci ). Recall that production levels of all firms are supposed to be unlimited. This proves part (ii). ✷ In the context of the resulting cooperative TU game, the following significant notions appear. For any non-trivial coalition T ⊆ N , T 6= ∅, let cT , c¯T , and cT respectively, denote the aggregate, average, and minimal cost of coalition T , that is cT =

X

ck

c¯T =

k∈T

c¯c¯T =

1 X · (ck )2 |T | k∈T

cT |T |

Note that

cT = min{ck |

k ∈ T }.

Moreover, (2.8)

2 X X ck − c¯T = (ck )2 − |T | · (¯ cT )2 (2.9)

k∈T

k∈T

Generally speaking, c¯T > cT and moreover, the equality is met only by identical marginal costs, that is, for any coalition, the average cost equals the minimum cost if and only if all the marginal costs of its members do not differ. By (2.9), c¯c¯T > (¯ cT )2 , that is the average of the squares of marginal costs covers the square of the average cost.

39


Theorem 2.2. Given the normal form game hN, (ck )k∈N , (wk )k∈N , ai of the noncooperative Stackelberg oligopoly situation with unlimited capacities (wi = +∞ for all i ∈ N ) and possibly distinct marginal costs, then the corresponding cooperative n-person Stackelberg oligopoly game hN, vi is determined by v(∅) = 0 and for all S ⊆ N , S 6= ∅, 2 2 a + cN \S − (n + 1 − s) · cS a + cN \S n+1−s = · − cS v(S) = (2.10) 4 · (n + 1 − s) 4 n+1−s

Here c∅ = 0. In case all marginal costs are identical, say ci = c for all i ∈ N , 2 1 then (2.10) reduces to v(S) = (a−c) · n+1−s for all S ⊆ N , S 6= ∅, and so, the 4 Stackelberg oligopoly game is a multiple of the symmetric n-person cooperative 1 , the imputation set of which degenerates into the single core game v(s) = n+1−s 1 allocation n · (1, 1, . . . , 1) ∈ Rn . 3.

1-Concavity of the dual game of Stackelberg oligopoly games

Assuming non-emptiness of its core, our goal is to study whether or not the 1convexity property applies to the Stackelberg oligopoly game. For that purpose, we are interested in the structure of the corresponding non-negative gap function. P Generally speaking, the validity of the inequality v(S) 6 v({k}) is equivalent k∈S P to the reversed inequality g v (S) > k∈S g v ({k}) for all S ⊆ N , S 6= N , S 6= ∅. Together with the non-negativity of the gap function g v , it follows that g v (S) > g v ({i}) whenever i ∈ S. In words, the gap function g v of the Stackelberg oligopoly game attains among non-trivial coalitions containing a given player its minimum either at the one-person coalition or the grand coalition N . According to the next proposition, the gap of the grand coalition is not minimal and hence, the 1-convexity property fails to hold for the Stackelberg oligopoly games. However, the solution concept called τ -value (cf. Tijs, 1981) agrees, concerning its allocation to any player i, with the efficient compromise between the marginal contribution mvi = v(N ) − v(N \{i}) and the stand-alone worth v({i}), i ∈ N (treated as upper and lower core bounds respectively). Proposition 3.1. Given the non-emptiness of the core of the Stackelberg oligopoly game hN, vi, the corresponding gap function g v satisfies g v (N ) > g v ({i}) for all i ∈ N.

Proof. Fix i ∈ N . Recall that g v (N \{k}) = g v (N ) for all k ∈ N . Fix j ∈ N , j 6= i. Due to the non-emptiness of the core, mvk > v({k}) for all k ∈ N . We conclude that g v (N ) − g v ({i}) = g v (N \{j}) − mvi + v({i}) X mvk − v(N \{j}) + v({i}) = k∈N \{i,j}

>

X

k∈N \{i,j}

=

X

k∈N \{i,j}

mvk −

X

v({k}) + v({i})

k∈N \{j}

(mvk − v({k})) > 0

40


Here we applied the inequality v(S) 6

P

k∈S

v({k}) to S = N \{j}).

✷

Now we arrive at the main result stating that the core of any Stackelberg oligopoly game coincides with its imputation set, provided its non-emptiness. By Theorem 1.3, the class of dual games of Stackelberg oligopoly savings games is a significant class of 1-concave (cost) games. The proof proceeds by checking the validity of (1.6). Note that the worth of any single player i ∈ N and the grand coalition N respectively, are given as follows:

v({i}) =

a + cN − (n + 1) · ci 4·n

2

for all i ∈ N , and v(N ) =

(a − cN )2 (3.11) 4

Theorem 3.2. The dual game hN, v ∗ i of the cooperative n-person Stackelberg oligopoly game hN, vi of the form (2.10) with distinct marginal costs is 1-concave only if the intercept a > 0 of the inverse demand function is large enough. On the one hand, ∗

g (v ) (N ) 6 0

if and only if

a>

L1 − cN 2

(3.12)

where the critical number L1 represents the lower bound given by

−1 2 2 L1 = c¯N − cN · (n + 1) · c¯c¯N − [cN + cN ] ∗

∗

g (v ) (S) 6 g (v ) (N )

On the other,

for all S ⊆ N , S 6= ∅

(3.13) (3.14)

Proof of Theorem 3.2. (The full proof consists of two parts.) Part 1. (v ∗ ) (N ) 6 0 or equivalently, by Proposition (1.1) 1.1, v(N ) > P Firstly, we check g v({i}). Put the substitution x = a + cN . By using (3.11), it holds that 4 · n · i∈N v({i}) = [x − (n + 1) · ci ]2 for all i ∈ N as well as 4 · n · v(N ) = n · [a − cN ]2 = n · [x − (cN + cN )]2 . Thus, we obtain the following chain of equalities:

4 · n · v(N ) −

X

i∈N

v({i})

2 X 2 = n · x − (cN + cN ) − x − (n + 1) · ci i∈N

2 2 = n · x − 2 · x · (cN + cN ) + cN + cN −

X

i∈N

2

2

2

x − 2 · x · (n + 1) · ci + (n + 1) · (ci )

= 2 · x · (n + 1) · cN − n · (cN + cN )

41


+ n · cN + cN

2

− (n + 1)2 ·

X

(ci )2

i∈N

2 = 2 · x · cN − n · cN + n · cN + cN − (n + 1)2 · n · c¯c¯N

= n · 2 · x · c¯N − cN

2 2 + cN + cN − (n + 1) · c¯c¯N

∗

So far, we conclude that g (v ) (N ) 6 0 if and only if

2

2 · x · c¯N − cN > (n + 1) · c¯c¯N − cN + cN

2

where

x = a + cN (3.15)

or equivalently, a > L21 − cN where the critical lower bound L1 is given by (3.13). Notice that the quadratic term x2 vanishes in the inequality (3.15). ✷ Remark 3.3. For future convenience, we treat an alternative proof of part 1 of Theorem 3.2 in the appendix. They differ in that this second proof is based on the variable a itself instead of the variable x. In the new setting, the description of the critcal lower bound (3.15) has to be replaced by a similar inequality: 2 · a · c¯N − cN > (n2 + 2 · n) · c¯c¯N − (¯ cN )2 + c¯c¯N − (cN )2

(3.16)

This second approach yields an alternative description of the same lower bound of the form −1 2 2 2 L2 = c¯N − cN · (n + 2 · n) · c¯c¯N − (¯ cN ) + c¯c¯N − (cN ) It is left to the reader to verify the validity of the equality L2 = L1 − 2 · cN .

(3.17) ✷

Proof of Theorem 3.2. (The full proof consists of two parts.) Part 2. ∗

∗

) Secondly, we check g (v P (S) 6 g (v ) (N ) for all S ⊆ N , S = 6 ∅, or equivalently, by Theorem 1.3, v(S) 6 v({i}) for all S ⊆ N , S = 6 N , S 6= ∅. Put the i∈S fundamental substitutions

x := a + cN

as well as

AS := cS + (n + 1 − s) · cS

From (2.10), we derive the following shortened notation for the worth of any multiperson coalition S as well as the one-person coalitions respectively, in the Stackelberg oligopoly game.

42


4 · n · v(S) =

2 n · x − AS n+1−s

=

4 · n · v({i}) = x − (n + 1) · ci 4·n·

X

v({i}) =

i∈S

X i∈S

n · x2 − 2 · x · AS + (AS )2 2

for all i ∈ N , and next

2

X i∈S

2

2

x − 2 · x · (n + 1) · ci + (n + 1) · (ci )

= s · x2 − 2 · x · (n + 1) · cS + (n + 1)2 · 4·n·

for all S ⊆ N ,

n+1−s

X

(ci )2

i∈S

v({i}) − v(S) = α2 · x2 + α1 · x + α0

(3.18)

Our main goal is to describe the coalitional notion of surplus in terms of a quadratic function of the variable x, say f (x) = α2 · x2 + α1 · x + α0 where α2 > 0. Definition 3.4. The three real numbers αk , k = 0, 1, 2, are given as follows: (n − s) · (s − 1) n = n+1−s n+1−s 1 α1 = · −2 · (n + 1 − s) · (n + 1) · cS + 2 · n · AS n+1−s

α2 = s −

=

(3.19) (3.20)

1 · 2 · n − (n + 1) · (n + 1 − s) · cS + 2 · n · (n + 1 − s) · cS(3.21) n+1−s

α0 =

X 1 (ci )2 − n · (AS )2 · (n + 1 − s) · (n + 1)2 · n+1−s

(3.22)

i∈S

Clearly, α2 > 0 since s 6= n, s 6= 1. Further, it holds that α1 < 0 due to c¯S > cS as well as n · (n + 1 − s) < s · (n + 1) · (n + 1 − s) − n

or equivalently, n < s · (n + 1 − s)

So far, we conclude that the quadratic function f (x) = α2 · x2 + α1 · x + α0 attains −α1 −α1 its minimum at x = 2·α and the corresponding minimal function values f ( 2·α )= 2 2 −(α1 )2 4·α2 (α1 )2 .

+ α0 . This minimal function value is non-negative if and only if 4 · α0 · α2 > For the sake of the forthcoming computational matters, recall that c¯T = ctT for all T ⊆ N , T 6= ∅, as well as (2.9). In order to apply shortened notation, put the substitution δs = n − (n + 1) · (n + 1 − s). As one out of two options for a possible

43


representation of α1 , we choose (3.21) to evaluate the square of α1 , as well as the product 4 · α2 · α0 . Finally, we arrive at a reasonable description of their difference as stated in the next lemma. Lemma 3.5. Consider the setting of Definition 3.4. Then the following equality holds: (i) α2 · α0 −

(α1 )2 = s · (s − 1) · (n − s) · (n + 1)2 · c¯c¯S − (¯ cS )2 4 2 2 − s · n · (n + 1 − s) · cS − c¯S

(3.23)

(ii) Moreover, a sufficient condition for 4·α2 ·α0 −(α1 )2 > 0 is given by the following inequality: 2 (s − 1) · (n − s) · (n + 1)2 · c¯c¯S − (¯ cS )2 > n · (n + 1 − s)2 · cS − c¯S (3.24) (iii) The sufficient condition (3.24) holds. Proof of Lemma 3.5. The current approach proceeds as follows. Firstly, we evaluate the square (α1 )2 and secondly, we study the two contributions within the · n · (AS )2 , while its product α2 · α0 , particularly the main contribution − (n−s)·(s−1) n+1−s second contribution will not be changed at all and kept till the end in the form

(n + 1)2 ·

(n − s) · (s − 1) X · (ci )2 n+1−s i∈S

that is

(n + 1)2 ·

(n − s) · (s − 1) · s · c¯c¯S n+1−s

1 In order to apply shortened notation, put ρs = n+1−s . Firstly, straightforward 2 calculations involving the relevant square (α1 ) and secondly, straightforward calculations involving the remaining part of the product α2 · α0 , yield the following:

2 (α1 )2 2 = (ρs ) · δs · cS + n · (n + 1 − s) · cS 4 2 2 2 2 2 2 = (ρs ) · (δs ) · (cS ) + n · (n + 1 − s) · (cS ) + 2 · δs · cS · n · (n + 1 − s) · cS 2 2 2 2 2 2 2 = (ρs ) · (δs ) · s · (¯ cS ) + n · (n + 1 − s) · (cS ) + 2 · δs · s · n · (n + 1 − s) · c¯S · cS

(3.25)

44

Theo Driessen, Aymeric Lardon, Dongshuang Hou In addition,

2 (n − s) · (s − 1) n · (AS )2 · = (ρs )2 · (n − s) · (s − 1) · n · cS + (n + 1 − s) · cS n+1−s n+1−s cS ·cS = (ρs )2 ·(n − s)·(s − 1)·n· s2 ·(¯ cS )2 + (n + 1 − s)2 ·(cS )2 + 2·s·(n + 1 − s)·¯ +

(3.26)

Summing up the two negative expressions (3.25)–(3.26) to be multiplied by the square (ρs )2 yields s2 · (δs )2 + (n − s) · (s − 1) · n · (¯ cS )2

(3.27)

+ (n + 1 − s) · n + (n − s) · (s − 1) · n · (cS )2

(3.28)

+ 2 · s · n · (n + 1 − s) · δs + (n − s) · (s − 1) · c¯s · cS

(3.29)

2

2

In order to simplify these calculations, we use the following simple equalities:

(n − s) · (s − 1) + n = s · (n + 1 − s) (n − s) · (s − 1) · n + n2 = s · n · (n + 1 − s) δs + (n − s) · (s − 1) = −(n + 1 − s)2 2 2 (3.30) −s (δs ) +(n−s)(s−1)n +n·s ·(n+1−s)3 = −(n+1−s)(n+1)2(n−s)(s−1)s The final computations are as follows:

−2

(ρs )

(α1 )2 · α2 · α0 − = −s · n · (n + 1 − s)3 · (cS )2 4 + 2 · s · n · (n + 1 − s)3 · c¯S · cS − s2 · (δs )2 + (n − s) · (s − 1) · n · (¯ cS )2 + (n + 1 − s) · (n + 1)2 · (n − s) · (s − 1) · s · c¯c¯S 2 = −n · s · (n + 1 − s)3 · cS − c¯S

45

Stackelberg Oligopoly Games − s2 · (δs )2 + (n − s)·(s − 1) · n ·(¯ cS )2 + n·s·(n + 1 − s)3 ·(¯ cS )2 + (n + 1 − s) · (n + 1)2 · (n − s) · (s − 1) · s · c¯c¯S 2 = −n · s · (n + 1 − s)3 · cS − c¯S + (n + 1 − s) · (n + 1)2 · (n − s) · (s − 1) · s · (¯ cc¯S − (¯ cS )2 )

The last equality is due to (3.30). 4.

✷

APPENDIX: Alternative Proofs.

Alternative proof of Theorem 3.2. The full proof consists of two parts. Part 1. (v ∗ ) (N ) 6 0 or equivalently, by Proposition 1.1, v(N ) > P Firstly, we check g i∈N v({i}). By using (3.11), we obtain the following chain of equalities: X 4 · n · v(N ) − v({i}) i∈N

2 X 2 a + cN − (n + 1) · ci = n · a − cN − i∈N

2 X 2 = n · a − cN − a2 + 2 · a · cN − (n + 1) · ci + cN − (n + 1) · ci i∈N

= n · a2 − 2 · a · cN + (cN )2 − n · a2 − 2 · a · n · cN − (n + 1) · cN

−

X

i∈N

2

2

2

(cN ) − 2 · (n + 1) · cN · ci + (n + 1) · (ci )

X = 2 · a · cN − n · cN + n · (cN )2 − n − 2 · (n + 1) · (cN )2 − (n + 1)2 · (ci )2 i∈N

= 2 · a · n · c¯N − cN + n · (cN )2 + (n + 2) · (cN )2 − (n + 1)2 · n · c¯c¯N

2

2

2

= n · 2 · a · c¯N − cN + (cN ) + n · (n + 2) · (¯ cN ) − (n + 1) · c¯c¯N

∗

So far, we conclude that g (v ) (N ) 6 0 if and only if cN )2 2 · a · c¯N − cN > (n + 1)2 · c¯c¯N − (cN )2 − n · (n + 2) · (¯

(4.31)

46


or equivalently, −1 1 L2 2 2 2 a > · c¯N − cN · (n + 1) · c¯c¯N − (cN ) − n · (n + 2) · (¯ cN ) = 2 2 Here the critical number L2 represents the lower bound given by

−1 2 2 2 L2 = c¯N − cN · (n + 1) · c¯c¯N − (cN ) − n · (n + 2) · (¯ cN ) Notice that

−1 2 2 2 · (n + 2 · n) · c¯c¯N − (¯ cN ) + c¯c¯N − (cN ) L2 = c¯N − cN

while

−1 2 L1 = c¯N − cN · (n + 1)2 · c¯c¯N − cN + cN

(4.32) (4.33)

Recall that c¯c¯N > (¯ cN )2 > (cN )2 . Thus, L2 > 0. In fact, it is left to the reader ✷ to verify the validity of the equality L22 = L21 − cN , that is L2 = L1 − 2 · cN . Alternative proof of Theorem 3.2. The full proof consists of two parts. Part 1. ∗ ∗ Secondly, we checkP g (v ) (S) 6 g (v ) (N ) for all S ⊆ N , S 6= ∅, or equivalently, by Theorem 1.3, v(S) 6 i∈S v({i}) for all S ⊆ N , S 6= N , S 6= ∅. This second proof differs from the first one in that it uses different fundamental substitutions as well as c(S, i) := cS − (n + 1) · ci

yS := a + cN \S

Fix S ⊆ N , S 6= N , S 6= ∅. Note that (2.10) it holds that

P

i∈S

for all i ∈ S,

instead of x = a + cN

c(S, i) = −(n + 1 − s) · cS . By using

2 n 4 · n · v(S) = · a + cN \S − (n + 1 − s) · cS n+1−s 2 n = · yS − (n + 1 − s) · cS Further, n+1−s 2 2 4 · n · v({i}) = a + cN − (n + 1) · ci = yS + c(S, i) for all i ∈ N Recall that

P

i∈S

chain of equalities:

c(S, i) = −(n + 1 − s) · cS for all S ⊆ N . We obtain the following

47


4·n·

X i∈S

X 2 v({i}) − v(S) = yS + c(S, i) i∈S

2 n · yS − (n + 1 − s) · cS − n+1−s X = (yS )2 + 2 · yS · c(S, i) + (c(S, i))2 i∈S

n 2 2 2 − · (yS ) − 2 · yS · (n + 1 − s) · cS + (n + 1 − s) · (cS ) n+1−s X X = s · (yS )2 + 2 · yS · c(S, i) + (c(S, i))2 i∈S

−

i∈S

n · (yS )2 + 2 · yS · n · cS − n · (n + 1 − s) · (cS )2 n+1−s

Our main goal is to describe the coalitional notion of surplus in terms of a quadratic function of the variable y, say g(y) = β2 · y 2 + β1 · y + β0 where β2 > 0. ✷ Definition 4.1. The three real numbers βk , k = 0, 1, 2, are given as follows: n (n − s) · (s − 1) = n+1−s n+1−s β1 = 2 · n · cS − s · (n + 1 − s) · c¯S

β2 = s −

β0 =

X i∈S

(c(S, i))2 − n · (n + 1 − s) · (cS )2

(4.34) (4.35) (4.36)

Clearly, β2 > 0 since s 6= n, s 6= 1. Further, it holds that β1 < 0 due to c¯S > cS as well as n < s · (n + 1 − s) since s · (s − 1) < n · (s − 1). So far, we conclude that the quadratic function g(y) = β2 · y 2 + β1 · y + β0 attains its minimum at −(β1 )2 −β1 1 y = 2·β and the corresponding minimal function values g( −β 2·β2 ) = 4·β2 + β0 . This 2 minimal function value is non-negative if and only if 4 · β0 · β2 > (β1 )2 . For the sake cT of the forthcoming computational matters, recall (3.11) as well as c¯T = |T | for all T ⊆ N , T 6= ∅. Recall the fundamental substitution c(S, i) := cS − (n + 1) · ci for all i ∈ S. Based upon (4.34)–(4.36), we evaluate the square of β1 , as well as the product 4 · β2 · β0 . Finally, we arrive at a reasonable description of their difference as stated in the next lemma.

48


Lemma 4.2. Consider the setting of Definition 4.1. Firstly, we evaluate β0 in the following form: β0 =

X i∈S

=

(c(S, i))2 − n · (n + 1 − s) · (cS )2

X i∈S

cS − (n + 1) · ci

2

− n · (n + 1 − s) · (cS )2

= s · (cS )2 − 2 · (n + 1) · (cS )2 + (n + 1)2 · s · c¯c¯S − n · (n + 1 − s) · (cS )2 = (−2 · n − 2 + s) · (cS )2 + (n + 1)2 · s · c¯c¯S − n · (n + 1 − s) · (cS )2 Secondly, we add the following chain of computations: (β1 )2 = β2 · β0 − n · cS − s · (n + 1 − s) · c¯S 4 (n − s) · (s − 1) = · (−2 · n − 2 + s) · s2 · (¯ cS )2 + (n + 1)2 · s · c¯c¯S n+1−s 2 − n · (n + 1 − s) · (cS ) − n2 · (cS )2 − 2 · n · s · (n + 1 − s) · cS · c¯S 2 2 2 + s · (n + 1 − s) · (¯ cS ) 2 = −n · (n − s) · (s − 1) − n · (cS )2 β2 · β0 −

(n − s) · (s − 1) 2 2 2 + cS )2 · (−2 · n − 2 + s) · s − s · (n + 1 − s) · (¯ n+1−s +

(n − s) · (s − 1) · (n + 1)2 · s · c¯c¯S + 2 · n · s · (n + 1 − s) · cS · c¯S n+1−s

= −n · s · (n + 1 − s) · (cS )2 + 2 · n · s · (n + 1 − s) · cS · c¯S (n − s) · (s − 1) + · (n + 1)2 · s · c¯c¯S n+1−s s2 3 − · (n − s) · (s − 1) · (2 · n + 2 − s) + (n + 1 − s) · (¯ cS )2 n+1−s 2 (n − s) · (s − 1) = −n · s · (n + 1 − s) · cS − c¯S + · (n + 1)2 · s · c¯c¯S n+1−s s2 + n · s · (n + 1 − s) − (n − s) · (s − 1) · (2 · n + 2 − s) n+1−s + (n + 1 − s)3 · (¯ cS )2

49

Stackelberg Oligopoly Games Thus, (β1 )2 (n + 1 − s) · β0 · β2 − 4 2 = −s · n · (n + 1 − s)2 · cS − c¯S + (s − 1) · (n − s) · (n + 1)2 · s · c¯c¯S + n · s · (n + 1 − s)2 − s2 · (n + 1 − s)3 − s2 · (n − s) · (s − 1) · (2 · n + 2 − s) · (¯ cS )2

2 = −s · n · (n + 1 − s)2 · cS − c¯S + s · (s − 1) · (n − s) · (n + 1)2 · c¯c¯S − s · (s − 1) · (n − s) · (n + 1)2 · (¯ cS )2 2 = −s · n · (n + 1 − s)2 · cS − c¯S + s · (s − 1) · (n − s) · (n + 1)2 · c¯c¯S − (¯ cS )2 It suffices to prove the next inequality: 2

2

(s − 1) · (n − s) · (n + 1) · c¯c¯S − (¯ cS )

2 > n · (n + 1 − s) · cS − c¯S (4.37) 2

or equivalently, by (2.9) 2 2 X (s − 1) · (n − s) · (n + 1)2 · ci − c¯S > s · n · (n + 1 − s)2 · cS − c¯S i∈S

By (4.37), we observe the same inequality as in Lemma 3.5 and hence, we may state the same sufficiency condition. ✷ References Chander, P., and H. Tulkens, (1997). The Core of an Economy with Multilateral Environmental Externalities. International Journal of Game Theory, 26, 379–401. Driessen, T. S. H. (1988). Cooperative Games, Solutions, and Applications. Kluwer Academic Publishers, Dordrecht, The Netherlands. Driessen, T. S. H., and H. I. Meinhardt (2005). Convexity of oligopoly games without transferable technologies, Mathematical Social Sciences, 50, 102–126. Driessen, T. S. H., Hou, Dongshuang, and Aymeric Lardon (2011a). Stackelberg Oligopoly TU-games: Characterization of the core and 1-concavity of the dual game. Working Paper, Department of Applied Mathematics, University of Twente, Enschede, The Netherlands. Driessen, T. S. H., Hou, Dongshuang, and Aymeric Lardon (2011b). A necessary and sufficient condition for the non-emptiness of the core in Stackelberg Oligopoly TU games. Working paper, University of Saint-Etienne, France.

50


Lardon, A. (2009). The γ-core of Cournot oligopoly TU games with capacity constraints. Working paper, University of Saint-Etienne, France. Lardon, A. (2011). PhD Thesis: Five Essays on Cooperative Oligopoly Games, University of Saint-Etienne, Universite Jean Monnet de Saint-Etienne, France, 13 October 2011. Norde, H., Do, K. H. P., and S. H. Tijs (2002). OLigopoly Games with and without transferable technologies. Mathematical Social Sciences, 43, 187–207. Sherali, H. D., Soyster, A. L., and F. H. Murphy (1983). Stackelberg–Nash–Cournot Equilibria, Characterizations and Computations. Operations Research, 31(2), 253–276. Tijs, S. H. (1981). Bounds for the core and the τ -value. In: Game Theory and Mathematical Economics, North-Holland Publishing Company, Amsterdam, 123–132. Zhao, J. (1999). A necessary and sufficient condition for the convexity in oligopoly games. Mathematical Social Sciences, 37, 187–207. Zhao, J. (1999). A β-core existence result and its application to oligopoly markets. Games and Economics Behavior, 27, 153–168.

On Uniqueness of Coalitional Equilibria Michael Finus,1 Pierre von Mouche2 and Bianca Rundshagen3 University of Bath, Department of Economics, Bath BA2 7AY, United Kingdom Email: [email protected] 2 Wageningen Universiteit, Hollandseweg 1, 6700 EW, Wageningen, The Netherlands E-mail: [email protected] 3 Universität Hagen, Department of Economics, Universitätsstrasse 11, 58097 Hagen, Germany E-mail: [email protected] 1

Abstract For the so-called ‘new approach’ of coalition formation it is important that coalitional equilibria are unique. Uniqueness comes down to existence and to semi-uniqueness, i.e. there exists at most one equilibrium. Although conditions for existence are not problematic, conditions for semiuniqueness are. We provide semi-uniqueness conditions by deriving a new equilibrium semi-uniqueness result for games in strategic form with higher dimensional action sets. The result applies in particular to Cournot-like games.

Key words: Coalition formation, Cournot oligopoly, equilibrium (semi-)uniqueness, game in strategic form, public good. 1.

Introduction

The analysis of coalition formation – in particular in the context of externalities – has become an important topic in economics. Examples do not only include firms that coordinate their output or prices in oligopolistic markets (cartels), jointly invest in research assets (R&D-agreements) or completely merge (joint ventures), but also countries that coordinate their tariffs (trade agreements and customs unions) or their environmental policy (international environmental agreements). Our article contributes to the so-called ‘new approach’ of coalition formation (see for instance Yi(1997) and Bloch(2003) for an extensive overview). The goal of this approach is to determine equilibrium coalition structures. As the approach consists of modelling coalition formation as a 2-stage game with simultaneous actions in each of both stages, it is important that for each possible coalition structure coalitional equilibria, i.e. equilibria in the second stage, are unique.1 So for the new approach it is important to have results that guarantee uniqueness of coalitional equilibria. Conditions should be such that they can be easily checked for the base games that appeared so far in these models, like Cournot and public good games. As far as we know, general uniqueness results for coalitional equilibria 1

Roughly speaking, in the first stage, the players choose a membership action which via a given member-ship rule leads to a coalition structure. In the second stage, the players are the coalitions in this coalition structure. Each of these coalitions chooses a ‘physical’ action for each of its members in a base game. See e.g. Finus and Rundshagen(2009) for more. Also see Bartl(2012).

52

Michael Finus, Pierre von Mouche, Bianca Rundshagen

of such games are not present in the literature. There it is just assumed that one deals with a situation where coalitional equilibria are unique or that one deals with a simple concrete example where uniqueness explicitly can be shown. Developing an abstract general uniqueness result is the main objective of this article. As shown in Section 3., a general equilibrium existence theorem guarantees for various common cases existence of coalitional equilibria. So existence is not a real issue, but equilibrium semi-uniqueness is. Especially as for coalitional equilibria one has to leave the comfortable usual setting of one dimensional action sets. Indeed: a coalition is formally treated as a meta-player whose action set is the Cartesian product of the action sets of the players in this coalition. In order to obtain our semi-uniqueness result for coalitional equilibria we develop a semi-uniqueness result for Nash equilibria of games in strategic form with higher dimensional action sets. This result, Theorem 1, can be considered as a variant of a result in Folmer and von Mouche(2004) to higher dimensions. It can handle various aggregative2 base-games with one-dimensional action sets. We identify a class of such games which contains Cournot and public good games and give with Corollary 3 a result that guarantees that for each possible coalition structure there exists a unique coalitional equilibrium. 2.

Coalitional equilibria

In this section, we fix the setting and notations and formally define the notion of coalitional equilibrium. 2.1.

Games in strategic form

A game in strategic form Γ is an ordered 3-tuple Γ = (I, (Xi )i∈I , (fi )i∈I ), where I is a non-empty finite set, every Xi is a non-empty set and every fi is a function Y fi : Xj → IR. j∈I

The elements of I are called players, Xi is called the action set of player i, the elements of Xi are called actions Q of player i, fi is called the payoff function of player i and the elements of j∈I Xj , being by I indexed families (xj )j∈I with xj ∈ Xj , are called action profiles. For i ∈ I, let ˆı := I \ {i}. Q (z) For i ∈ I and z = (zj )j∈ˆı ∈ j∈ˆı Xj , the conditional payoff function fi : Xi → IR is defined by (z)

fi (xi ) := fi (xi ; z);

here (xi ; z) is the by I indexed family with xi for the element with index Q i and zj for the element with index j 6= i. An action profile x = (xj )j∈I ∈ j∈I Xj is a 2

I.e. games where the payoff P function of each player i is a function of his own action xi and of a weighted sum l γl xl of all actions.

53

On Uniqueness of Coalitional Equilibria

(Nash) equilibrium if, for all i ∈ N , writing again x = (xi ; z), xi is a maximiser of (z) fi . We denote by E the set of equilibria of Γ . We need some further notations for the sequel. For C ⊆ I, let Y XC := Xj . j∈C

So an element ξ C of XC is a by C indexed family (ξC;i )i∈C with Q ξC;i ∈ Xi ; for i ∈ N , we identify X{i} with Xi . And an element of element of C∈C XC is a by C indexed family ξ = (ξ C )C∈C = ((ξC;i )i∈C )C∈C . 2.2.

Notion of coalitional equilibrium

Suppose given a game in strategic form Γ = (I, (Xi )i∈I , (fi )i∈I ). A coalition is a subset of I and a coalition structure of I is a partition of I, i.e. a set with as elements non-empty disjoint coalitions whose union is I. Given a coalition structure C, we denote for i ∈ I by Ci the unique element of C with i ∈ Ci Q Q and define the mapping J C : C∈C XC → j∈I Xj by J C ((ξ C )C∈C ) := (ξCj ;j )j∈I .

For a subset D of I the function fD : fD :=

Q

X

i∈D

C∈C

XC → IR is defined by

fi ◦ J C .

Having these notations, the next definition formalizes the intended notion of coalitional equilibrium (with base game Γ ) as outlined in section 1.. Definition 1. Given a game in strategic form Γ = (I, (Xi )i∈I , (fi )i∈I ) and a coalition structure C of I, the (with C associated ) game in strategic form ΓC is defined by ΓC := (C, (XC )C∈C , (fC )C∈C ). ⋄ A Nash equilibrium of ΓC also is called a coalitional equilibrium of Γ ; more precisely we speak of a C-equilibrium of Γ . We also will refer to the elements of C as metaplayers. The action sets XC of ΓC are typically more dimensional. Note that if C = {{1}, {2}, . . . {n}}, then ΓC = Γ and a C-equilibrium of Γ is nothing else than a Nash equilibrium of Γ . And if C = {I}, P a C-equilibrium is nothing else than a maximizer of the total payoff function i∈I fi .

54 3.

Michael Finus, Pierre von Mouche, Bianca Rundshagen Existence of coalitional equilibria

General equilibrium existence and semi-uniqueness results for games in strategic form have immediate counterparts regarding coalitional equilibria if they allow for higher-dimensional action sets. A powerful standard existence result in Tan et al.(1995)Tan, Yu, and Yuan leads to the following sufficient conditions for the game ΓC := (C, (XC )C∈C , (fC )C∈C ) to have a Nash equilibrium: I. each action set XC is a compact convex subset of a Hausdorff topological linear space; II. each payoff function fC is upper-semi-continuous; III. every fC is lower-semi-continuous in the variable related to XCˆ ; IV. every fC is quasi-concave in ξ C ∈ XC . It may be useful to note that if each function fi is quasi-concave in (its own action) xi , this does not necessarily imply that IV holds. Even assuming that each function fi is concave in each variable is not sufficient.3 A natural question is to ask for simple sufficient conditions such that for each coalition structure C a C-equilibrium exists. As can be easily verified by the above existence result, such conditions are for instance: each action set Xi is a segment of IR, each payoff function fi is continuous and concave. 4.

A higher dimensional equilibrium semi-uniqueness result

In this section we consider a game in Q strategic form Γ = (I, (Xi )i∈I , (fi )i∈I ) where each player i ∈ I has action set Xi = j∈Mi Xi;j with Mi a non-empty set and the Xi;j proper intervals of IR. Theorem 1. For i ∈ I let Ei := {ei | e ∈ E} and Ei;j := {ei;j | e ∈ E} (j ∈ Mi ). Suppose the following conditions I-III hold. I. The partial derivatives

∂fi ∂xi;j

of IR := IR ∪ {−∞, +∞}. II. There exist functions

(i ∈ I, j ∈ Mi ) exist at every e ∈ E as an element

Φi : Ei → IR (i ∈ I), Θi : {(Φl (el ))l∈I | e ∈ E} → IR (i ∈ I), and, with Ψi : E → IR (i ∈ I) defined by Ψi (e) := Θi ((Φl (el ))l∈I ), functions Ti;j : Ei;j × Φi (Ei ) × Ψi (E) → IR (i ∈ I, j ∈ Mi ), such that for all i ∈ I and j ∈ Mi ∂fi a. ∂x (e) = Ti;j (ei;j , Φi (ei ), Ψi (e)) (e ∈ E); i;j b. Ti;j is decreasing in each of its three variables, and strictly decreasing in the first or second. 3

It is worth noting that the sum of quasi-concave functions may fail to be quasi-concave and a function that is concave in each of its variables may fail to be concave.


55

III. a. For all i ∈ I: Φi and Θi are increasing.4 b. For all a, b ∈ E: Ψi (a) ≥ Ψi (b) (i ∈ I) or Ψi (b) ≥ Ψi (a) (i ∈ I). 1. For all a, b ∈ E: Ψi (a) = Ψi (b) (i ∈ I) and even Φi (ai ) = Φi (bi ) (i ∈ I). 2. If every Ti;j is strictly decreasing in the first variable, then #E ≤ 1. ⋄ Proof. 1. Suppose a, b ∈ E. Step 1: Ψi (a) ≥ Ψi (b) (i ∈ I) ⇒ Φi (ai ) ≤ Φi (bi ) (i ∈ I). Proof: by contradiction assume Ψi (a) ≥ Ψi (b) (i ∈ I) and for some m ∈ I Φm (am ) > Φm (bm ). With J the set of elements j ∈ Mm for which am;j is a left boundary point of Xm;j or bm;j is a right boundary point of Xm;j , we have am;j ≤ bm;j (j ∈ J). Now suppose j ∈ Mm \ J. Because a is an equilibrium and am;j is not a left boundary point of Xm;j , it follows by condition I that Dm;j fm (a) ≥ 0. And, by the same arguments, Dm;j fm (b) ≤ 0. So by condition IIa we have Tm;j (am;j , Φm (am ), Ψm (a)) ≥ 0 ≥ Tm;j (bm;j , Φm (bm ), Ψm (b)).

(1)

As Ψm (a) ≥ Ψm (b) and Φm (am ) > Φm (bm ), condition IIb implies Tm;j (am;j , Φm (am ), Ψm (a)) ≤ Tm;j (am;j , Φm (bm ), Ψm (b)),

(2)

with strict inequality if Tm;j is strictly decreasing in the second variable. (1) and (2) imply Tm;j (am;j , Φm (bm ), Ψm (b)) ≥ Tm;j (bm;j , Φm (bm ), Ψm (b)), with strict inequality if Tm;j is strictly decreasing in the second variable. As Tm;j is decreasing, and strictly decreasing in the first or second variable, it follows that am;j ≤ bm;j . Hence, we proved am;j ≤ bm;j (j ∈ Mm ), i.e. am ≤ bm . By condition IIIa this implies Φm (am ) ≤ Φm (bm ), a contradiction. Step 2: Ψi (a) ≥ Ψi (b) (i ∈ I) ⇒ Ψi (a) = Ψi (b) (i ∈ I) Proof: suppose Ψi (a) ≥ Ψi (b) (i ∈ I). By Step 1: Φi (ai ) ≤ Φi (bi ) (i ∈ I). This implies, as Θi is increasing, Ψi (a) ≤ Ψi (b). Thus Ψi (a) = Ψi (b). Step 3: Ψi (a) = Ψi (b) (i ∈ I). 4

Q Given a finite product r∈J Zr of subsets of IR the relation ≥ (and its dual ≤) on Q br (r ∈ J). And a function r∈J QZr is defined by: (ar )r∈J ≥ (br )r∈J means ar ≥ Q f : r∈J Zr → IR is called increasing if for all a, b ∈ r∈J Zr one has a ≥ b ⇒ f (a) ≥ f (b).

56


Proof: by condition IIb we have Ψi (a) ≥ Ψi (b) (i ∈ I) or Ψi (b) ≥ Ψi (a) (i ∈ I). Without loss of generality we may assume that Ψi (a) ≥ Ψi (b) (i ∈ I). Step 3 implies Ψi (a) = Ψi (b) (i ∈ I). Step 4: Φi (ai ) = Φi (bi ) (i ∈ I). Proof: by Step 3 we have Ψi (a) = Ψi (b) (i ∈ I). Now apply Step 1. 2. By contradiction suppose #E ≥ 2. Fix a, b ∈ E and i ∈ I and j ∈ Mi such that ai;j 6= bi;j . We may assume that ai;j > bi;j . By part 1, Ψi (a) = Ψi (b) =: yi and Φi (ai ) = Φi (bi ) =: wi . As a is an equilibrium and ai;j is not a left boundary point of Xi;j , it follows that Di;j fi (a) ≥ 0. And, by the same arguments, Di;j fi (b) ≤ 0. By condition IIa Ti;j (ai;j , wi , yi ) ≥ 0 ≥ Ti;j (bi;j , wi , yi ). As Ti;j is strictly decreasing in its first variable, this implies a contradiction.

⊔ ⊓

Taking, in Theorem 1, mi = 1 (i ∈ I) and Φi = Id leads to: Corollary 1. For i ∈ I let Ei := {ei | e ∈ E}. Sufficient for #E ≤ 1 is that the following conditions I-III hold. I. The partial derivatives II. There exist functions

∂fi ∂xi

(i ∈ I) exist at every e ∈ E as an element of IR. ϕi : E → IR (i ∈ I),

and functions

ti : Ei × ϕi (E) → IR (i ∈ I),

such that for all i ∈ I ∂fi a. ∂x (e) = ti (ei , ϕi (e)) (e ∈ E); i b. ti is decreasing in each of its two variables, and strictly decreasing in the first. III. a. Every ϕi is increasing. b. For all a, b ∈ E: ϕi (a) ≥ ϕi (b) (i ∈ I) or ϕi (b) ≥ ϕi (a) (i ∈ I). ⋄ And here is a more practical variant of Theorem 1: 5 Corollary P 2. For i ∈ I, let ti;j ≥ 0 (j ∈ Mi ), ri ≥ 0, si > 0, Wi := Yi := si k∈I rk Wk and define Φi : Xi → IR and Ψi : XI → IR by X X Φi (xi ) := ti;j xi;j , Ψi (x) := si rk Φk (xk ). j∈Mi

P

j∈Mi ti;j Xi;j ,

k∈I

Suppose the following conditions I, IIa and IIb hold. I. Each player i’s payoff function fi is partially differentiable with respect to each variable xi;j . II. There exist functions Ti;j : Xi;j × Wi × Yi → IR (i ∈ I, j ∈ Mi ), such that for all i ∈ I and j ∈ Mi 5

The sum in Wi and Yi is a Minkowski-sum.

57


∂fi a. ∂x (x) = Ti;j (xi;j , Φi (xi ), Ψi (x)) (x ∈ XI ); i;j b. Ti;j is decreasing in each of its three variables, and strictly decreasing in the first or second.

1. For all a, b ∈ E: Ψi (a) = Ψi (b) (i ∈ I) and even Φi (ai ) = Φi (bi ) (i ∈ I). 2. If every Ti;j is strictly decreasing in the first variable, then #E ≤ 1. ⋄ 5.

Uniqueness of coalitional equilibria for Cournot-like games

In the following definition a class of games in strategic form is introduced for which we provide sufficient conditions for uniqueness of coalitional equilibria. Definition 2. A Cournot-like game is a game in strategic form Γ = (N, (Ki )i∈N , (πi )i∈N ) where every Ki is a proper interval of IR with 0 ∈ Ki ⊆ IR+ and X πi (x) = ai (xi ) − xβi i bi ( γl xl ) l∈N

where, with Y := – – – –

ai : Ki → IR; βi ∈ {0, 1}; γi > 0; bi : Y → IR. ⋄

P

l∈N

γl K l ,

In case Ki is bounded, i.e. where Ki = [0, mi ] or Ki = [0, mi [ we say that player i has a capacity constraint. Note that some players may have a capacity constraint while others may not have. The class of Cournot-like games contains various heterogeneous Cournot oligopoly games: take every βi = 1. It contains6 all homogeneous Cournot oligopoly games: take in addition all bi equal and each γ = 1. It also contains various public good games: take every βi = 0. We call βl the type of player l. In the next theorem and proposition we consider a Cournot-like game Γ and fix a coalition structure C of N . We suppose that all players belonging to a same coalition C ∈ C are of the same type βC . Also we suppose for every C ∈ C that γl = γl′ (l, l′ ∈ C) and in case βC = 1 that bl , bl′ (l, l′ ∈ C). Theorem 2. Suppose that each function ai and bi is differentiable. Consider the with the coalition structure C associated game

For C ∈ C, let WC := IR (j ∈ C) by

P

ΓC = (C, (KC )C∈C , (πC )C∈C ). l∈C

γl Kl and define the functions TC;j : Kj × WC × Y →

TC;j (xj , w, y) := Daj (xj ) −

X w β C γj Dbi (y) − βC bj (y). #C · βC + (1 − βC ) i∈C

Suppose every TC;j is decreasing in each of its three variables and strictly decreasing in its first or second variable. 6

Disregarding Cournot oligopoly games with finite action sets.

58


1. For all Nash equilibria η, µ of ΓC one has XX XX ηC;i = µC;i C∈C i∈C

P

C∈C i∈C

P

and even i∈C ηC;i = i∈C µC;i (C ∈ C). 2. If every TC;j is strictly decreasing in its first variable, then ΓC has at most one Nash equilibrium. ⋄ Proof. Let γl =: γC (l ∈ C) and in case βC = 1, let bl =: bC (l ∈ C). Consider the game ΓC . The payoff function of player C ∈ C is X X X β πC (ξ) = (πi ◦ J C )(ξ) = ai (ξC;i ) − (ξC;i ) C bi ( γm ξCm ;m ) i∈C

i∈C

m∈N

X X X β = ai (ξC;i ) − (ξC;i ) C bi ( γm ξA;m ) i∈C

A∈C m∈A

X X X β = ai (ξC;i ) − (ξC;i ) C bi ( γA ξA;m ) . i∈C

If βC = 0, then πC (ξ) = fore for j ∈ C

P

i∈C

ai (ξC;i ) −

P

A∈C

m∈A

P P i∈C bi ( A∈C γA m∈A ξA;m ) and there-

X X X ∂πC (ξ) = Daj (ξC;j ) − γj Dbi ( γA ξA;m ). ∂ξC;j i∈C

A∈C

m∈A

If βC = 1, then

πC (ξ) =

X i∈C

and therefore for j ∈ C

X X X ai (ξC;i ) − ( ξC;i )bC ( γA ξA;m ) i∈C

A∈C

m∈A

X X X X X ∂πC (ξ) = Daj (ξC;j ) − bC ( γA ξA;m ) − γj ( ξC;i )DbC ( γA ξA;m ). ∂ξC;j A∈C

m∈A

i∈C

A∈C

m∈A

Noting that for the above functions TC;j one has

TC;j (xj , w, y) := Daj (xj ) − γj pC (w, y) − βC bj (y) where pC : WC × Y → IR is defined by P i∈C Dbi (y) if βC = 0, pC (w, y) = wDbC (y) if βC = 1, we obtain

X X X ∂πC (ξ) = TC;j (ξC;j , ξC;i , γA ξA;m ). ∂ξC;j i∈C

A∈C

m∈A

Having the above, we can apply Corollary 2 which implies the desired results. ⊓ ⊔ 7

7

Q Taking I = C, MC = C (C ∈ C), XC;j = Kj (C ∈ C, j ∈ C), XC = j∈MC XC;j = K tC;j = 1 (C ∈ C, j ∈ MC ), rC = γC (C ∈ C), sC = P 1 (C ∈ C), fC = PC (C ∈ C), C C (C ∈ C), TC;j = TC;j (C ∈ C, j ∈ MC ), ΦC (ξ C ) = i∈C πi ◦J = πP j∈MC ξC;j (C ∈ C) and ΨC (ξ) = D∈C γD ΦD (ξD ) (C ∈ C).

59


Remark: sufficient for every TC;j to be decreasing in each of its three variables and strictly decreasing in its first variable is that the following practical condition holds: every ai is strictly concave and every bi is increasing and convex. Proposition 1. Consider the with the coalition structure C associated game ΓC = (C, (KC )C∈C , (πC )C∈C ). Given C ∈ C, the following condition guarantees strict concavity of all conditional payoff functions of player C: the function aC : KC → IR given by X aC (ξ C ) := ai (ξC;i ) i∈C

is concave and P a. if βC = 0, then the function i∈C bi is strictly convex or aC is strictly concave; b. if βC = 1, then the function y 7→ ybC (y) is convex, and this function is strictly convex or aC is strictly concave. ⋄ (ξ ) Proof. With ξCˆ ∈ KCˆ , the conditional payoff function πC Cˆ : KC → IR reads

X X X (ξ ) β ξC;m + z), πC Cˆ (ξ C ) = ai (ξC;i ) + −(ξC;i ) C bi (γC i∈C

P

m∈C

i∈C

P

where z = A∈C\C γA m∈A ξA;m The first sum in this expression is by assumption a concave function. Case βC = 0: the second equals X X −bi (γC ξC;m + z) i∈C

m∈C

(ξ ) and also is concave. As the first or second sum is strictly concave, πC Cˆ is strictly concave. Case βC = 1: the second sum equals X X −bC (γC ξC;m + z) ξC;m m∈C

m∈C

(ξ ) and also is concave. As the first or second sum is strictly concave, πC Cˆ is strictly concave. ⊔ ⊓

The last paragraph in Section 2, the remark after Theorem 2 and Proposition 1 imply: Corollary 3. Let Γ be a Cournot-like game with compact action sets, βi = β (i ∈ N ), γi = γi′ (i, i′ ∈ N ) and β = 1 ⇒ bi = bi′ (i, i′ ∈ N ). Suppose each function ai is differentiable and strictly concave and each function bi is differentiable, increasing and convex. Then for every coalition structure C the game Γ has a unique C-equilibrium. ⋄

60


References Bartl, D. (2012). Application of cooperative game solution concepts to a collusive oligopoly game. In School of Business Administration in Karviná, editor, Proceedings of the 30th International Conference Mathematical Methods in Economics, pages 14–19. Bloch, F. (2003). Non–cooperative models of coalition formation in games with spillovers. In C. Carraro, editor, Endogenous Formation of Economic Coalitions, chapter 2, pages 35–79. Edward Elgar, Cheltenham. Finus, M. and B. Rundshagen (2009). Membership rules and stability of coalition structures in positive externality games. Social Choice and Welfare, 32(0), 389–406. Folmer, H. and P. H. M. von Mouche (2004). On a less known Nash equilibrium uniqueness result. Journal of Mathematical Sociology, 28(0), 67–80. Tan, K. J. Yu, and X. Yuan (1995). Existence theorems of Nash equilibria for noncooperative n-person games. International Journal of Game Theory, 24(0), 217–222. Yi, S. (1997). Stable coalition structures with externalities. Games and Economic Behavior, 20(0), 201–237.

Quality Level Choice Model under Oligopoly Competition on a Fitness Service Market Margarita A. Gladkova1 , Maria A. Kazantseva2 and Nikolay A. Zenkevich1 St.Petersburg State University, Graduate School of Management, Volkhovsky per. 2, St.Petersburg, 199004, Russia 2 Vienna University of Economics and Business, Augasse 2-6, 1090 Vienna, Austria E-mail: [email protected] [email protected] [email protected] 1

Abstract The growth of complexity of business conditions causes the necessity of innovative approaches to strategic decision-making, instruments and tools that help them to reach the leading position in mid-term and longterm perspective. One of the instruments that allow increasing company’s competitiveness is the improvement of the service quality. The goal of the research is to develop theoretical basis (models) and practical methods of the service quality level evaluation and choice which is made by the service provider. Research objectives are: analysis of consumer satisfaction with the service, development of game-theoretical models of service providers’ interaction, definition of the strategy of service quality level choice, development of practical recommendations for Russian companies to implement the strategy. Keywords: quality choice, willingness to pay, exponential distribution, twostage game, Nash equilibrium, optimal quality differentiation, fitness industry.

1.

Introduction

The growth of complexity of an external environment and business conditions, namely, high development of information and communication technologies and competition boost, predefines the identification of new sources of development of companies competitive abilities and ways to increase management effectiveness. This fact causes the necessity in development of innovative approaches to strategic decisionmaking, instruments and tools that may help companies to reach the leading position in mid-term and long-term perspective. One of the instruments that allow increasing company’s competitiveness is the improvement of the service quality. Contemporary approaches to company management are based on the analysis of the adding value framework. The value of the service for the consumer is highly defined by its quality. Therefore, the problem of quality level choice under competition is a very important element of strategic management. The appropriate choice of service quality level and price provides a company with necessary conditions to maintain high competitiveness and stable development. At the same time service quality level is defined by consumer satisfaction. The aim of this paper is to propose evaluated theoretical and applied methodology of quality management under

62

Margarita A. Gladkova, Maria A. Kazantseva, Nikolay A. Zenkevich

competition which is correlated with the study the consumer satisfaction and companies strategic interaction in the market. In this paper we will define quality from the point of view of customers of the investigated service. Customer involvement in the production of the services creates additional argument for the importance of quality evaluation from the customers point. In the paper quality of service is a quantitative estimation of the level of consumer satisfaction with this service. Assume as well that service quality is a complex notion defined through its characteristics. The characteristics of quality should be measurable, accurate and reliable. They are measured on the basis of customers’ opinion. Thus, in this paper the quality is evaluated as a cumulative value assigned for the service quality characteristics and is cumulative measure of consumer’s satisfaction with service quality. Therefore the higher is consumer satisfaction the higher is the service quality level. After consumer satisfaction with the service is analyzed and service quality is evaluated, the goal of the research is to develop theoretical basis (models) and practical methods of the service quality level evaluation and choice which is made by the service provider. Therefore, the theoretical research objectives are: – development of game-theoretical models of service providers’ interaction, – definition of the strategy of service quality level choice. The empirical part of the research to suggest practical recommendations for Russian companies to implement the "quality price" strategy that may increase companies payoff under competition. The survey was conducted in St. Petersburg in fitness industry. Consumer preferences and satisfaction were defined, game-theoretical analysis of St. Petersburg industrial market allowed finding the current state and equilibrium service quality levels. the change in market shares for Fitness clubs of St. Petersburg was evaluated in equilibrium. 2.

Game-Theoretical Model of Duopoly

Suppose that there are 2 firms on the market which produce homogeneous services differentiated by quality. Let firm i produces goods with the quality si , and let s1 < s2 . Assume, that the values of si are known to both firms and consumers. According to the model, firms use Bertrand price competition. In this case pi is a price of firm i for the goods with quality si . The game-theoretical model is presented as dynamic game which consists of the following stages: a) each firm i chooses its service quality levels si ; b) firms compete in prices pi . Consumers differ in their willingness to pay for quality level s, which is described by the parameter θ ∈ [0, ∞) . This parameter is called inclination to quality. The utility of a consumer with a willingness to pay for quality θ (consumer θ) when buying a service of quality s at a price p is equal to: θs − p, p 6 θs Uθ (p, s) = (1) 0, p > θs It is clear that the consumer θ will purchase the product of quality s at price p if Uθ (p, s) > 0 and won”t buy a product otherwise.

Quality Level Choice Model under Oligopoly Competition

63

The investigated industrial market is considered to be partially covered. The model suggests that inclination to quality is exponentially distributed. This means that the majority of consumers have the willingness to buy services with the critical level of quality. The case when consumers are eager to buy the lowest level of quality is considered, but it may be extended to the situation with the highest level of quality. Therefore, in the model it is assumed that the parameter of inclination to quality θ is a random variable and has exponential distribution with density function: f (x) =

0, x 6 0 λe−λx , x > 0

(2)

The payoff function of the firm i which provides a service of quality si , where si ∈ [s, s], is the following: Ri (p1 , p2 , s1 , s2 ) = pi (s1 , s2 ) × Di (p1 , p2 , s1 , s2 ), i = 1, 2 where pi (s1 , s2 ) is the price of the service of the firm i, Di (p1 , p2 , s1 , s2 ) – the demand function for the service of quality si , which is specified. Introduce the following variables: θ1 and θ2 . Consumer with inclination to quality θ is indifferent to the purchase of goods with the quality s1 and price p1 , if θs1 − p1 = 0

(3)

θ1 = θ1 (p1 , s1 ) = p1 /s1

(4)

Then we can find that:

θ1 characterizes a consumer, who is equally ready to buy a service with the quality s1 and price p1 or refuse to buy this service. Consumer with inclination to quality θ is indifferent to the purchase of services with quality s1 , s2 and prices p1 , p2 respectively, if: θs1 − p1 = θs2 − p2

(5)

Therefore, θ2 is equal: θ2 = θ2 (p1 , p2 , s1 , s2 ) =

p2 − p1 s2 − s1

(6)

θ2 characterizes a consumer, who is indifferent to buy a good with the quality s1 and price p1 and a good with the quality s2 and price p2 .

64


Then, demand function Di (p1 , p2 , s1 , s2 ) for firms 1 and 2 can be presented as following:  θ2 (p1 , pR2 , s1 , s2 )     D (p , p , s , s ) = f (θ)dθ = F (θ2 (p1 , p2 , s1 , s2 )) − F (θ1 (p1 , s1 ));  1 1 2 1 2  θ1 (p1 , s1 ) ∞  R   D (p , p , s , s ) = f (θ)dθ = 1 − F (θ2 (p1 , p2 , s1 , s2 )).  2 1 2 1 2   θ2 (p1 , p2 , s1 , s2 ) Then the payoffs of each three firms will be evaluated by the sales return function: R1 (p1 , p2 , s1 , s2 ) = p1 × D1 (p1 , p2 , s1 , s2 ) R2 (p1 , p2 , s1 , s2 ) = p2 × D2 (p1 , p2 , s1 , s2 , )

where pi (s1 , s2 ) is the price of the service of the firm i with quality si Game theoretical model of quality choice is a two stages model, when the choice on each stage is made simultaneously. – On the first stage firms i choose quality levels si ; – On the second stage firms i compete in prices pi . It is assumed, that after the first stage all quality levels are known to both companies and consumers. This game theoretical model should be solved using the backward induction method. It means that Nash equilibrium is fined in two steps. On the first step assuming that the quality levels are known we find prices p∗i (s1 , s2 ) for services offered by each firm. On the second step, when the prices p∗i (s1 , s2 ) are known we find quality levels s∗1 , s∗2 in Nash equilibrium for firms 1 and 2 correspondingly. Taking into account the exponential distribution of inclination to quality θ , we can rewrite payoff functions as following:  p2 − p1 p   R (p , p , s , s ) = p × (e−λθ1 − e−λθ2 ) = p × e−λ s11 − e−λ s2 − s1 1 1 2 1 2 1 1 p − p1  −λ 2  R2 (p1 , p2 , s1 , s2 ) = p2 × e−λθ2 = p2 × e s2 − s1 To find equilibrium prices, use the first order condition:  p − p1 p1  −λ 2   ∂R1 = e−λ s1 1 − λ p1 − e s2 − s1 1 + λ p1 s1 s2 − s1 = 0 ∂p1 p − p 1  −λ 2   ∂R2 = e s2 − s1 1 − λ p2 s2 − s1 = 0 ∂p2

The obtained system of equations has unique solution, which may be calculated numerically using MATLAB algorithm, where an optimal strategy of the second s2 − s1 company p∗2 = , and optimal strategy of the first company p∗1 are defined λ from the first equation of the system and are unique. Thus, for instance, if we have information that s1 = 100, s2 = 150, λ = 0, 15 then one can obtain the following equilibrium prices: p∗1 = 115, p∗2 = 333 . On the second stage of the analysis the companies compete in quality. As the solution on the first stage is obtained numerically, on this stage we will also use


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MATLAB to get the optimal quality strategies and numerical values for market shares and payoffs in equilibrium. First analyze how the payoff functions change with quality. It can be shown numerically that payoff function of second company increases with its quality for any fixed service quality level of the first company. Therefore, the optimal service quality level strategy of second company is the highest possible quality level: s∗2 = s.

Fig. 1: Companies payoff function with respect to second company service quality level.

Figure 1 presents the payoff values for both companies when company 2 service quality level is increasing.

Fig. 2: Company 1 payoff function with respect to its service quality level.

Next, determine the quality levels, where the maximum value function of the first company is achieved. For this purpose, knowing the quality level of second company, with a predetermined pitch changing the quality service level of the first company. For each pair of service qualities from the first-order conditions the prices are obtained in equilibrium. Figure 2 represents the change in payoff of the first company with respect to its service quality level (see Fig. 2). The graph shows that there exists a unique quality level of the first company where the company payoff achieves its maximum value. When quality levels are calculated, equilibrium price , demand and revenue can be found. Table 1 presents an example of an optimal numerical solution for a described game when the input parameters are: λ = 0, 15; s = 770. 3.

Game-Theoretical Model of Oligopoly

Suppose now that there are 4 firms on some industrial market. Similarly to the previous section, let firm i produce services of quality si , and lets s1 < s2 < s3 < s4 . Assume, that the values of si are known to all firms and consumers. According to the model, firms use Bertrand price competition. In this case pi is a price of firm i

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Table 1: Numerical solution in duopoly and exponential distribution of inclination to quality. s1

s2 p 1

p2

D1 D2 R1 R2

460 770 626 2067 0,32 0,50 199 1029

for the goods with quality si . The game-theoretical model is presented as dynamic game which consists of the following stages: a) each firm i chooses its service quality levels si ; b) firms compete in prices pi . Consumers differ in their willingness to pay for quality level s, which is described by the parameter θ ∈ [0, ∞) . This parameter is called inclination to quality. The utility of a consumer is defined as in the previous section. The investigated industrial market is again considered to be partially covered. Now again the model when inclination to quality is exponentially distributed is analyzed. This means that the majority of consumers have the willingness to buy services with the critical level of quality. The case when consumers are eager to buy the lowest level of quality is considered, but it may be extended to the situation with the highest level of quality. Suppose that there are 4 firms on the market which produce homogeneous services differentiated by quality. The payoff function of the firm i which provides a service of quality si , where si ∈ [s, s], is the following: Ri (p, s) = pi (s) × Di (p, s), i = 1, 4 where pi (s) = pi (s1 , s2 , s3 , s4 ) is the price of the service of the firm i, Di (p, s) = Di (p1 , p2 , p3 , p4 , s1 , s2 , s3 , s4 )âĹŠ the demand function for the service of quality si , which is specified. Introduce the following variables: θ1 , θ2 , θ3 and θ4 Consumer with inclination to quality θ is indifferent to the purchase of goods with the quality s1 and price p1 , if θs1 − p1 = 0 Then we can find that: θ1 = θ1 (p1 , s1 ) = p1 /s1 θ1 characterizes a consumer, who is equally ready to buy a service with the quality s1 and price p1 or refuse to buy this service. Consumer with inclination to quality θ is indifferent to the purchase of services with quality s1 , s2 and prices p1 , p2 respectively, if:

θs1 − p1 = θs2 − p2


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Therefore, θ2 is equal:

θ2 = θ2 (p1 , p2 , s1 , s2 ) =

p2 − p1 s2 − s1

θ2 characterizes a consumer, who is indifferent to buy a good with the quality s1 and price p1 and a good with the quality s2 and price p2 . Consumer with inclination to quality θ is indifferent to the purchase of goods with quality s2 , s3 and prices p2 , p3 respectively, if:

θs2 − p2 = θs3 − p3 Therefore, θ3 is equal:

θ3 = θ3 (p2 , p3 , s2 , s3 ) =

p3 − p2 s3 − s2

θ3 characterizes a consumer, who indifferent to buy a good with the quality s2 and price p2 and a good with the quality s3 and price p3 . Consumer with inclination to quality θ is indifferent to the purchase of goods with quality s3 , s4 and prices p3 , p4 respectively, if:

θs3 − p3 = θs4 − p4 Therefore, θ4 is equal:

θ4 = θ4 (p3 , p4 , s3 , s4 ) =

p4 − p3 s4 − s3

and characterizes a consumer, who indifferent to buy a good with the quality s3 and price p3 and a good with the quality s4 and price p4 . Then, demand function Di (p1 , p2 , p3 , p4 , s1 , s2 , s3 , s4 ) for firms 1, 2, 3 and 4 can be presented as following:

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 θ2 (p1 , pR2 , s1 , s2 )     D (p , p , s , s ) = f (θ)dθ = F (θ2 (p1 , p2 , s1 , s2 )) − F (θ1 (p1 , s1 ));  1 1 2 1 2    θ (p , s )  1 1 1    θ3 (p2 , pR3 , s2 , s3 )     D (p , p , s , s ) = f (θ)dθ = F (θ3 (p2 , p3 , s2 , s3 ))  2 1 2 1 2    θ (p , p , s , s )  2 1 2 1 2  −F (θ2 (p1 , p2 , s1 , s2 ));  θ4 (p3 , pR4 , s3 , s4 )     D (p , p , p , s , s , s ) = f (θ)dθ = F (θ4 (p3 , p4 , s3 , s4 ))  3 1 2 3 1 2 3    θ (p , p , s , s )  3 2 3 2 3    −F (θ3 (p2 , p3 , s2 , s3 ));    ∞  R   f (θ)dθ = 1 − F (θ4 (p3 , p4 , s3 , s4 )). D4 (p3 , p4 , s3 , s4 ) =    θ4 (p3 , p4 , s3 , s4 ) Then the payoffs of each firm will be evaluated by the sales return function:  R1 (p1 , p2 , s1 , s2 ) = p1 × D1 (p1 , p2 , s1 , s2 )    R2 (p1 , p2 , p3 , s1 , s2 , s3 ) = p2 × D2 (p1 , p2 , p3 , s1 , s2 , s3 ) R  3 (p2 , p3 , p4 , s2 , s3 , p4 ) = p3 × D3 (p2 , p3 , p4 , s2 , s3 , p4 )   R4 (p3 , p4 , s3 , p4 ) = p4 × D4 (p3 , p4 , s3 , p4 )

where pi (s) is the price of the service of the firm i. Game theoretical model of quality choice is a two stages model, when the choice on each stage is done simultaneously. – On the first stage firms i choose quality levels si ; – On the second stage firms i compete in prices pi . It is assumed, that after the first stage all quality levels are known to both companies and consumers. The choice on the first stage is made subsequently and on the second stage simultaneously. This game theoretical model should be solved using the backward induction method. It means that Nash equilibrium is fined in two steps. On the first step assuming that the quality levels are known we find prices p∗i (s) for services offered by each firm. On the second step, when the prices p∗i (s) are known we find quality levels s∗1 , s∗2 , s∗3 , s∗4 in Nash equilibrium for firms 1, 2, 3, 4 correspondingly. To solve the problem MATLAB algorithm similar to the one described in Section 2 is used. Here the Service quality is chosen in 4 subsequent steps. 4.

Quality Estimation

In this research the quality is managed using game-theoretical approach which leads to the problem of quality measurement. The quality is observed from customer point of view. We introduce integrated service quality which means the composite index of consumer satisfaction with the service. The quality may have any value from the unit interval [0,1]. In situation when a customer is totally satisfied with received service, the quality of service is equal to one. Service is represented as a set of characteristics, which should be measurable, precise and reliable. If characteristics are measurable, it is possible to predict them,


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choose, plan, control and therefore manage. Only in this case the total quality can be objectively calculated and can be used to provide managerial recommendations. In order to calculate service quality in the current state the results of questionnaire the program ASPID 3W by Hovanov (2004) is used. ASPID 3W is based on the method of summary measures. This method is universal and can be used both for product and service quality evaluation. The main idea of this method is to summarize all assessments of one complicated object into one united estimate, which will characterize the quality of this object. The method can be applied to any multivariate object: complicated technical systems, different versions of managerial, organizational and investment decisions, consumers’ goods and services, etc. The main steps of quality calculation using ASPID 3W are: 1. All initial characteristics are summarized in vector x = (x1 , . . . , xm ). Each of these characteristics is essential for quality calculation, but they became useful only after summarizing in one united indicator. 2. After that, vector q = (q1 , . . . , qm ) is formed form individual indicators, representing the function qi = q(xi ; i), qi = q(xi ; i) corresponding to the initial characteristics and evaluating the tested object using m different criteria. 3. The type of synthesized function Q(q) is chosen which is corresponded with vector q = (q1 , . . . , qm ). Function Q(q) is depended on vector w = (w1 , . . . , wm ) of non-negative parameters which determine relevance of independent indicators for aggregated estimation: Q = Q(q) = Q(q; w). 4. The meaning of parameters w = (w1 , . . . , wm ) is determined. These parameters are interpreted as the weights which show the influence of independent characteristics q1 , . . . , qm on Q. Assume that w1 + · · · + wm = 1 . To sum up, the quality of services offered by each mobile operator is calculated as weighted sum of all characteristics of services (coverage area, speed and quality of data communication, quality of voice transmission, availability of mobile services offices and payment points, number of additional services, availability of tariffs and their diversity, technical support) multiplied on average price for this service. Weights are calculated using the results of the survey and are based on customers’ satisfaction. 5.

Experimental Section

5.1. Fitness Industry in St. Petersburg The main aim of the empirical study is to test theoretical models for some industrial market. To do that first fitness industry in St. Petersburg, Russia is analyzed and we find out the quality levels of services offered by the companies in this market. For this purpose the questionnaire is used. The main research tool is questionnaire and it was conducted in St. Petersburg. St. Petersburg and Moscow - two capitals of fitness industry (Moscow accounts more than 53 % Russia’s national turnover, St. Petersburg - 17 %). In the end of 2008 (pre-crisis period) the center "Evolution - Sports Consulting" estimated that in St. Petersburg, there were 377 fitness clubs, but in the crisis period their number decreased dramatically (Fitness market in Russia. The results of 2010 and forecast for 2014. Analysis of price dynamics). At the moment the market of fitness services in St. Petersburg has about 350 fitness clubs. According to the Fitness Faculty Company, in St. Petersburg since

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the crisis the number of fitness units increased by 20 %, while only in 2012 this figure raised by another 40 %. In general, the potential growth of the market in the North-West region, according to the forecast of "TOP Fitness" is estimated in to be approximately 25-35 %. At the moment, the St. Petersburg fitness industry is represented by chain and non-chain clubs, among them a leader in the market are chain players. It should be noted that this market is fairly strong with respect to local chains and clubs, which consequently reduces the number of representatives of Moscow (and thus the Russian leaders) presented in St. Petersburg. For example, the leaders of the market share in Russia - Russian Fitness Group (World Class clubs and FizKult) and Strata Partners (Orange Fitness and CityFitness) have, respectively, 5 ( 3 and 2 ) and 1 ( 0 and 1) clubs on the market. In addition, most of Moscow chains significantly weakened during the crisis. Here is the list of the main leaders of the market, i.e. main chain clubs, which leadership is determined by the number of clubs on the market of St. Petersburg: – – – – – –

FITNESS HOUSE SPORT LIFE ALEX Fitness Fitness Planet OLYMP Extra Sport

However, given the territoriality of competition, it must be mentioned that most chain clubs are widespread may lose to single clubs or chains with fewer clubs. Among the well-known medium-sized but successful regional players are: – – – – –

FIT FASHION TAURAS FITNESS WORLD CLASS The Flying Dutchman Neptune and others.

5.2.

Service Quality Levels Evaluation

In order to evaluate current fitness club service quality levels the questionnaire was used as a main source of information. Dr. Harrington highlights in his works the essential role of questionnaires in quality evaluation (Harrington 1991) as they help to estimate the level of customers’ satisfaction with the offered quality of services. The questionnaire was developed by authors and based on the SERVPERF approach to service quality evaluation. In the paper we suppose that the fitness service has five characteristics which influence the satisfaction of consumers and their choice of the fitness club: – – – – –

Fitness club image, Gym, Rooms for group training, Timetable and variety of classes, Administration.


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Table 2: Quality characteristics: club image component. Code M 1.1 M 1.2 M 1.3 M 1.4 C1.1 H1.2 U 1.1 H1.2 M 1.4

Description Nice and comfortable location Comfortable parking space Pool is big and clean Interior is comfortable and pleasant Useful and informative web site High quality (professional) reputation Atmosphere of trust and understanding between clients and club workers The promises on service quality were fulfilled Variety of additional services (fitness-bar, medical service, etc.)

Each component is evaluated according to five quality dimensions of SERVPERF approach: tangibles, reliability, responsiveness, willingness to help customers and provide prompt service, assurance, empathy. 1. Club image or general characteristic of the club. This component describes the provider of the service, its location, reputation, reliability and other. 2. Gym and rooms for group training are described with the characteristics listed in the Table 3 and evaluated by the consumers only if they attend the gym and rooms for group training correspondingly. Table 3: Quality characteristics: gym and rooms for group training component. Code Gym Group rooms M 2.1 M 3.1 H2.1 H3.1 U 2.1 U 3.1 O2.1 O3.1 M 2.2 M 3.2 C2.2 C3.2

Description Modern equipment is used Safety level is good enough Personnel is professional and has high competences Personnel is attentive to clients’ interests Personnel is good-looking Personnel has individual approach to each client during the training class

3. Timetable and variety of classes component may significantly influence the decision of the client to attend group training. That’s why this component is considered separately. 4. Administration includes administrators of the club and sales departments employee. Thus, the questionnaire comprehensively evaluates the perceived service quality by identifying and assessing the importance of each component of the service from the point of view of the client, and the level of satisfaction and importance of each quality characteristics. To check the questionnaire on the criteria of clarity and accuracy the pilot test was conducted on the 15 respondents. Testing was made in the form of personal

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Margarita A. Gladkova, Maria A. Kazantseva, Nikolay A. Zenkevich Table 4: Quality characteristics: timetable and variety of classes. Code Description M 3.1 Group trainings are interesting and different C3.1 Timetable is appropriate C3.2 Timetable is prepared with respect to clientsâĂŹ demands and desires O3.1 Demands on timetable changes are considered quickly

Table 5: Quality characteristics: administration component. Code M 4.1 U 4.1 C4.1 O4.1 H4.1

Description Personnel is good-looking Personnel is polite Personnel has individual approach to each client problem Personnel solves clients’ problems quickly Clients are informed quickly and on time

interviews, which made it possible to compile a complete set of requiring corrections or clarifications of the final questionnaire. Sample description. The clients of the following four clubs participated in the interview: 1. 2. 3. 4.

OLYMP FITNES FAMILY FITNESS HOUSE SPORT-LIFE

120 customers of the clubs participated in the survey (30 people from each club). The sample is uniform in the sense that there were almost equal number of men (40 %) and women (60 % ), family (42 %) and single (58 %) customers. This allows us to understand better the difference among client groups , as well as to carry out a comparative analysis. It was also found that the biggest age group of customers of fitness services in St. Petersburg are the customers in the age of 20-29 years (42 %) and 30-39 years (34 %) with middle -income (54 %) and above average (37 %). It is interesting to notice that most of the customers of fitness services are occupied in the top management (24 %). Based on the structure of the questionnaire during an interview respondents rated the quality of fitness services as a set of certain service characteristics. Therefore we present an assessment of service quality as a generalized quantitative characteristic, which is the aggregate indicator of quality, and thus determines the quality of fitness services in general as an integral quality (Gladkova, Zenkevich, Sorokina, 2011). In order to calculate the service quality we used ASPID-3W (Hovanov, 1996), and conducted quality evaluation process, as follows: 1. The evaluation of composite indicators of perceived quality for each of the components of the four clubs: the image of the club, the gym, group training classes, the variety of classes and timetable, and administrative staff;


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2. The calculation of integral measures of perceived quality of the components for each of the four clubs 3. The calculation of the composite indicator of the perceived service quality for each club : α1 -OLYMP, α2 -FITNESS FAMILY, α3 -FITNESS HOUSE, α4 SPORT LIFE, where αi : 0 < αi 6 1 (respectively, αi = 1 characterizes the quality perceived by the consumer as the maximum and gives him the most satisfaction). The weights for processing ASPID-3W are the averages of the importance of each characteristic (for step 1) and the average of the importance of components (for step 2 ). In the Table 6 the average importance of each characteristic and component of the service are presented. As the prices for the fitness services in each club, we take the standard figures listed in the price page on the websites and calculate the average for each club (see Table 7). As a result of the procedures described above were obtained the evaluation of integrated fitness service in each club (see Table 8). 5.3. Service Quality Level Choice under Competition It is assumed that each customer is characterized by inclination to quality (Gladkova, Zenkevich, Sorokina, 2011) , which can be calculated from the information that we got from the questionnaire. For this purpose the respondents were asked about the maximum prices they are ready to pay for the fitness service. In order to apply game-theoretic modeling to determine optimal strategies for service quality for competing clubs on the investigated geographic market, it is necessary to formulate and test hypotheses about the distribution of customers of fitness clubs by the inclination to quality. Using the Kolmogorov-Smirnov test the hypothesis on the exponential distribution was tested and accepted at the level of significance 0.05. Also from the price information the respondents are ready to pay for the services we found that the distribution parameter equal to the reciprocal of the sample mean is equal to 0.0666. As a result, it was found that the characteristic penchant for quality has an exponential distribution. In the case of the exponential distribution of inclination to quality the game-theoretic model presented in the Section 2 can be applied to the of oligopoly competition of fitness clubs on the investigated geographic market. This model is a development of a game-theoretical model of duopoly in a vertical differentiation J. Tirole (Tirole, 1988) , which is applicable in the case of a uniform distribution of inclination to quality. In addition to the basic model of J. Tirole, there are also various modifications and improvements of the model for different cases. The solution of the model is realized in MATLAB as follows: 1. Arrange all the players according to their current quality levels (see Table 8)? Where the first player is the one with highest current integrated quality level. • OLYMP - current quality = 0,639, • FITNESS FAMILY - current quality = 0,597, • FITNESS HOUSE - current quality = 0,583, • SPORT LIFE - current quality = 0,614. 2. Using the value of the distribution parameter = 0.0666, we calculate the equilibrium quality strategies of the players, while setting the desired quality for

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Margarita A. Gladkova, Maria A. Kazantseva, Nikolay A. Zenkevich Table 6: Average importance of each characteristic and component of the service. Characteristic Name Nice and comfortable location Comfortable parking space Pool is big and clean Interior is comfortable and pleasant Useful and informative web site High quality (professional) reputation Atmosphere of trust and understanding The promises on service quality were fulfilled Variety of additional services (fitnessbar, medical service, etc.) Modern equipment is used in the gym Safety level in the gym is good enough Group training personnel is professional and has high competences Group training personnel is attentive to clients’ interests Group training personnel is goodlooking Group training personnel has individual approach Group trainings are interesting and different Timetable is appropriate Timetable is prepared with respect to clients’ demands and desires Demands on timetable changes are considered quickly Administrative personnel is goodlooking Administrative personnel is polite Administrative personnel has individual approach Administrative personnel solves clients’ problems quickly Clients are informed quickly and on time

Importance Group Name 6,53 5,40 6,25 5,75 5,00 5,92 6,36 6,32

Club Image

Group Importance 2,929

Gym

3,541

Timetable

3,035

Administration

2,518

6,08 5,96 5,58 4,95 4,79 4,48 4,84 5,23 5,40 5,11 4,93 6,24 6,49 6,22 6,38 60,34

the leading player 0.9. Note that according to the model the quality is established (achieved) by the players in the order selected on the first step. Then, the equilibrium service quality strategies are as follows: • • • •

OLYMP - equilibrium quality = 0,9, FITNESS FAMILY - equilibrium quality = 0,77, FITNESS HOUSE - equilibrium quality = 0,71, SPORT LIFE - equilibrium quality = 0,66.


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Table 7: Prices of fitness services in each club, rub. Club Service price, RUB OLYMP 20 638 FITNESS FAMILY 16 453 FITNESS HOUSE 16 830 SPORT LIFE 14 500

Table 8: Integrated fitness service quality. Club Integrated quality level αi OLYMP 0,639 FITNESS FAMILY 0,597 FITNESS HOUSE 0,583 SPORT LIFE 0,614

3. Further, in accordance with the equilibrium quality players simultaneously set prices for their services. Prices are set by the players in accordance with the laws of competition: the player with the highest quality has the right to charge a higher price, then the player with the lowest level of quality in setting a high price only to lose their customers. Table 9: Modeling results. Club

Integrated Average Equilibrium Equilibrium quality current price price quality OLYMP 0,639 20,64 22,64 0,900 FITNESS FAMILY 0,597 17,09 17,94 0,758 FITNESS HOUSE 0,583 16,83 16,33 0,684 SPORT LIFE 0,614 14,50 15,09 0,764

The equilibrium price and the equilibrium service qualities are the optimal strategies for the competition on the considered geographic market. 4. The use of optimal price and quality strategies will lead the companies to the following market shares: • OLYMP - 43 %, • FITNESS FAMILY - 34 %, • FITNESS HOUSE - 18 %, • SPORT LIFE - 5 %. It should be noted that in this model we assume that the players are rational and tend to increase (or maintain) its own market share and increase revenue (or maintain the level of revenue). 6.

Results and Discussion

First of all, it is interesting to investigate the market shares of the fitness clubs on the geographic market. Exploring the fitness clubs, it was found that the behavior

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Fig. 3: Respondents distribution by inclination to quality.

of these clubs on the regional market can be described by means of four competitive roles (Kotler et al , 2007, 489–521). The leader of this market is the territorial club OLYMP, which appeared earlier than the others in the market, has a large customer base and is often a leader in the introduction of price changes, the presentation of new programs, sporting goods and other services in their package. FITNESS FAMILY is a contender for the leadership without any doubts, who uses violent methods to fight for market share. The follower is the club FITNESS HOUSE, who used soft policy to retain its market share. Finally, SPORT LIFE which is characterized by the limited services provided for customers (SPORT LIFE has no pool, also has a small area, only one hall for group training and gym), focuses on a niche segment, attracting students and low-income clients to the club (a student club cards). However, despite the various competitive roles, the major players except SPORT LIFE have equal shares of presence in the market, which is a consequence of undifferentiated quality and exceptional price competition, which is wide spread among fitness. SPORT LIFE retains share of 10 % due to low capacity. According to the modeling results, players will need to improve the services quality following the leader, as well as to amend the prices (in accordance with the objective maximization of the revenue). This will lead to the change in their future market shares. It should be noted that the redistribution of shares will occur only as a result of specific programs aimed to improve the service quality of leader club, which should allow differentiating the leader from other players without price differentiation. Differentiation in quality is a major problem to retain players’ market shares. It can be seen that despite the decline in the shares, OLYMP’s competitors still have to improve the quality of services at little change in prices. For example, FITNESS HOUSE club in order to retain its share of 17 % will have to improve the quality by 20 % (according to the evaluation of the integrated quality), while reducing the cost. An important factor in the evaluation of the results is the ability to set the optimal strategy of the player. Given that this club does not have the information obtained by us during this simulation, we can assume that the strategy will not be implemented, or will not be held lower prices, i.e. strategy will be implemented partially. In connection with this, the FITNESS HOUSE club’s market share and revenue will be even smaller, and the rested share and capital will be distributed among the other competitors. According to the modeling results it was shown that almost without changing the quality and prices (changes to 7 % and 4 %, respectively) SPORT LIFE club


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is able to secure a share of 4 % to settle for a niche market. Note that the player SPORT LIFE in comparison to its competitors is fairly weak player and supports its existence only on thee niche market (in practice it acts in a democratic segment). The share of player SPORT LIFE is rather dependent on the policy of FITNESS HOUSE. If FITNESS HOUSE doesn’t hold its share the customers can switch to the SPORT LIFE club and this will increase the competitiveness of the latter. However, if the implementation of the new policy is successful and if the quality reached the optimal level, the player FITNESS HOUSE can even win the share of SPORT LIFE (take its niche consumers the students by offering student card at a comparable price). This may raise a question of the existence of the fourth player. As a result of strong competition and service quality improvement, the very existence of this chain on the market is under consideration with the club gets under question. The market share of SPORT LIFE reduced almost by half, which is quite possible, and this will make it unattractive for business owners who may decide to close this club or remodel it entirely. When SPORT LIFE leaves the market its market share will be distributed among the remaining players. Player 3 and 4 are similar enough in the sense of their possible ways of development. As a result of OLYMP’s actions, their market shares will be reduced significantly despite the improvement of the service quality (the fact that the share will grow with the clients of players 3 and 4 should be considered when developing a marketing strategy). The vast number of customers of these clubs may not decrease at all or just slightly due to the development of the fitness industry and the appearance of new customers. Consequently, in the long term these clubs will lose its positions on the market and either leave the market or make a total renovation. Finally, the main competitor, according to the staff of OLYMP, is FITNESS FAMILY club (quite big club with a low fullness level). It has considerable potential to increase its market share. The market share value of this club in thee equilibrium, which was obtained in Section 3, restrained only by a successful and increasing differentiation of the club OLYMP. FITNESS FAMILY will increase its share from the "mediocre" position of the above players 3 and 4. Its quality should to be improved by 29 % when the price only by 5 %, but given the "weak position" in quality, which are the characteristics of the personnel (gym and group training) as well as the variety of proposed activities, it can be said that these shortcomings in the service typical enough for a new player in the fitness service market. As practice shows, the cubs quickly improve these disadvantages and may become a leader of the investigated market. The situation for OLYMP club is rather complicated: it is necessary to improve the service quality by 40 % , raising the price by 10 % only. It is understandable that this strategy is aimed not only to the increase of market share but to hold the aggressive competitor FITNESS FAMILY. OLYMP must devote all its efforts to achieve the optimal level of service quality to avoid the unfavorable scenario. 7.

Conclusion

Therefore as a result of the research we developed the best strategies for each player of the investigated territorial market. On the basis of competitive analysis we evaluated the reality of these strategies implementation for each player. In addition, we also confirmed the competitiveness and potential of leadership of the club FITNESS FAMILY and its "danger" as a competitor for the club OLYMP. Therefore,

78

Margarita A. Gladkova, Maria A. Kazantseva, Nikolay A. Zenkevich Table 10: Strategic quality and price changes for equilibrium achievement. Change of integrated quality Club Value % OLYMP 0,26 40,85 % FITNESS FAMILY 0,17 28,64 % FITNESS HOUSE 0,13 21,44 % SPORT LIFE 0,04 6,84 %

Change in price strategy Value % 1,997 9,68 % 0,8523 4,99 % -0,4978 -2,96 % 0,5944 4,10 %

we reiterate the need to improve the service quality of the club OLYMP to preserve leadership on this market. References Abbott, L. (1955). Quality and competition. New York: Columbia University Press. Aoki, R. (1994). Sequential vs. simultaneous quality choices with Bertrand. Mimeo (State University of New York), pp. 377–397. Aoki, R., Pursa, T. J. (1996). Sequential versus simultaneous choice with endogenous quality. International Journal of Industrial Organization, 15, 103–121. Balkyte, A., Tvaronaviciene, Ì. (2010). Perception of competitiveness in the context of sustainable development: facets of "sustainable competitiveness". Journal of Business Economics and Management, 11 (2), 341–365. Barnes, B. R., Sheys, T., Morris, D. (2006). Analysing Service Quality: The Case of a US Military Club. BTotal Quality Management, 16(8-9), 955–967. Benassi, C., Chirco, A., Colombo, C. (2006). Vertical differentiation and distribution of income. Bulletin of Economic Research, 58(4), 307–378. Chelladurai, P., Fiona, L. Scott, J. (1987). Haywood-Farmer Dimensions of Fitness Services:Development of a Model.. Journal of Sport Management, 1, 159–172. Chen, Y.-T., Chou, T.-Y.. (2011). Applying GRA and QFD to Improve Library Service Quality.. The Journal of Academic Librarianship, 37(3), 237–245. Chen L.-S., C.-H. Liu, C.-C. Hsu, and C.-S. Lin. (2010). C-Kano model: a novel approach for discovering attractive quality elements. Total Quality Management, 1189–1214. Cronin J. and Taylor S. (1992). Measuring service quality: reexamination and extension. Journal of marketing, pp. 55–68. Donnenfeld, S. and Weber, S. (1992). Vertical product differentiation with entry. International Journal of Industrial Organization, 10, 449–472. Edmonds, T. (2000). Regional competitiveness and the role of the knowledge economy, house of commons library. Research paper. London: Research Publications Office, pp. 55–73. Feurer, R., Chaharbaghi, K. (1994). Defining Competitiveness: a Holistic Approach. Management Decision, 32(2), 49–58. Frascatore, M. (2002). On vertical differentiation under Bertrand and Cournot: When input quality and upward sloping supply. The B.E. Journals in Economic Analysis and Policy (Berkeley Electronic Press), 2(1), 1–17. Gayer, A. (2007). Oligopoly, endogenous monopolist and product quality. The B.E. journal of theoretical economics, 7(1). Gladkova, M. A., Zenkevich, N. A. (2007).Game-theoretical model "quality-price" under competition on the industry market. Vestnik of Saint-Petersburg University. Management Series, 4, 3–31 (in Russian). Gladkova, M., Zenkevich, N. (2009). Quality Competition: Uniform vs. Non-uniform Consumer Distribution.Contributions to Game Theory and Management. Vol II. Collected


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papers presented on the Second International Conference "Game Theory and Management"/ Editors Leon A. Petrosjan, Nikolay A. Zenkevich - SPb, Graduate School of Management, SPbU, pp. 111–124. Gold’s Gym Celebrates Decade in Russia (2007) . Fitness Business Pro, 23(2). Gronroos, C. (2006). Service management and marketing: a customer relationship management approach. Wiley: Chichester, UK. Hovanov, K. N., Hovanov, N. V. (1996).DSSS "ASPID-3W". Decision Support System’s Shell "Analysis and Synthesis of Parameters under Information Deficiency - 3rd modification for Windows". "ASPID-3W" registered by Federal Agency for Computer Programs Copyright Protection (Russian Federation, Num.960087). Hovanov, N., Yudaeva, M., Hovanov, K. (2009). Multicriteria estimation of probabilities on basis of expert non-numeric, non-exact and non-complete knowledge. European Journal of Operational Research, 195, 857–863. Hoyle, D. (2003). ISO 9000:2000 an A-Z guide. Oxford: Butterworth-Heinemann An imprint of Elsevier Science. (2011). Industry Profile. Fitness Equipment. Johnson, G., Scholes, K. (2002). Exploring Corporate Strategy. 6th ed. London: Prentice Hall. Kadocsa, G. (2006). Research of Competitiveness Factors of SME. . Acta Polytechnica Hungarica, 3(4), 71–84. Kano, N., Seraku, N., Takahashi, F. and Tsuji, S. (1984). Attractive quality and must be quality. Journal of Japanese Society for Quality Control, 39–48. Kano, N. (2001). Life cycle and creation of attractive quality. Proceedings of the 4th QMOD Conference, 18–36. Lambertini, L. and Tedeschi, P. (2007). Would you like to enter first with low-quality good?. Bulletin of Economic Research (Blackwell Publishing), 59(3) , 269–282. Lovelock, C., Wirtz, J. (2007). Services Marketing: People, Technology, Strategy. 6th ed. Pearson Prentice Hall. Lutz, S. (1997). Vertical product differentiation and entry deterrence. Journal of Economics, 65(1), 79–102. Mohammad, J., Moshref, H., Razieh, G. (2011). Service Quality Evaluation in General Department of Health Insurance of Fars Province Using a SERVQUAL Model (Case Study: Shiraz). Interdisciplinary Journal of Contemporary Research in Business, 3(4), 118–125. Motta, M. (1993). Endogenous quality choice: price vs. Quantity competition. The journal of industrial economics, XLI(2), 113–131. Noh, Y.-H., Moschini, G. (2006). Vertical product differentiation, entry-deter-rence strategies, and entry qualities. Review of Industrial Organization, 29, 227–252. Pakdil, F., AydÄśn, O. (2007). Expectations and perceptions in airline services: An analysis using weighted SERVQUAL scores Original Research Article.. Journal of Air Transport Management, 13(4), 229–237. Parasuraman, A., Berry, L. and Zeithaml, V. (1985). Vertical product differentiation, entrydeter-rence strategies, and entry qualities. Review of Industrial Organization, 29, 227– 252. Parasuraman, A., Berry, L. and Zeithaml, V. (1985). Quality counts in services, too. Business horizon, 44–53. Parasuraman, A., Berry, L. and Zeithaml, V. (1988). SERVQUAL: a multiple-item scale for measuring consumer perceptions of service quality . Journal of retailing, 12–40. Petrosyan, L. A., Zenkevich, N. A., Semina, E. A. (1998).Game theory: University textbook.Vyssh, shkola. Knizhniy dom "Universitet". Moscow. (in Russian). Pettinger, R. (2004). Contemporary strategic management. Tasingstoke, Hampshire: Palgrave Macmillan. Simchi-Levi, D., Kaminsky, P., Simchi-Levi, E. (2004). Managing the Supply Chain.. New York: McGraw-Hill.

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A Problem of Purpose Resource Use in Two-Level Control Systems ⋆ Olga I. Gorbaneva1 and Guennady A. Ougolnitsky2 1 Southern Federal University, Faculty of Mathematics, Mechanics, and Computer Sciences, Milchakova St. 8A, Rostov-on-Don, 344090, Russia E-mail: [email protected] 2 E-mail: [email protected]

Abstract The system including two level players–top and bottom–is considered in the paper. Each of the players have public (purpose) and private (non-purpose) interests. Both players take part of payoff from purpose resource use. The model of resource allocation among the purpose and nonpurpose using is investigated for different payoff function classes and for three public gain distribution types. A problem is presented in the form of hierarchical game where the Stackelberg equilibrium is found. Keywords: resource allocation, two-level control system, purpose use, nonpurpose resource use, Stackelberg equilibrium.

1.

Introduction

A wide set of social and economic development problems is solved due to budget financing, which is performed in different forms (grants, subventions, assignments, credits) and always has a strictly purpose character, i.e., allocated funds should be spent only on prescribed needs. Article 289 of the Budget Code of the Russian Federation and Article 15.14 RF Code on Administrative Offences make provisions on responsibility for non-purpose use of budget funds. Nevertheless, non-purpose use of budget financing is widespread and can be considered as a kind of opportunistic behavior corresponding to the private interests of the active agents (Willamson, 1981). Non-purpose use of resources is linked to corruption, especially to “kickbacks”, when budget funds are allocated to an agent in exchange for a bribe and only partially used appropriately. They are largely spent on private agent-briber interests. It is naturally for the resource use problem to be treated in terms of the interest concordance in hierarchical control systems. This allows for a mathematical apparatus of hierarchical game theory (Basar, 1999), of contract theory (Laffont, 2002), information theory of hierarchical systems (Gorelik, 1991), active system theory (Novikov, 2013a) and organizational system theory (Novikov, 2013b). Simultaneously, resource allocation models in hierarchical systems with regard to their misuse are little studied (Germeyer, 1974) and are analyzed in authors’ investigation line (Gorbaneva and Ougolnitsky, 2009-2013). This article is focused on the question how resource allocation among purpose and non-purpose directions is depended on different public and private payoff function classes of distributor and resource recipients. ⋆

This work was supported by the the Russian Foundation for Basic Research, project # 12-01-00017

82 2.

Olga I. Gorbaneva, Guennady A. Ougolnitsky Structure of investigation

We consider a two-level control system which consists of one top level element A1 (resource distributor) and one bottom level element A2 (resource recipient). The top level has some resource amount (which we assume to be a unit). The distributor assigns a part of resources to the recipient for purpose use, and the rest for his own interests. The bottom level assigns in his turn a part of obtained resources for his own interests (non-purpose use), and the rest for the public interests (purpose use). Both levels take part in purpose activity profit and have their payoff functions (Fig. 1).

Fig. 1: The structure of modeled system.

The model is built as a hierarchical two-person game in which a Stackelberg equilibrium is sought (Basar, 1999). A payoff function of each player consists of two summands: non-purpose activity profit and a part of the system purpose activity profit. The payoff functions are: g1 (u1 , u2 ) = a1 (1 − u1 , u2 ) + b(u1 , u2 ) · c(u1 , u2 ) → max; u1

g2 (u1 , u2 ) = a2 (u1 , 1 − u2 ) + b(u1 , u2 ) · c(u1 , u2 ) → max . u2

subject to 0 ≤ ui ≤ 1, i = 1, 2, and conditions on functions a, b and c ai ≥ 0;

∂ai ∂ai ≤ 0, ≥ 0, i = 1, 2, ∂ui ∂uj6=i ∂bi bi ≥ 0; ≥ 0, i = 1, 2, ∂ui ∂c ≥ 0, i = 1, 2. ∂ui

Here index 1 relates to the top level attributes (a leading player), index 2 relates to the bottom level attributes (a following player); - ui is a share of resources assigned by i-th level to the purpose use (correspondingly,

A Problem of Purpose Resource Use in Two-Level Control Systems

83

1 − ui remains on non-purpose resource use in private interests); - gi is a payoff function of i-th level; - ai is a payoff function of i-th level private interest; - bi is a share of purpose activity profit obtained by i-th level; - c is a payoff function of purpose system activity (society, organization). Power, linear, exponential and logarithmic functions are considered as functions a and c. These functions depend on variables u1 , u2 , and they are cumulative ones, i.e. a1 = a1 (1 − u1 ), a1 = a2 (u1 (1 − u2 )), c = c(u1 u2 ). In this case a share of resources being is assigned to the public aims. The relations a1 = a1 (1 − u1 ), a1 = a2 (u1 (1 − u2 )), reflect the hierarchical structure of the system. The non-purpose activity income of top level does not depend on the part of the funds the bottom level assigned for the public aims but the non-purpose activity income of bottom level depends on the part of the funds the top level gives him. Three income types of purpose income distribution b are considered: 1) uniform one, in which the shares in purpose activity income are the same for both players, in particular, if n = 2 bi =

1 , i = 1, 2, 2

2) proportional one, in which the shares in income are proportional to the shares assigned to the public aims by the corresponding level, i.e. u1 , u1 + u2 u2 b2 = ; u1 + u2

b1 =

3) constant one, in which: b1 = b, b2 = 1 − b; The player strategy is a share ui of available resources assigned to the public aims. The top-level player u1 defines and informs the bottom level about it. Then the second player chooses the optimal value u2 knowing the strategy of the first player. The investigation aim is to study how the relation of functions a1 , a2 , b1 , b1 , c effects on the game solution (Stackelberg equilibrium). The next functions are taken as a non-purpose payoff function: - power with an exponent less than one (a(x) = axα , α < 1, a > 0), - linear (a(x) = ax, a particular case of power function with an exponent equaled to one), - power with an exponent greater than one, (a(x) = axk , k > 1, a > 0); - exponential (a(x) = a(1 − exp −λx), λ > 0, a > 0); - logarithmic (a(x) = alog(1 + x), a > 0). ∂a ∂2a As a rule, functions are chosen with constraints ∂x ≥ 0, ∂x 2 ≤ 0. The first condition is satisfied by all functions, the second condition is not satisfied only by function a(x) = axk , k > 1. The first and the second functions are production functions. The last two functions are not production ones since the property of scaling production returns does not hold.

84

Olga I. Gorbaneva, Guennady A. Ougolnitsky

Similarly, the next functions are taken as a purpose payoff function: - power with an exponent less than one (c(x) = cxα , α < 1, c > 0); - linear (c(x) = cx); - power with an exponent greater than one, (c(x) = cxk , k > 1, c > 0); - exponential (c(x) = c(1 − exp −λx), λ > 0, c > 0); - logarithmic (c(x) = clog(1 + x), c > 0). Thirteen of twenty five possible combinations are solved analytically: 1) combinations of similar functions, when a and c are either power, or exponential, or logarithmic ones; 2) combinations of any non-purpose use function and linear purpose use function; 3) combinations of linear non-purpose use function and any purpose use function. Six of the rest cases are investigated numerically. 3.

Analytical investigation of different model classes

We consider the case when a1 (u1 , u2 ) = 1 − u1 , a2 (u1 , u2 ) = u1 (1 − u2 ) ,c2(u1 , u2 ) = u1 u2 , b1 = 12 , b2 = 21 . Then: u1 u2 → max, u1 2 u1 u2 g2 (u1 , u2 ) = u1 − → max u2 2

g1 (u1 , u2 ) = 1 − u1 +

(1) (2)

This is the game with constant sum. Function g2 decreases in u2 , therefore the optimal value u2 ∗ = 0, at which g1 (u1 , 0) = 1 − ui . Function g1 decreases on u1 , therefore the value u1 ∗ = 0 is optimal. So, Stackelberg equilibrium in the game is ST1 = {(0; 0)} , while the player gains are g1 = a1 , g2 = 0 , i.e. both players use strategy of egoism (assign all available resources for private aims), but the top level gets maximum, while the bottom level gets zero. Consider the case when a1 = a1 (1 − u1 ), a2 = a2 u1 (1 − u2 ), c = (u1 u2 )k is the production power function. There may be two fundamentally different cases: 1) k = 1 (linear resource use function); Then, g1 (u1 , u2 ) = a1 (1 − u1) + b1 u1 u2 , g2 (u1 , u2 ) = a2 u1 (1 − u2) + b2 u1 u2 . We find optimal strategy of the bottom level: ∂g2 = (b2 − a2 )u1 , ∂u2 u∗2

=

1, b2 > a2 , 0, b2 < a2 .

The top level optimizes his gain function: a1 (1 − u1 ) + b1 u1 u2 , b2 > a2 , g1 (u1 , u∗2 ) = a1 (1 − u1 ), b 2 < a2 . ∂g1 = ∂u1

b 1 − a1 , b 2 > a2 , −a1 , b2 < a2 .

A Problem of Purpose Resource Use in Two-Level Control Systems Thus, (Fig. 2), u∗1

=

85

1, (b2 > a2 ) ∧ (b1 > a1 ), 0, (b2 < a2 ) ∨ (b1 < a1 ).

If b2 > a2 and b1 > a1 then both players apply altruistic strategy (u1 ∗ = u2 ∗ = 1), and g1 = b1 , g2 = b2 . In other cases the leading player behaves egoistically (u1 ∗ = 0), then g1 = a1 , g2 = 0.

Fig. 2: Game outcomes (3.1)-(3.2).

2) 0 < k < 1 (power resource use function). Then, g1 (u1 , u2 ) = a1 (1 − u1 ) + b1 (u1 u2 )k , g2 (u1 , u2 ) = a1 u1 (1 − u2 ) + b2 (u1 u2 )k . We find the bottom level optimal strategy: ∂g2 = −a2 u1 + kb2 (u1 u2 )k−1 = 0, ∂u2 1

∗

u2 =

a2 k−1 ( kb ) 2

u1

.

The top level optimizes his payoff function: g1 (u1 , u2 ∗ ) = b1 (

k a2 k−1 ) + a1 (1 − u1 ). kb2

Since function g1 decreases on u1 , then u1 ∗ = 0. We consider the case when the payoff function from non-purpose activity is linear, the payoff function from purpose activity is logarithmic, and a share of the purpose activity profit is constant for both levels: a1 (u1 , u2 ) = a1 (1 − u1 ), a2 = a2 u1 (1 − u2 ),

c = c log2 (1 + u1 u2 ), b1 = b, b2 = 1 − b.

86


Then gain functions are g1 (u1 , u2 ) = a1 (1 − u1 ) + bc log2 (1 + u1 u2 ) → max,

(3)

g2 (u1 , u2 ) = a2 u1 (1 − u2 ) + (1 − b)c log2 (1 + u1 u2 ) → max,

(4)

u1

u2

subject to 0 ≤ ui ≤ 1, i = 1, 2. Find the Stackelberg equilibrium. We divide this process into two phases and describe in detail now. 1) First, we solve a bottom level optimization problem. Suppose the value u1 is known. We find the derivative of g2 with respect to u2 and equate it to zero: ∂g2 (1 − b)cu1 (u1 , u2 ) = −a2 u1 + = 0. ∂u2 (1 + u1 u2 ) ln 2 We solve the equation. The case u1 = 0 has no practical interest, therefore we can − 1 . Finding divide both parts of equation by u1 and express u2 : u2 ∗ = u11 (1−b)c a2 ln 2 the second derivative of the function g2 with respect to u2 , we see that the point u2 ∗ is a maximum point: ∂ 2 g2 (1 − b)cu1 2 (u , u ) = − < 0. 1 2 ∂u2 2 (1 + u1 u2 )2 ln 2 Taking into account the restriction on u2 , note that the optimal strategy of the bottom level player is   a2 ln 2 ≥ (1 − b)c,  0, (1−b)c (1−b)c 1 1 ∗ u2 = u1 a2 ln 2 − 1 , 0 < u1 a2 ln 2 − 1 < 1,   (1−b)c 1, a2 ln 2 ≥ 1 + u1 ,

2) Solve a top level problem if the bottom level answer is known. Consider three cases: a) u2 ∗ = 0 . In this case g1 (u1 , 0) = a1 (1 − u1 ) + bc log2 1 = a1 (1 − u1 ). Since g1 decreases in u1 , the top level optimal strategy is u1 ∗ = 0, i.e. if top level knows that the bottom level assigns all available resources for the private aims, then he gives no resources to the bottom level andassigns the resources for his private aims. 1 ∗ b) u2 = u1 (1−b)c a2 ln 2 − 1 .

Then, g1 (u1 , u2 ∗ ) = a1 (1 − u1 ) + bc log2 (1−b)c a2 ln 2 . Here, similar to the previous case, the function g1 decreases with respect to u1 . Note that the bottom level chooses his strategy so that the constant value of resources is assigned for the public aims. Hence, the more resource is given to the bottom level by the top one, the more may be spent on the bottom level private aims (as the difference between resources, which were given by the top level, and constant value u1 u2 = (1−b)c a2 ln 2 − 1, which were assigned for the public aims by the bottom level). And conversely, the less resource is given to the bottom level by the top one, the less may be spent on bottom level private aims. Hence, taking into account

A Problem of Purpose Resource Use in Two-Level Control Systems

87

the decreasing of function in u1 , it is profitable for the top level to assign as little as possible resource for the public aims, hence the bottom level assigns as little as possible for the public aims. So, it is profitable for the bottom level to assign for the public aims as much resources as the bottom level assigns for the public aims, namely u1 ∗ = (1−b)c a2 ln 2 − 1, thereby causing the lower level to spend all the resources on public aims, i.e. u2 = 1. c) u2 ∗ = 1. In this case g1 (u1 , 1) = a1 (1 − u1 ) + bc log2 (1 + u1 ). Maximize this function taking into account the restriction 0 ≤ u1 ≤ 1. From the first order conditions ∂g1 bc (u1 , 1) = −a1 + = 0. ∂u1 (1 + u1 ) ln 2 we obtain: u1 ∗ =

bc − 1. a1 ln 2

Finding the second derivative of g1 with respect to u1 , we can see that the point u1 ∗ is a point of maximum: bc ∂ 2 g1 (u1 , u2 ∗ (u1 )) = − < 0. ∂u1 2 (1 + u1 )2 ln 2 Taking into account the restriction on u1, the optimal strategy of the bottom level is  0, a1 ln 2 ≥ bc,  u∗1 = a1bcln 2 − 1, 0 < a1bcln 2 − 1 < 1,  bc 1, a1 ln 2 − 1 ≥ u1 ,

So, the Stackelberg equilibrium is   (0; 0), a2       (1; 1), a2 u¯ = bc  − 1; 1 , a2    a1 ln 2    (1−b)c − 1; 1 , a2 a2 ln 2

> < < >

(1−b)c bc ln 2 or a1 > ln 2 , (1−b)c bc 2 ln 2 and a1 < 2 ln 2 , a1 (1−b) and 2 bc < a1 < lnbc2 , b ln 2 a1 (1−b) (1−b)c (1−b)c and < a < . 2 b 2 ln 2 ln 2

As can be seen from this formula, if assigning of some resource part for the public aims is profitable for the bottom level then the top level can enforce the bottom level to assign all the resources for the public aims. I.e., the bottom level assigns all the resources either only for public aims or only for private aims. Consider each branch of the Stackelberg equilibrium: I. u = (0; 0) if a2 > (1−b)c or a1 > lnbc2 (Fig.2). In this case for one or two of the ln 2 players the private activity gives much more profit than the public activity. It is not profitable for this player to assign the resources for the public aims, but then another player either has no incentive to assign resources to the public aims (for the top level) or has no resources (for the bottom level). The players’ gains are g1 = a1 , g2 = 0.

88


bc II. u = (1; 1) if a2 < (1−b)c 2 ln 2 and a1 < 2 ln 2 (Fig.3). In this case for both players the public activity gives much more profit than the private activity, therefore each of them assigns all the resources for the public aims. The players’ gains are

g1 = bc, g2 = (1 − b)c.

bc III. u = − 1; 1) if a2 < a1 (1−b) and 2 bc b ln 2 < a1 < ln 2 (Fig.3). In this case for the top level it is profitable to assign only a part of resources for the public aims (since the both activities profits are comparable) while for the bottom level it is profitable to assign all the resources for the public aims. The players’ gains are bc bc bc + bc log2 , g2 = (1 − b)c log2 . g1 = 2a1 − ln 2 a1 ln 2 a1 ln 2

( a1bcln 2

a1 (1−b) (1−b)c IV. u = ( (1−b)c and (1−b)c a2 ln 2 − 1; 1) if a2 > b 2 ln 2 < a2 < ln 2 (Fig.3). In this case for both players it is profitable to assign a part of the resources for the public aims, since the both activities profits are comparable. The bottom level is going to assign a fixed value of resources for the public aims and to leave the rest for the private aims. But the top level gives only this fixed value of resources to the bottom level thereby he enforces the bottom level to assign all the resources for the public aims. The players’ payoffs are (1 − b)c (1 − b)c a1 (1 − b)c + bc log2 , g2 = (1 − b)c log2 . g1 = 2a1 − a2 ln 2 a2 ln 2 a2 ln 2

Fig. 3: Game outcomes (3.3)-(3.4)

Finally, we consider the case when purpose and non-purpose activity functions are power with an exponent less than one: α

a1 = a1 (1 − u1 )α , a2 = a2 (u1 (1 − u2 )) ,

c = (u1 u2 )α , b1 = b, b2 = 1 − b.

89

A Problem of Purpose Resource Use in Two-Level Control Systems Then gain functions are g1 (u1 , u2 ) = a1 (1 − u1 )α + bc(u1 u2 )α → max, u1

α

g2 (u1 , u2 ) = a2 (u1 (1 − u2 )) + (1 − b)c(u1 u2 )α → max, u2

The Stackelberg equilibrium is (Fig. 4):  q u¯ = 

1−α

a1

b(1−b) 1−α c 1−α q √ α 1 √ α 1−α 1−α (1−b)c+ 1−α a2 + b(1−b) 1−α c 1−α

√

1−α

√ 1−α

(1−b)c

(1−b)c+

√

1−α

(6)

1

α

1−α

r

(5)

a2

;

We omit the players’ payoffs in this case. All the thirteen considered cases can be grouped together on the number of outcomes of the game: 1) One outcome, when public and private payoff functions are power with an exponent less than one. In this case for both players it is profitable to assign a part of resources for the public aims, and another part for the private aims. 2) Two outcomes (0; 0) and (1;1) (Fig. 2), when: a. The private payoff function is power with an exponent less than one and the public payoff function is linear; b. The public and private payoff functions are either linear or power with an exponent greater than one in any combinations. 3) Three outcomes, when private activity function is linear and public payoff function is power with an exponent less than one. In this case for one of the player it is profitable to assign all the resources for the public aims. 4) Four outcomes (Fig. 3), when one of the functions (either private or public payoff) is linear and another function is logarithmic. 5) Five outcomes (Fig. 4), when a. Public and private payoff functions are linear or exponential in any combinations except the case when both the functions are linear. b. Public and private payoff functions are logarithmic. 4.

Numerical investigation of different model classes

We use a numerical investigation for a few cases that could not be solved analytically. At first we consider a case when the purpose activity function is exponential and the non-purpose activity function is power with an exponent less than one, purpose activity profit share is constant for the players: a1 = a1 (1 − u1 )α , a2 = a2 (u1 (1 − u2 ))α ,

c = c(1 − e−λu1 u2 ), b1 = b, b2 = 1 − b.

In this case the payoff functions are

g1 (u1 , u2 ) = a1 (1 − u1 )α + bc(1 − e−λu1 u2 ) → max,

(7)

g2 (u1 , u2 ) = a2 (u1 (1 − u2 ))α + (1 − b)c(1 − e−λu1 u2 ) → max,

(8)

u1

u2

subject to 0 ≤ ui ≤ 1, i = 1, 2.

90


Fig. 4: One of the possible cases of the considered game with five outcomes

To find the bottom level optimal strategy we calculate the derivative of g2 with respect to u2 and equate it to zero: ∂g2 a2 αu1 α (u1 , u2 ) = − + λu1 (1 − b)ce−λu1 u2 = 0. ∂u2 (1 − u2 )1−α

(9)

Prove that the bisection method may be applied for solving this equation. Note that the second derivative of g2 with respect to u2 is negative, ∂ 2 g2 a2 α(1 − α)u1 α (u , u ) = − λ2 u21 (1 − b)ce−λu1 u2 < 0, 1 2 ∂u2 2 (1 − u2 )2−α ∂g2 (u1 , u2 ) is monotone. therefore, the function ∂u 2 ∂g2 Then find signs of ∂u2 (u1 , u2 ) at the endpoints of [0,1].

∂g2 ∂u2 (u1 , u2 )

∂g2 α ∂u2 (u1 , 0) = −a2 αu1 + λu1 (1 − b)c, α 1 →u2 →1_ − a2 αu + λu1 (1 − b)ce−λu1 u2 0+

(10) →u2 →1_ −∞.

(11)

If (10) is positive, then the equation may be solved by the bisection method, and the solution obtained is a maximum point since the second derivative is negative. If (10) is negative, then bisection method is not applied, but the left part of equation is monotone then it is negative at the segment [0, 1], hence, function g2 decreases, then the maximum point is u2 = 0. That is, 0, −a2 αu1 α + λu1 (1 − b)c < 0, ∗ u2 = ∈ (0; 1), −a2 αu1 α + λu1 (1 − b)c > 0,

The top level can use this information to enforce the bottom level to choose non-zero strategy. For the bottom level to choose the positive strategy u2 > 0, it is necessary to satisfy the condition −a2 αu1 α + λu1 (1 − b)c > 0. When the inequality have been

A Problem of Purpose Resource Use in Two-Level Control Systems solved for the variable u1 , we obtain u1 >

q

1−α

91

a2 α λ(1−b)c

For the bottom level not to spend all the resources onqprivate aims, it is recoma2 α mended for the top level to choose the strategy u1 > 1−α λ(1−b)c . But he can do it 1−α q a2 α a2 α only if 1−α λ(1−b)c < 1, which is equivalent to a2 < λ(1−b)c .

If the top level cannot use this strategy or this strategy is not profitable for him then the bottom level choose the strategy u2 = 0. Find then the optimal top level behavior and his payoff g1 (u1 , 0) = a1 (1 − u1 )α . As can be seen, the function g1 decreases in u1 , therefore, u1 = 0. Draw some conclusions: 1−α a2 α I. If a2 > λ(1−b)c then the top level cannot effect on the bottom one, in this case u2 = 0, and therefore u1 = 0. This occurs when the capacity of the bottom level of non-purpose activity is significantly more than production capacity of purpose activity. 1−α a2 α then the top level can enforce the bottom level to spend II. If a2 < λ(1−b)c q a2 α some part of resources on the public aims assigning u1 > 1−α λ(1−b)c . This occurs when the capacity of the bottom level of purpose activity is significantly more than production capacity of non-purpose activity. 5.

Conclusion

In this paper a problem of non-purpose resource use is treated in terms of analysis of control mechanism properties providing the concordance of interests in hierarchical (two-level) control systems. The interests of players are described by their payoff functions including two summands: purpose and non-purpose resource use profits. Different classes of these functions are considered. The top level subject (resource distributor) is treated as a leading player and the bottom level (resource recipient) subject is treated as a following player. This leads to the Stackelberg equilibrium concept. Performed analytical and numerical investigation permits to make the next conclusions. In the case when the payoff functions for purpose and non-purpose activities are power with an exponent less than one it is profitable to assign only a part of resources for the public aims and another part of them for the private aims for both players. In the case when one of the payoff functions for purpose or non-purpose activities is power with an exponent greater than one and another of them is either linear or power with an exponent greater than one it is profitable to assign all the resources for only public aims (“egoism” strategy) or for only private aims (“altruism” strategy). In other cases the next situations may occur: A) if the effect of the private activities of a player is much more than effect of the public activity then for a player the “egoism” strategy is profitable; B) if the effect of the private activities of a player is much less than effect of the public activity then for a player the “altruism” strategy is profitable; C) if the effects of the private and public activities of a player are comparable then for any player it is profitable to assign only a part of resources for the public aims and the other part for the private aims.

92


References Willamson, O. (1981). Firms and markets . Modern economical thought, 211-297 (in Russian). Basar, T. and Olsder, G. Y. (1999). Dynamic Noncooperative Game Theory. SIAM. Laffont, J.-J. and Martimort, D. (2002). Theory of Incentives. The Principal-Agent Model. Princeton. Gorelik, V. A., Gorelov, M. A., and Kononenko, A. F. (1991). Analysis of conflict situations in control systems (in Russian). Mechanism design and management: Mathematical methods for smart organizations.Ed. by Prof. D. Novikov. N.Y.: Nova Science Publishers, 2013. Novikov, D. (2013). Theory of control in organizations. N.Y.: Nova Science Publishers. Germeyer, Yu. B. and Vatel I. A. (1974). Games with hierarchical interests vector. Izvestiya AN SSSR. Technical cybernetics, 3, 54–69 (in Russian). Gorbaneva, O. I. and Ougolnitsky, G. A. (2009). Resource allocation models in the hierarchical systems of river water quality control. Large-scale Systems Control, 26, 64–80 (in Russian). Gorbaneva, O. I. (2010). Resource allocation game-models in the hierarchical systems of river water quality control. Mathematical game theory and its applications, 2(10), 27–46 (in Russian). Gorbaneva, O. I. and Ougolnitsky, G. A. (2013). Static models of corruption in resource allocation mechanisms for three-level control systems. Large-scale Systems Control, 42, 195-216 (in Russian). Gorbaneva, O. and Ougolnitsky, G. (2013). Purpose and Non-Purpose Resource Use Models in Two-Level Control Systems. Advances in Systems Science and Application, 13(4), 378-390 (in Russian).

Multicriteria Coalitional Model of Decision-making over the Set of Projects with Constant Payoff Matrix in the Noncooperative Game Xeniya Grigorieva St.Petersburg State University, Faculty of Applied Mathematics and Control Processes, University pr. 35, St.Petersburg, 198504, Russia E-mail: [email protected] WWW home page: http://www.apmath.spbu.ru/ru/staff/grigorieva/

Abstract Let N be the set of players and M the set of projects. The multicriteria coalitional model of decision-making over the set of projects is formalized as family of games with different fixed coalitional partitions for each project that required the adoption of a positive or negative decision by each of the players. The players’ strategies are decisions about each of the project. The vector-function of payoffs for each player is defined on the set situations in the initial noncooperative game. We reduce the multicriteria noncooperative game to a noncooperative game with scalar payoffs by using the minimax method of multicriteria optimization. Players forms coalitions in order to obtain higher income. Thus, for each project a coalitional game is defined. In each coalitional game it is required to find in some sense optimal solution. Solving successively each of the coalitional games, we get the set of optimal n-tuples for all coalitional games. It is required to find a compromise solution for the choice of a project, i. e. it is required to find a compromise coalitional partition. As an optimality principles are accepted generalized PMS-vector (Grigorieva and Mamkina, 2009; Petrosjan and Mamkina, 2006) and its modifications, and compromise solution. Keywords: coalitional game, PMS-vector, compromise solution, multicriteria model.

1.

Introduction

The set of agents N and the set of projects M are given. Each agent fixed his participation or not participation in the project by one or zero choice. The participation in the project is connected with incomes or losses by different parametres which the agents wants to maximize or minimize. This gives us an optimization problem which can be modeled as multicriteria noncooperative game. We reduce the multicriteria noncooperative game to a noncooperative game with scalar payoffs by using the minimax method of multicriteria optimization. Agents may form coalitions. This problem we will call as multicriteria coalitional model of decision-making. Denote the players by i ∈ N and the projects by j ∈ M . The family M of different games are considered. In each game Gj , j ∈ M the player i has two strategies accept or reject the project. The payoff of the player in each game is determined by the strategies chosen by all players in this game Gj . As it was mentioned before the players can form coalitions to increase the payoffs components. In each

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Xeniya Grigorieva

game Gj coalitional partition is formed. The problem is to find the optimal strategies for coalitions and the imputation of the coalitional payoff between the members of the coalition. The games G1 , . . . , Gm are solved by using the PMS-vector (Grigorieva and Mamkina, 2009; Petrosjan and Mamkina, 2006) and its modifications. Then having the solutions of games Gj , j = 1, m the optimality principle - “the compromise solution" is proposed to select the best projects j ∗ ∈ M . The problem is illustrated by example of the interaction of three players. 2.

State of the problem

Consider the following problem. Suppose – N = {1 , . . . , n} is the set of players; – Xi = {0 ; 1} is the set of pure strategies xi of player i , i = 1, n. The strategy xi can take the following values: xi = 0 as a negative decision for the some project and xi = 1 as a positive decision; – li = 2 is the number of pure strategies of player i; – x = (x1Q , . . . , xn ) is the n-tuple of pure strategies chosen by the players; – X= Xi is the set of n-tuples; i=1 , n P – µi = {ξ (xi )}xi ∈Xi , ξ (xi ) > 0 ∀ xi ∈ Xi , ξ (xi ) = 1 is the mixed strategy xi ∈Xi of player i; will be used denotation too µi = ξi0 , ξi1 , where ξi0 is the probability of making negative decision by the player i for some project, and ξi1 is the probability of making positive decision correspondingly; – Mi is the set of mixed strategies of the i-th player; – µ is theQn-tuple of mixed strategies chosen by players for some project; – M= Mi is the set of n-tuples in mixed strategies for some project; i=1, n

– Ki : X → Rr is the vector-function of payoff defined on the set X for each player i , i = 1, n . ˜ ( x): Thus, we have multicriteria noncooperative n-person game G D E ˜ (x) = N, {Xi } G , {K (x)} i i=1 , n i=1 , n , x∈X .

(1)

Using the minimax method of multicriteria optimization, we reduce the nonco˜ operative n-person game G(x) to a noncooperative game G(x) with scalar payoffs: E D G (x) = N, {Xi }i=1 , n , {Ki (x)}i=1 , n , x∈X , (2) where

Ki (x) = max Kis (x) , Kis (x) ∈ Ki (x) , x ∈ X ,

(3)

s=1 , r

Ei (µ) =

X

x1 ∈X1

...

X

[Ki (x) ξ (x1 ) . . . ξ (xn )] , i = 1 , n .

(4)

xn ∈Xn

Now suppose M = {1 , . . . , m} is the set of projects, which require making positive or negative decision by n players.

95

Multicriteria Coalitional Model of Decision-making A coalitional partitions Σ j of the set N is defined for all j = 1 , m: l n o [ Σ j = S1j , . . . , Slj , l 6 n , n = |N | , Skj ∩ Sqj = ∅ ∀ k 6= q, Skj = N . k=1

Then we have m simultaneous l-person coalitional games Gj (xΣ j ) , j = 1 , m , in normal form associated with the game G (x): n o n o ˜ j ˜ j (xΣ j ) Gj (xΣ j ) = N, X , H , j = 1, m. Sk

k=1 , l , Skj ∈Σ j

Sk

k=1 , l , Skj ∈Σ j

(5)

Here for all j = 1 , m: – x ˜S j = {xi }i∈S j is the l-tuple of strategies of players from coalition Skj , k = 1, l; k k ˜ j = Q Xi is the set of strategies x – X ˜S j of coalition Skj , k = 1, l, i. e. CarteS k

– – – –

k

i∈Skj

sian product of the sets of players’ strategies, which are included into coalition Skj ; ˜ x ˜ j , k = 1, l is the l-tuple of strategies xΣ j = x ˜S j , . . . , x ˜S j ∈ X, ˜S j ∈ X Sk 1 l k of all coalitions; ˜ j is the set of l-tuples in the game Gj (xΣ j ); ˜= Q X X Sk l k=1, Q ˜ lS j = X li is the number of pure strategies of coalition Skj ; Skj = k j i∈Sk Q lΣ j = lS j is the number of l-tuples in pure strategies in the game Gj (xΣ j ). k

k=1,l

˜ j is the set of mixed strategies µ – M ˜S j of the coalition Skj , k = 1, l; Sk k l j S l j k P S – µ ˜S j = µ ˜1S j , ... , µ ˜S jk , µ ˜ξS j > 0 , ξ = 1, lS j , µ ˜ξS j = 1, is the mixed k

k

k

k

k

ξ=1

k

strategy, that is the set of mixed strategies of players from coalition Skj , k = 1, l; ˜ µ ˜ j , k = 1, l, is the l-tuple of mixed – µΣ j = µ ˜S j , . . . , µ ˜S j ∈ M, ˜S j ∈ M Sk 1 l k strategies; ˜ = Q M ˜ j is the set of l-tuples in mixed strategies; – M S k

k=1, l

˜S (˜ – E µ) is the payoff function of coalition Skj in mixed strategies and defined as k i X X h ˜S (˜ ˜ S (xΣ j ) ξ˜ x˜ j . . . ξ˜ x E µ) = ... H ˜ j , i = 1 , n . (6) k k S S ˜

j ∈X j S1 S1

x ˜

x ˜

S

˜

1

1

j ∈X j S l l

From the definition of strategy x˜S j of coalition Skj it follows that k xΣ j = x ˜S j , . . . , x ˜S j and x = (x1 , . . . , xn ) are the same n-tuples in the games 1

l

G(x) and Gj (xΣ j ). However it does not mean that µ = µΣ j . ˜ j : X ˜ → R1 of coalition S j for the fixed projects j, j = Payoff function H k S k

1, m, and for the coalitional partition Σ j is defined under condition that: X ˜ j (xΣ j ) > H j (xΣ j ) = Ki (x) , k = 1 , l , j = 1 , m , Skj ∈ Σ j , H S S k

k

i∈Skj

(7)

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Xeniya Grigorieva

where Ki (x) , i ∈ Skj , is the payoff function of player i in the n-tuple xΣ j . Definition 1. A set of m coalitional l-person games defined by (5) and associated with noncooperative games defined by (1)−(2) is called multicriteria coalitional model of decision-making with constant matrix of payoffs in the noncooperative game. Definition 2. Solution of the multicriteria coalitional model of decision-making with constant matrix of payoffs in the noncooperative game in pure strategies is x∗Σ j∗ , that is Nash equilibrium (NE) in a pure strategies in l-person game Gj ∗ (xΣ j∗ ), with ∗ ∗ the coalitional partition Σ j , where coalitional partition Σ j is the compromise coalitional partition (see 3.2). Definition 3. Solution of the multicriteria coalitional model of decision-making with constant matrix of payoffs in the noncooperative game in mixed strategies is µ∗Σ j∗ , that is Nash equilibrium (NE) in a mixed strategies in l-person game ∗ ∗ Gj ∗ (µΣ j∗ ), with the coalitional partition Σ j , where coalitional partition Σ j is the compromise coalitional partition (see 3.2). Generalized PMS-vector is used as the coalitional imputation (Grigorieva and Mamkina, 2009; Petrosjan and Mamkina, 2006). 3.

Algorithm for solving the problem

3.1.

Algorithm of constructing the generalized PMS-vector in a coalitional game.

Remind the algorithm of constructing the generalized PMS-vector in a coalitional game (Grigorieva and Mamkina, 2009; Petrosjan and Mamkina, 2006). ˜ j (xΣ j ) for all coalitions S j ∈ Σ j , k = 1, l , 1. Calculate the values of payoff H k Sk for coalitional game Gj (xΣ j ) by using formula (3). 2. Find NE (Nash, 1951) x∗Σ j or µ∗Σ j (one or more) in the game Gj (xΣ j ). The n j o ∗ . payoffs’ vector of coalitions in NE in mixed strategies E µΣ j = v Sk k=1, l

Denote a payoff of coalition Skj in NE in mixed strategies by lΣ j X ˜ j (x∗ j ), k = 1, lΣ j , v Skj = pτ, j H Σ τ, S τ =1

k

where ˜ j x∗ j is the payoff of coalition S j , when coalitions choose their pure – H k τ, Sk Σ strategies x ˜∗S j in NE in mixed strategies µ∗Σ j . Q kξk – pτ, j = µ ˜S j , ξk = 1, lS j , τ = 1, lΣ j , is probability of the payoff’s realization ˜ j H τ, S

k

k=1,l x∗Σ j

k

k

of coalition Skj .

˜ j x∗ j is a random variable. There could be many l-tuple of NE The value H τ, Sk Σ in the game, therefore, v S1j , ...., v Slj , are not uniquely defined. ∗ The payoff of each coalition in NE according to Shapley’s E µΣ j is divided j j value (Shapley, 1953) Sh (Sk ) = Sh Sk : 1 , ... , Sh Sk : s :

97

Multicriteria Coalitional Model of Decision-making

X (s′ −1) ! (s−s′ ) ! [v (S ′ ) − v (S ′ \ {i})] ∀ i = 1, s , Sh Skj : i = s ! j ′

(8)

S ⊂Sk S ′ ∋i

where s = Skj (s′ = |S ′ |) is the number of elements of sets Skj (S ′ ), and v (S ′ ) is the maximal guaranteed payoff of S ′ ⊂ Sk . Moreover s X v Skj = Sh Skj : i . i=1

Then PMS-vector in the NE in mixed strategies µ∗Σ j in the game Gj (xΣ j ) is defined as PMSj (µ∗Σ j ) = PMSj1 (µ∗Σ j ) , ..., PMSjn (µ∗Σ j ) , where 3.2.

PMSji (µ∗Σ j ) = Sh Skj : i , i ∈ Skj , k = 1, l.

Algorithm for finding a set of compromise solutions.

Remind the algorithm (Malafeyev, 2001; p.18).

for

finding

a

set

of

compromise

solutions

j j CPMS (M ) = arg min max max PMSi − PMSi . j

i

j

∗

Step 1. Construct the ideal vector R = (R1 , . . . , Rn ) , where Ri = PMSji = max PMSji is the maximal value of payoff functions of player i in NE on the set M , j

and j is the number of project j ∈ M :   PMS11 ... PMS1n  ... ... ...  m PMSm 1 ... PMSn ↓ ... ↓ ∗ j1∗ PMS1 ... PMSjnn Step 2. For each j find deviation of payoff function values for other players from the maximal value, that is ∆ji = Ri − PMSji , i = 1 , n:   R1 − PMS11 ... Rn − PMS1n . ∆=  ... ... ... m R1 − PMSm ... R − PMS n 1 n ∆ji∗ j

Step 3. From the found deviations ∆ji for each j select the maximal deviation = max ∆ji among all players i: i



  1  → ∆1i∗1 ∆1 ... ∆1n R1 − PMS11 ... Rn − PMS1n   =  ... ... ...  ... . ... ... ... m m m ∆m → ∆m R1 − PMS1 ... Rn − PMSn 1 ... ∆n i∗ m

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Xeniya Grigorieva

Step 4. Choose the minimal deviation for all j from all the maximal deviations ∗ among all players i ∆ji∗∗ = min ∆ji∗ = min max ∆ji . j

j

j

j

i

The project j ∗ ∈ CPMS (M ) , on which the minimum is reached is a compromise solution of the game Gj (xΣ j ) for all players. 3.3.

Algorithm for solving the multicriteria coalitional model of decision-making over the set of projects with constant matrix of payoffs in the noncooperative game.

Thus, we have an algorithm for solving the problem. ˜ 1. Reduce the multicriteria noncooperative n-person game G(x) (see (1)) to a noncooperative game G(x) with scalar payoffs (see (2)) using the minimax method of multicriteria optimization. 2. Fix a j , j = 1 , m. 3. Construct the coalitional Gj (xΣ j ) associated with the noncooperative game G(x) for the fixed j. 2. Find the NE µ∗Σ j in the coalitional game Gj (xΣ j ) and find imputation in NE, that is PMSj µ∗Σ j . 3. Repeat iterations 1-2 for all other j , j = 1 , m.

4. Find compromise solution j ∗ , that is j ∗ ∈ CPMS (M ).

4.

Example

Consider the set M = {j}j=1, 5 and the set N = {I1 , I2 , I3 } of three players, each ˜ (x): xi = 1 is “yes" and having 2 strategies in multicriteria noncooperative game G ˜ (x) are xi = 0 is “no" for all i = 1 , 3. The payoff functions of players in the game G determined by the table 1. Table 1: The payoffs of players. The I1 1 1 1 1 0 0 0 0

strategies I2 I3 1 1 1 0 0 1 0 0 1 1 1 0 0 1 0 0

The payoffs of players I1 I2 I3 (4, 3.43, 1.71) (2, 1.71, 0.86) (1, 0.86, 0.43) (1, 0.86, 0.43) (2, 1.71, 0.86) (2, 1.71, 0.86) (3, 2.57, 1.29) (1, 0.86, 0.43) (5, 4.29, 2.14) (5, 4.29, 2.14) (1, 0.86, 0.43) (3, 2.57, 1.29) (5, 4.29, 2.14) (3, 2.57, 1.29) (1, 0.86, 0.43) (1, 1, 0.43) (1.14, 2, 0.86) (1.86, 2, 2) (0, 0, 0) (4, 3.43, 1.71) (3, 3, 1.29) (0, 0, 0) (4, 3.43, 1.71) (2, 2, 0.86)

˜ Reduce the multicriteria noncooperative n-person game G(x) to a noncooperative game G(x) with scalar payoffs using the minimax method of multicriteria optimization (see (3)). The values of payoff functions of players in the game G (x) are in the table 2.


99

Table 2: The payoffs of players. The I1 1 1 1 1 0 0 0 0

strategies I2 I3 1 1 1 0 0 1 0 0 1 1 1 0 0 1 0 0

The I1 4 1 3 5 5 1 0 0

payoffs I2 I3 2 1 2 2 1 5 1 3 3 1 2 2 4 3 4 2

The payoffs of coalition {I1 , I2 } {I2 , I3 } {I1 , I3 } {I1 , I2 , I3 } 6 3 5 7 3 4 3 5 4 6 8 9 6 4 8 9 8 4 6 9 3 4 3 5 4 7 3 7 4 6 2 6

1. Compose and solve the coalitional game G2 (xΣ 2 ) , Σ2 = {{I1 , I2 } , I3 }, i. e. find NE in mixed strategies in the game: η = 3/7 1 − η = 4/7 1 0 0 (1, 1) [6, 1] [3, 2] 0 (0, 0) [4, 3] [4, 2] ξ = 1/3 (1, 0) [4, 5] [6, 3] 1 − ξ = 2/3 (0, 1) [8, 1] [3, 2] . It’s clear, that first matrix row is dominated by the last one and the second is dominated by third. One can easily calculate NE and we have y = 3/7 4/7 , x = 0 0 1/3 2/3 .

Then the probabilities of payoffs’s realization of the coalitions S = {I1 , I2 } and N \S = {I3 } in mixed strategies (in NE) are as follows: ξ1 ξ2 ξ3 ξ4

η1 η2 0 0 0 0 . 1/ 4/ 7 21 2/ 8/ 7 21

The Nash value of the game in mixed strategies is calculated by formula: 1 2 4 8 36 7 1 1 E (x, y) = [4, 5] + [8, 1] + [6, 3] + [3, 2] = , = 5 , 2 . 7 7 21 21 7 3 7 3 In the table 3 pure strategies of coalition N \S and its mixed strategy y are given horizontally at the right side. Pure strategies of coalition S and its mixed strategy x are given vertically. Inside the table players’ payoffs from the coalition S and players’ payoffs from the coalition N \S are given at the right side. Divide the game’s Nash value in mixed strategies according to Shapley value (8): Sh1 = v (I1 ) + Sh2 = v (I2 ) +

1 2 1 2

[v (I1 , I2 ) − v (I2 ) − v (I1 )] , [v (I1 , I2 ) − v (I2 ) − v (I1 )] .

100

Xeniya Grigorieva Table 3: The maximal guaranteed payoffs of players I1 and I2 .

Math. Expectation 2.286 4.143 2.714 0.000 v (I1 ) min 1 2.286 min 2 0.000 max 2.286

2.000 1.000 2.429 4.000 v (I1 ) 2.000 1.000 2.000

The strategies of N \ S, the payoffs of S and the payoffs of N \ S η = 0.43 +1 0 − (1 , 1) (4 , 2) ξ = 0.33 + (1 , 2)   (3 , 1) 1 − ξ = 0.67 + (2, 1)  (5 , 3) 0 − (2 , 2) (0 , 4)

1 − η = 0.57 +2 (1 , 2) (5 , 1)   (1 , 2)  (0 , 4)

Find the maximal guaranteed payoffs v (I1 ) and v (I2 ) of players I1 and I2 . For this purpose fix a NE strategy of a third player as y¯ = 3/7 4/7 .

Denote mathematical expectations of players’ payoff from coalition S when mixed NE strategies are used by coalition N \S by ES(i, j) (¯ y) , i, j = 1, 2. In the table 3 the mathematical expectations are located at the left, and values are obtained by using the following formulas: ES(1, 1) (¯ y ) = 37 · 4 + 47 · 1 ; 37 · 2 + 47 · 2 ; 73 · 1 + 47 · 2 = 2 72 ; 2 ; 1 74 ; ES(1, 2) (¯ y ) = 37 · 3 + 47 · 5 ; 37 · 1 + 47 · 1 ; 73 · 5 + 47 · 3 = 4 71 ; 1 ; 3 76 ; ES(2, 1) (¯ y ) = 37 · 5 + 47 · 1; 73 · 3 + 47 · 2; 37 · 1 + 47 · 2 = 2 57 ; 2 37 ; 147 ; ES(2,2) (¯ y) = 37 · 0 + 47 · 0 ; 37 · 4 + 47 · 4 ; 37 · 3 + 47 · 2 = 0; 4 ; 2 73 . Third element here is the mathematical expectation of payoff of the player I3 (see table 2 too). Then, look at the table 2 or table 3, min H1 (x1 = 1, x2 , y¯) = min 2 27 ; 4 17 = 2 72 ; v (I1 ) = max 2 72 ; 0 = 2 27 ; 5 min H1 (x1 = 0, x2 , y¯) = min 2 7 ; 0 = 0; min H2 (x1 , x2 = 1, y¯) = min 2; 2 37 = 2 ; v (I2 ) = max {2; 1} = 2. min H2 (x1 , x2 = 0, y¯) = min {1; 4} = 1;

Thus, maxmin payoff for player I1 is v (I1 ) = 2 72 and for player I2 is v (I2 ) = 2. Hence, Sh1 (¯ y ) = v (I1 ) + 12 5 71 − v (I1 ) − v (I2 ) = 2 27 + 12 5 71 − 2 27 − 2 = 2 75 ; Sh2 (¯ y) = 2 + 73 = 2 37 . Thus, PMS-vector is equal: 5 3 1 PMS1 = 2 ; PMS2 = 2 ; PMS3 = 2 . 7 7 3


101

Table 4: Shapley’s value in the cooperative game. The strategies of players I1 I2 I3 1 1 1 1 1 2 1 2 1 1 2 2 2 1 1 2 1 2 2 2 1 2 2 2

The payoffs of players I1 I2 I3 4 2 1 1 2 2 3 1 5 5 1 3 5 3 1 1 2 2 0 4 3 0 4 2

The payoff Shapley’s of coalition value HN (I1 , I2 , I3 ) λ1 HN λ2 HN λ3 HN 7 5 9 2.5 3.5 3 9 2.5 3.5 3 9 2.5 3.5 3 5 7 6

2. Solve the cooperative game G5 (xΣ 5 ), Σ5 = {N = {I1 , I2 , I3 }}, see table 4. Find the maximal payoff HN of coalition N and divide him according to Shapley value (8), (Shapley, 1953): Sh1 =

1 1 [v (I1 , I2 ) + v (I1 , I3 ) − v (I2 ) − v (I3 )] + [v (N ) − v (I2 , I3 ) + v (I1 )] ; 6 3

1 1 [v (I2 , I1 ) + v (I2 , I3 ) − v (I1 ) − v (I3 )] + [v (N ) − v (I1 , I3 ) + v (I2 )] ; 6 3 1 1 Sh3 = [v (I3 , I1 ) + v (I3 , I2 ) − v (I1 ) − v (I2 )] + [v (N ) − v (I1 , I2 ) + v (I3 )] . 6 3 Find the guaranteed payoffs:

Sh2 =

v (I1 , I2 ) = max {4, 3} = 4; v (I1 , I3 ) = max {3, 2} = 3; v (I2 , I3 ) = max {3, 4} = 4 ; v (I1 ) = max {1, 0} = 1 ; v (I2 ) = max {2, 1} = 2; v (I3 ) = max {1, 2} = 2 . Then

(2, 1, 1)

Sh1

(1, 2, 2)

= Sh1

(1, 2, 1)

= Sh1

=

1 1 1 1 1 1 5 1 1 + + [9 − 4] + = + + + = 2 , 3 6 3 3 3 6 3 3 2

1 1 1 2 1 1 6 2 1 + + [9 − 3] + = + + + = 3 , 2 3 3 3 2 3 3 3 2 1 1 1 2 1 1 5 2 (2, 1, 1) (1, 2, 2) (1, 2, 1) Sh3 = Sh3 = Sh3 = + + [9 − 4] + = + + + = 3. 3 3 3 3 3 3 3 3 3. Solve noncooperative game G1 (xΣ 1 ), Σ1 = {S1 = {I1 } , S2 = {I2 } , S3 = { I3 }}. In pure strategies NE not exist. From p. 3 it follows that the guaranteed payoffs v (I1 ) = 1 ; v (I2 ) = 2; v (I3 ) = 2 . Find the optimal strategies with Nash arbitration scheme, see table 5. Then optimal n-tuple are ((1) , (1) , (2)) and ((2) , (1) , (2)), the payoff in NE equals ((1) , (2) , (2)). A detailed solution of games for various cases of coalitional partition of players is provided in (Grigorieva, 2009). Present the obtained solution in (Grigorieva, 2009) in the table 6. (2, 1, 1)

Sh2

(1, 2, 2)

= Sh2

(1, 2, 1)

= Sh2

=

102

Xeniya Grigorieva Table 5: Solution of noncooperative game.

The strategies of players I1 I2 I3 1 1 1 1 1 2 1 2 1 1 2 2 2 1 1 2 1 2 2 2 1 2 2 2

The payoffs of players I1 I2 I3 4 2 1 1 2 2 3 1 5 5 1 3 5 3 1 1 2 2 0 4 3 0 4 2

Pareto-optimality (P) and Nash arbitration scheme Nash arbitration scheme P (4 − 1) (2 − 2) (1 − 2) < 0 (1 − 1) (2 − 2) (2 − 2) = 0 + (3 − 1) (1 − 2) (5 − 2) < 0 (5 − 1) (1 − 2) (3 − 2) < 0 (5 − 1) (3 − 2) (1 − 2) < 0 (1 − 1) (2 − 2) (2 − 2) = 0 + (0 − 1) (4 − 2) (3 − 2) < 0 (0 − 1) (4 − 2) (2 − 2) < 0 -

Table 6: Payoffs of players in NE for various cases of the coalitional partitions of players. Project

Coalitional partitions

The n-tuple of NE (I1 , I2 , I3 )

1

Σ1 = {{I1 } {I2 } {I3 }}

2

Σ2 = {{I1 , I2 } {I3 }}

3

Σ3 = {{I1 , I3 } {I2 }}

4

Σ4 = {{I2 , I3 } {I1 }}

5

Σ5 = {I1 , I2 , I3 }

((1) , (1) , (0)) ((0) , (1) , (0)) ((1, 0) , 1) ((1, 0) , 0) ((0, 1) , 1) ((0, 1) , 0) (1, (1) , 1) (1, (0) , 1) (0, (1) , 1) (0, (0) , 1) (1, (0, 1)) (1, 0, 1) (1, 0, 0) (0, 1, 1)

Probability of realization NE 1 1/7 4/21 2/7 8/21 5/12 1/12 5/12 1/12 1 1 1 1

Payoffs of players in NE ((1) , (2) , (2))

((2.71, 2.43) , 2.33)

(2.59, (2.5) , 2.91)

(3, (3, 3)) (2.5, 3.5, 3)

Table 7: The set of compromise coalitional partitions.

Σ1 = {{I1 } {I2 } {I3 }} Σ2 = {{I1 , I2 } {I3 }} Σ3 = {{I1 , I3 } {I2 }} Σ4 = {{I2 , I3 } {I1 }} Σ5 = {I1 , I2 , I3 } R

I1 1 2.71 2.59 3 2.5 3

I2 2 2.43 2.5 3 3.5 3.5

I3 2 2.33 2.91 3 3 3

I1 I2 I3 ∆ {{I1 } {I2 } {I3 }} 2 1.5 1 2 ∆ {{I1 , I2 } {I3 }} 0.29 1.07 0.67 1.07 ∆ {{I1 , I3 } {I2 }} 0.41 1 0.09 1 ∆ {{I2 , I3 } {I1 }} 0 0.5 0 0.5 ∆ {I1 , I2 , I3 } 0.5 0 0 0.5


103

Applying the algorithm for finding a compromise solution, we get the set of compromise coalitional partitions (table 7). Therefore, compromise imputation are PMS-vector in coalitional game with the coalition partition Σ4 in NE (1 , (0 , 1)) in pure strategies with payoffs (3 , (3 , 3)) and Shapley value in the cooperative game in NE ((1 , 0 , 1) , (1 , 0 , 0) , (0 , 1 , 1) – cooperative strategies) with the payoffs (2.5 , 3.5 , 3). Moreover, in situation, for example, (1 , (0 , 1)) the first and third players give a positive decision for corresponding project. In other words, if the first and third players give a positive decision for corresponding project, and the second does not, then payoff of players will be optimal in terms of corresponding coalitional interaction. 5.

Conclusion

A multicriteria coalitional model of decision-making over the set of projects with constant payoff matrix in the noncooperative game and algorithm for finding optimal solution are constructed in this paper, the numerical example is given. References Grigorieva, X., Mamkina, S. (2009). Solutions of Bimatrix Coalitional Games. Contributions to game and management. Collected papers printed on the Second International Conference “Game Theory and Management" [GTM’2008]/ Edited by Leon A. Petrosjan, Nikolay A. Zenkevich. - SPb.: Graduate School of Management, SpbSU, 2009, pp. 147–153. Petrosjan, L., Mamkina, S. (2006). Dynamic Games with Coalitional Structures. Intersectional Game Theory Review, 8(2), 295–307. Nash, J. (1951). Non-cooperative Games. Ann. Mathematics 54, 286–295. Shapley, L. S. (1953). A Value for n-Person Games. In: Contributions to the Theory of Games( Kuhn, H. W. and A. W. Tucker, eds.), pp. 307–317. Princeton University Press. Grigorieva, X. V. (2009). Dynamic approach with elements of local optimization in a class of stochastic games of coalition. In: Interuniversity thematic collection of works of St. Petersburg State University of Civil Engineering (Ed. Dr., prof. B. G. Wager). Vol. 16. Pp. 104–138. Malafeyev, O. A. (2001). Control system of conflict. SPb.: St. Petersburg State University, 2001.

Differential Games with Random Duration: A Hybrid Systems Formulation Dmitry Gromov1 and Ekaterina Gromova2 Faculty of Applied Mathematics, St. Petersburg State University, St.Petersburg, Russia E-mail: [email protected] 2 Faculty of Applied Mathematics, St. Petersburg State University, St.Petersburg, Russia E-mail: [email protected] 1

Abstract The contribution of this paper is two-fold. First, a new class of differential games with random duration and a composite cumulative distribution function is introduced. Second, it is shown that these games can be well defined within the hybrid systems framework and that the problem of finding the optimal strategy can be posed and solved with the methods of hybrid optimal control theory. An illustrative example is given. Keywords: games, hybrid, etc.

1.

Introduction

Game theory as a branch of mathematics investigates conflict processes controlled by many participants (players). These processes are referred to as games. In this paper we focus on the duration of games. In differential game theory it is common to consider games with a fixed duration (finite time horizon) or games with an infinite time horizon. However, in many real-life applications the duration of a game can not be determined a priori but depends on a number of unknown factors and thus is not deterministic any longer. To take account of this phenomenon, a finite-horizon model with random terminal time is considered. For the first time the class of differential games with random duration was introduced in (Petrosyan and Murzov, 1966) for a particular case of a zero-sum pursuit game with terminal payoffs at random terminal time. Later, the general formulation of the differential games with random duration was given in (Petrosyan and Shevkoplyas, 2003). Section 2. provides a brief overview of these results. Apparently, Boukas, Haurie and Michel, in (Boukas et al., 1990), were first to consider an optimal control problem with a random stopping time. Apart from that, in the optimal control theory there have also been papers exploring the idea of random terminal time applied to non-game-theoretical problems. In particular, the problem of the consumer’s life insurance under condition of the random moment of death was discussed in (Yaari, 1965, Chang, 2004). In many cases the probability density function of the terminal time may change depending on some conditions, which can be expressed as a function of time and state. Consider, for instance, the example of the development of a mineral deposit. The probability of a breakdown may depend on the development stage. At the

Differential Games with Random Duration: A Hybrid Systems Formulation

105

initial stage this probability is higher than during the routine mining operation. Therefore one needs to define a composite distribution function for the terminal time as described in Sec. 3. To the best of our knowledge, this formulation has never been considered before despite its obvious practical appeal. In our view, this is caused by the limitations of the generally adopted technique for the computation of optimal strategies. In non-cooperative differential games players solve the optimal control problem of the payoff maximization. One of the basic techniques for solving the optimal control problem is the Hamilton-Jacobi-Bellman equation (Dockner et al., 2000). However, in the above described case a solution (i.e., a differentiable value function) to the HJB equation may not exist. In this case a generalized solution is sought for (the interested reader is referred to Bardi and Capuzzo-Dolcetta, 1997, Vinter, 2000). An alternative to the HJB equation is the celebrated Pontryagin Maximum Principle (Pontryagin et al., 1963) which was recently generalized to a class of hybrid optimal control problems (see, e.g., Riedinger et al., 2003, Shaikh and Caines, 2007, Azhmyakov et al., 2007). In Sec. 3., we show that the optimization problem for a differential game with random terminal time and composite distribution function can be formulated and solved within the hybrid control systems framework. Finally, in the last section an application of our theoretical results is presented. We investigate one simple model of non-renewable resource extraction, where the termination time is a random variable with a composite distribution function. Two different switching rules are studied and a qualitative analysis of the obtained results is presented. 2.

Differential Game Formulation

Consider an N -person differential game Γ (t0 , x0 ) starting at the time instant t0 from the initial state x0 , and with duration T − t0 . Here the random variable T with a cumulative distribution function (CDF) F (t), t ∈ [t0 , ∞), is the time instant at which the game Γ (t0 , x0 ) ends. The CDF F (t) is assumed to be an absolutely continuous nondecreasing function satisfying the following conditions: C1. F (t0 ) = 0, C2. lim F (t) = 1. t→∞

Furthermore, there exists an a.e. continuous function f (t) = F ′ (t), called the probability density function (PDF), such that (Royden, 1988) F (t) =

Zt

t0

f (τ )dτ

∀t ∈ [t0 , ∞).

Let the system dynamics be described by the following ODEs: x˙ = g(x, u1 , . . . , uN ), x ∈ Rm , ui ∈ U ⊆ comp(R),

x(t0 ) = x0 ,

(1)

where g : Rm × RN → Rm is a vector-valued function satisfying the standard existence and uniqueness requirements (see, e.g., Lee and Markus, 1967, Ch. 4). The instantaneous payoff of the i-th player at the moment τ , τ ∈ [t0 , ∞) is defined as hi (x(τ ), ui (τ )). Then the expected integral payoff of the player i, where

106

Dmitry Gromov, Ekaterina Gromova

i = 1, . . . , N is evaluated by the formula Ki (t0 , x, u) =

Z∞ Zt

hi (x(τ ), ui (τ ))dτ dF (t) =

t0 t0

Z∞ Zt

hi (x(τ ), ui (τ ))dτ f (t)dt.

(2)

t0 t0

The Pareto optimal strategy in the game Γ (t0 , x0 ) is defined as the n-tuple of controls u∗ (t) = (u∗1 (t), . . . , u∗n (t)) maximizing the joint expected payoff of players: (u∗1 (t), . . . , u∗n (t))

= argmax u

n X

Ki (t0 , x, u).

(3)

i=1

Hence, the Pareto optimal solution of Γ (t0 , x0 ) is (x∗ (t), u∗ (t)) and the total optimal payoff V (x0 ) is V (x0 , t0 ) =

n X

∗

∗

Ki (t0 , x , u ) =

i=1

n Z∞ Zt X i=1 t

0

hi (x∗ (τ ), u∗i (τ ))dτ f (t)dt.

(4)

t0

For the set of subgames Γ (ϑ, x∗ (ϑ)), with ϑ > t0 , occurring along the optimal trajectory x∗ (ϑ) one can similarly define the expected total integral payoff in the cooperative game Γ (ϑ, x∗ (ϑ)): ∗

V (x (ϑ), ϑ) =

n Z∞ Z t X

hi (x∗ (τ ), u∗i (τ ))dτ dFϑ (t),

(5)

i=1 ϑ ϑ

where Fϑ (t) is a conditional cumulative distribution function defined as Fϑ (t) =

F (t) − F (ϑ) , 1 − F (ϑ)

t ∈ [ϑ, ∞).

(6)

and the conditional probability density function has the following form: fϑ (t) =

f (t) , 1 − F (ϑ)

t ∈ [ϑ, ∞).

(7)

2.1. Transformation of the Integral Functional Below, the transformation procedure of the double integral functional (2) and its reduction to a single integral is described. We obtain this result by changing the order of integration; alternative approaches were presented in, e.g., (Boukas et al., 1990, Chang, 2004). In the following, we assume that the expression under the integral sign is such that the order of integration in (2) is immaterial. Note that in general this is not true (see, for example, 1). A detailed account on this issue is presented in (Kostyunin and Shevkoplyas, 2011). From now on, without loss of generality we set t0 = 0. Consider the integral functional of the i-th player: Z∞ Zt 0

0

hi (τ ) dτ f (t)dt,


107

where hi (τ ) is a shorthand for hi (x(τ ), ui (τ )). Define function a(t, τ ) as follows:

a(t, τ ) = f (t)hi (τ ) · χ{τ 6t} =

f (t)hi (τ ), τ 6 t; 0, τ >t

Taking into account the above mentioned assumption, we interchange the variables of integration in the double integral. Then we get: Z∞ 0

dt

Zt 0

=

f (t)hi (τ )dτ =

Z∞ 0

dτ

Z∞ τ

Z∞ 0

dt

Z∞

a(t, τ )dτ =

0

Z∞ f (t)hi (τ )dt = (1 − F (τ ))hi (τ )dτ. 0

In the general case, the expected payoff of the player i in the game Γ (t0 , x0 ) can be rewritten as: Z∞ Ki (t0 , x, u) = (1 − F (τ ))hi (x(τ ), ui (τ ))dτ. (8) t0

In the same way we get the expression for expected payoff of the player in the subgame Γ (ϑ, x(ϑ)): 1 Ki (ϑ, x, u) = 1 − F (ϑ) 3.

Z∞ (1 − F (τ ))hi (x(τ ), ui (τ ))dτ.

(9)

ϑ

Hybrid Formulation of a Differential Game

In this section we give the definition of a hybrid control problem and the associated hybrid optimal control problem. It is shown that the differential game with a composite CDF (CCDF) introduced in Subsection 3.2. fits perfectly in the hybrid framework. Hence, a hybrid differential game as well as a number of particular cases are considered, the respective optimal control problems are defined, and the solution strategies are proposed. 3.1. Hybrid Optimal Control Problem Below, we give the definition of a hybrid system. For more details, the interested reader is referred to (Riedinger et al., 2003, Shaikh and Caines, 2007), as well as (Azhmyakov et al., 2007). Definition 1. The hybrid system HS is defined as a tuple HS = (Q, X, U, f, γ, Φ, q0 , x0 ), where – Q = {1, . . . , N } is the set of discrete states, XRl is the continuous state, Uq ⊂ Rm , q ∈ Q are the admissible control sets, which are compact and convex, and Uq := {u(·) ∈ Lm ∞ (0, tf ) : u(t) ∈ Uq , a.e. on[0, tf ]} represent the sets of admissible control signals.

108


– q0 ∈ Q and x0 ∈ X are the initial conditions. – fq : X × U → X is the function that associates to each discrete state q ∈ Q a differential equation of the form x(t) ˙ = fq (x(t), u(t)).

(10)

– γq,q′ : X → Rk is the function that triggers the change of discrete state. Let q ∈ Q be the current discrete state and x(t) be the state trajectory evolving according to the respective differential equation (10). The transition to the discrete state q ′ ∈ Q occurs at the moment χ when γq,q′ (x(χ)) = 0. The set Γq,q′ = {x ∈ X|γq,q′ (x) = 0} is referred to as the switching manifold. – When the discrete state changes from q to q ′ , the continuous state might change discontinuously. This change is described by the jump function Φq,q′ : X → X. Let χ be the time at which the discrete state changes from q to q ′ , then the continuous state at t = χ is described as x(χ) = Φq,q′ (x(χ− )), where x(χ− ) = lim x(t). t→χ−0

Definition 2. A hybrid trajectory of HS is a triple X = (x, {qi }, τ ), where x(·) : [0, T ] → Rn , {qi }i=1,...,r is a finite sequence of locations and τ is the corresponding sequence of switching times 0 = t0 < · · · < tr = T such that for each i = 1, . . . , r there exists ui (·) ∈ Ui such that:

– x(0) = x0 and xi (·) = x(·)|[ti−1 ,ti ) is an absolutely continuous function in [ti−1 , ti ) continuously extendable to [ti−1 , ti ], i = 1, . . . , r. – x˙ i (t) = fqi (xi (t), ui (t)) for almost all t ∈ [ti−1 , ti ], i = 1, . . . , r. – The switching condition xi (ti ) ∈ Γqi ,qi+1 along with the jump condition xi+1 (ti ) = Φqi ,qi+1 (xi (ti )) are satisfied for each i = 1, . . . , r−1.

Using the introduced notation we can state a hybrid optimal control problem and characterize an optimal solution to this problem. Let the overall performance of HS be evaluated by the following functional criterion: J(x0 , q0 , u) =

Zti r X

Lqi (xi (t), ui (t), t)dt,

(11)

i=1 t i−1

where Lqi : X × U × R>0 , qi ∈ Q, are twice continuously differentiable functions. Assume that the sequence of discrete states q ∗ is given. Then the necessary conditions for a solution (x∗ , q ∗ , τ, u∗ ) to HS to minimize (11) is given by the following theorem. Theorem 1 (Riedinger et al., 2003). If u∗ (t) and (x∗ (t), q ∗ (t), τ ) are the optimal control and the corresponding hybrid trajectory for HS, then there exists a piecewise absolutely continuous curve p∗ (t) and a constant p∗0 > 0, (p∗ , p∗0 ) 6= (0, 0) such that – The tuple (x∗ (t), q ∗ (t), p∗ (t), u∗ (t), τ ) satisfies the associated Hamiltonian system ∂H x(t) ˙ = ∂pqi (x∗ (t), p∗ (t), u∗ (t)), p(t) ˙ =−

∂Hqi ∂x

(x∗ (t), p∗ (t), u∗ (t)),

t ∈ [ti−1 , ti ], i = 1, . . . , r

(12)

Differential Games with Random Duration: A Hybrid Systems Formulation where

109

Hqi (x∗ (t), p∗ (t), u∗ (t)) = = p∗0 Lqi (xi (t), ui (t), t) + p∗ (t)fqi (xi (t), ui (t)).

– At any time t ∈ [ti−1 , ti ), the following maximization condition holds: Hqi (x∗ (t), p∗ (t), u∗ (t)) = sup Hqi (x∗ (t), p∗ (t), u(t)).

(13)

u(t)∈U

– At the switching time ti , there exists a vector π ∈ Rn such that the following transversality conditions are satisfied: p∗ (t− i )=

n P

pk (ti )

k=1

Hqi−1 (t− i )

∂Φk q

i ,qi+1

∂xj

= Hqi (ti ) −

−

n P

k=1

πki

n P

(t− i )+

n P

πki

k ∂γi,i+1 − ∂xj (ti ),

k=1 ∂Φk i,i+1 pk (ti ) ∂t (t− i )−

(14)

k=1 − ∂t (ti )

k ∂γi,i+1

3.2. Composite Cumulative Distribution Function Let t0 be the initial time, Fi (t), i = 1, . . . , N be a set of CDFs characterizing different modes of operation and satisfying, along with C1 and C2, the following property: C3. The CDFs Fi (t) are assumed to be absolutely continuous nondecreasing functions such that each CDF converges to 1 asymptotically, i.e., Fi (t) < 1 ∀t < ∞.

Furthermore, let τ = {τi } s.t. t0 = τ0 < τ1 < · · · < τN −1 < τN = ∞ be an ordered sequence of time instants at which the switches between individual CDFs occur. The composite CDF Fσ (t) is defined as follows:  t ∈ [τ0 , τ1 ),   F1 (t), Fσ (t) = αi (τi )Fi+1 (t) + βi (τi ), t ∈ [τi , τi+1 ), (15)   1 6 i 6 N − 1,

where αi (τi ) =

Fσ (τi− )−1 Fi+1 (τi )−1 ,

and βi (τi ) = 1 −

Fσ (τi− )−1 Fi+1 (τi )−1 .

the left limit of Fσ (t) at t = τi− , i.e., Fσ (τi− ) =

lim

t→(τi −0) Fσ′ (t) and

Here, Fσ (τi− ) is defined as

Fσ (t).

The composite PDF is defined as fσ (t) = has the following form:   t ∈ [τ0 , τ1 ),  f1 (t), fσ (t) = αi (τi )fi+1 (t), t ∈ [τi , τi+1 ), (16)   1 6 i 6 N − 1.

Proposition 1. Given a set of CDFs Fi (t), 1 6 i 6 N , such that C1-C3 hold for each Fi (t). Then the composite CDF Fσ defined by (15) satisfies C1-C3. Proof. See Appendix.

From Lemma 1 it follows that fσ (t) has well-defined finite left and right limits at points τi , 1 6 i 6 N − 1, fσ (τi− ) =

lim

t→(τi −0)

fσ (t),

fσ (τi+ ) =

lim

t→(τi +0)

fσ (t),

110


which are not necessarily equal, and is continuous otherwise. The optimization problem (3) for the CCDF (15) can be written taking into account the transformation (8): u∗ (t) = argmax u

= argmax u

n X

Ki (x0 , t0 , u1 , . . . , un ) =

i=1 n ZτN X i=1 τ

(17)

(1 − Fσ (τ ))hi (x(τ ), ui (τ ))dτ.

0

3.3. Hybrid Differential Game The optimization problem (1), (17) can hardly be solved in a straightforward way due to the special structure of the composite CDF Fσ (t). However, this problem can be readily formulated as a hybrid optimal control problem. We know that Fσ (t) is defined by a number of elementary CDFs, (15), and the switching instants τi , i = 1, . . . , N . There are two types of switching instants τi corresponding to a) Time-dependent switches; b) State-dependent switches. In the first case, the sequence τ is given; the remaining degrees of freedom are the values of the state at the switching times τi , i.e., x(τi ). In the second case, the switching times τi are determined as the solutions to the equations γi (xi (τi− )) = 0, i.e., the regime changes as the state crosses the switching manifold defined by the map γ : Rn×l → Rk . We assume that the sequence of operation modes (i.e., discrete states) is fixed a priori. Therefore, there is no need in performing any combinatorial optimization and the problem of determining the optimal strategy can be completely formulated within the framework of hybrid optimal control as shown below. Time-dependent case To apply the results of Theorem 1 the problem (1), (17) has to be modified. Namely, we extend the system (1) by one differential equation modelling the CDF Fσ . Thus, on each interval [τi−1 , τi ) the differential equations the payoff function are written as x˙ = g(x, u), x˙ σ = f¯i (t), Ki (t0 , x, u) =

Rτi

τi−1

where h(x(t), u(t)) =

n P

(18) (1 − xσ (t))h(x(t), u(t))dt,

hi (xi (t), ui (t)) is the total instantaneous payoff, and

i=1

f¯i = fσ [τi−1 ,τi ) .

Note that since the switching times are fixed a priory, functions f¯i are well defined. The respective Hamiltonian functions are Hi (xt , u, p0 , pt ) = p0 (1 − xσ )h(x, u) + hp, g(x, u)i + pσ f¯i (t), where p0 = −1, xt (t) = [x(t), xσ (t)]′ , and pt (t) = [p(t), pσ (t)]′ . Solving the Hamiltonian equations (12) together with the maximization condition (13) one obtains


111

a solution to (18). To solve (12), a number of boundary conditions has to be defined. First, these are initial and end point conditions x(τ0 ) = x0 , x(∞) = 0, and xσ (τ0 ) = 0, xσ (∞) = 1. Second, there are constraints imposed on the state and adjoint variables at switching times τi , i = 1, . . . , N − 1: x(τi− ) = x(τi ), p(τi− ) = p(τi ), xσ (τi− ) = xσ (τi ), pσ (τi− ) = pσ (τi ), Hi−1 (xt (τi− ), pt (τi− ))

(19)

= Hi (xt (τi ), pt (τi )).

With these conditions the problem becomes well-defined. We note that the righthand sides of the differential equations in (18) depend on t. Therefore, on each interval [τi−1 , τi ), an additional condition H(x(τi ), u∗ (τi ), p(τi )) = 0 has to be added. For details on the time-variant Maximum Principle see, e.g., (Pontryagin et al., 1963, Ch. 1). State-dependent case This case is slightly more involved compared to the previous one. The problem is that the switching instants τi are defined from the solution of the switching condition γi,i+1 = x(τi ) − x ˜i = 0 and, thus, not defined a priori. Looking at (16) one can notice that the composite PDF depends on τi which means that the functions f¯i (t) in (18) are not well-defined. Therefore, the equations (18) are modified as shown below x˙ = g(x, u), x˙ σ = xα fi (t), x˙ α = 0,

xα (τ0 ) = 1 Rτi Ki (t0 , x, u) = (1 − xσ (t))h(x(t), u(t))dt.

(20)

τi−1

with Hamiltonian functions modified accordingly

Hi (xt , u, pt ) = p0 (1 − xσ )h(x, u) + hp, g(x, u)i + pσ xα fi (t). The particularity of this model is that along with the mentioned switching condition γi,i+1 = x(t)−˜ xi , there is a jump function associated with xα . When a switching between discrete states occurs, the state changes discontinuously according to the jump function [x(τi ), xσ (τi ), xα (τi )] = Φi,i+1 (x, xσ ) = h i xσ (τi− )−1 = x(τi− ), xσ (τi− ), Fi+1 (τi )−1 . The intermediate conditions (19) have to be rewritten to take into account the switching and jump functions: x(τi ) = x(τi− ),

p(τi− ) = p(τi ) + π,

xσ (τi ) = xσ (τi− ),

pσ (τi− ) =

xα (τi ) =

xσ (τi− )−1 Fi+1 (τi )−1 ,

pσ (τi ) Fi+1 (τi )−1 ,

pα (τi− ) = 0.

Furthermore, the Hamiltonian function is not continuous any longer since the jump function is time-variant. The condition on the Hamiltonian at switching instants τi

112


is hence Hqi−1 (τi− ) = Hqi (τi ) + pα (τi )

(xσ (τi ) − 1)fi+1 (τi ) − (ti ). (Fi+1 (τi ) − 1)2

The end point conditions remain unchanged. With all conditions imposed, the optimization problem (20) becomes well-defined and can be solved using standard procedures as illustrated in the following section. 4.

Example

To illustrate the presented approach we consider a simple example of finding a Pareto optimal solution in the game of resource extraction with N players and two operation modes. Note that despite its obvious simplicity, this example can demonstrare rather non-trivial behaviour. The two CDFs are F1 (t) = 1 − exp(−λ1 t) and F2 (t) = 1 − exp(−λ2 t) with λ1 , λ2 > 0 and the switching time τ . The resulting CCDF Fσ (t) is defined as Fσ (t) =

(

1 − exp(−λ1 t), 1−

exp(−λ1 τ ) exp(−λ2 τ )

t ∈ [0, τ ),

exp(−λ2 t), t ∈ [τ, ∞).

(21)

We consider two exponential CDF with rate parameters λ = 0.01 and λ = 0.1. The corresponding CDFs are shown in Fig. 1.

Fig. 1: Two exponential distributions


113

The system dynamics is described by a first order DE: x(t) ˙ =−

N X

ui (t),

i=1

x(0) = x0 , x(∞) = 0, ui (·) ∈ [0, umax],

(22)

where u(t) is the rate of extraction. The initial amount of resource is set to x(0)=100 and x(∞) is routinely defined as x(∞) = lim x(t). t→∞

The instanteneous payoff function is chosen as hi (x(t), u(t)) = ln(ui (t)). The optimal control problem is thus defined to be min

N X i=1

Z∞ N X Ki (x, u) = − (1 − Fσ (s)) ln(ui (s))ds.

(23)

i=1

0

Before proceeding to the hybrid formulation, we present the solution to the optimal control problem (23) defined over a single interval. This result is of independent interest, as this class of optimization problems is fairly common for a wide range of resource extraction applications (see, e.g., Dockner et al., 2000). 4.1. Optimal Solution to a Single Mode Optimal Control Problem Consider a more general version of the problem (22), (23) on the interval [t0 , tf ] ⊂ [0, ∞) ∪ {∞} with the boundary conditions x(t0 ) = x0 , x(tf ) = xf , x0 > xf . Moreover, we assume that there is one single (non-composite) CDF F (t) such that F (tf ) = 1. The Hamiltonian is written as H = −ψ

N X i=1

ui (t) + ψ0 (1 − F (t))

N X

ln(ui (t)),

ψ0 = 1.

i=1

The differential equation for the adjoint variable ψ is ∂H = 0, ψ˙ = − ∂x whence we conclude that ψ(t) = ψ ∗ = const for all t. The optimal controls u∗i are found from the first order extremality condition ∂H ∂ui = 0: 1 u∗i (ψ, t) = ∗ (1 − F (t)). ψ 2

Moreover, u∗i maximize H as follows from ∂∂uH2 = −ψ0 (1 − F (t)) u12 < 0. i i The value of ψ ∗ is determined from the boundary condition x(tf ) = xf . Solving (22) and taking into account this condition we find ψ ∗ as N ψ = x0 − xf ∗

Ztf

t0

(1 − F (t))dt

and hence, the optimal controls take the following form: u∗i (t) = N

Rtf

t0

x0 − xf (1 − F (τ ))dτ

(1 − F (t)).

(24)

114


The state x(t) of the system (22) with the control (24) is ∗

x (t) = x0 −

Zt

t0

N

Rtf

t0

x0 − xf (1 − F (τ ))dτ

(1 − F (s))ds.

Note that the optimal control u∗ (t) exists if the integral in the denominator conRt verges, i.e. t0f (1 − F (τ ))dτ < ∞ (which might not be the case if tf = ∞). Taking into account the Bellman optimality principle, the optimal controls u∗i (t) can be expressed as functions of the current state: u∗i (t, x(t)) =

x(t) − xf (1 − F (t)). Rtf N (1 − F (τ ))dτ

(25)

t

Hence, from (9) and (25), the value function V (t, x(t)) is given by I(t) V (t, x(t)) = − ln 1 − F (t) where I(t) = N

Rtf t

x(t) − xf I(t)

1 − 1 − F (t)

Ztf

(1−F (s)) ln(1−F (s))ds, (26)

t

(1 − F (τ ))dτ .

Finally, in the framework of the resource extraction problem one may need to compute the expectation of the state x(t) at the end of the exploration process:   E(x(t)) =

Rtf

t0

Rt  f (t) x0 −

t0 N

= xf + (x0 − xf )

tf R

t0 tf R

x0 −xf

(1−F (τ ))dτ

 (1 − F (s))ds dt =

F (t)(1−F (t))dt

t0

N

tf R

. (1−F (τ ))dτ

t0

In the following, we will assume N = 1 to simplify the notation. 4.2. Time-Dependent Case We assume that the switching time τ is fixed and equal to τs and the state at time τs is x(τs ) = xs . Hence, the optimal control problem can be decomposed into two problems, on the intervals I1 = [0, τs ) and I2 = [τs , ∞). The optimal control on the first interval [0, τ ) is u∗ (t) =

(x0 − xs ) (x0 − xs )λ1 (1 − Fσ (t)) = exp(−λ1 t), Rτs (1 − exp(−λ1 τs )) (1 − Fσ (s))ds

t ∈ I1 .

0

In the same way we define the optimal control on the second interval: xs xs λ2 u∗ (t) = R∞ (1 − Fσ (t)) = exp(−λ2 t), exp(−λ2 τs ) (1 − Fσ (s))ds τs

t ∈ I2 .


115

Both expressions contain the unknown switching state xs . Solving the optimal control problem, xs is found as a function of the switching time τs : xs =

λ1 x0 . λ2 exp(λ1 τs ) − (λ2 − λ1 )

In Fig. 2, the dependence of the switching state xs on the switching time τs is shown for two sequences of operation modes.

Fig. 2: Dependence of the optimal switching state x∗s = x(τs ) on the switching time τs . The continuous line corresponds to the case λ1 = 0.01, λ2 = 0.1, the dotted one – λ1 = 0.1, λ2 = 0.01

Informally, one can describe these two cases as the "safe" mode first (λ1 = 0.01, λ2 = 0.1), and the "dangerous" mode first (λ1 = 0.1, λ2 = 0.01). The second case is of particular interest. It turns out that for a small τs the optimal strategy is to preferrably extract during the "safe" mode. However, as τs grows, the risk that the system breaks down grows and so, the expected gain in the payoff is compensated by the risk of an abrupt interruption of the game. As τs reaches a certain value the optimal strategy becomes to extact as much as possible during the "dangerous" phase as there is only a slight hope that the process will "survive" until the switching time τs . Interesting to note that the switching time at which the optimal strategy changes is determined from the equation λ21 (exp(λ2 τs ) − 1) − λ22 (exp(λ1 τs ) − 1) = 0.

116


4.3. State-Dependent Case In the second case, we assume that the switching time τ is determined from the condition x(τ ) = ax0 , a ∈ [0, 1], where the parameter a describes the extent of exploration at which the regime changes (i.e., a switching occurs). As in the previous case, the optimal control problem can be decomposed into two problems, on the intervals [0, τ ) and [τ, ∞). Now consider the first interval [0, τ ). The optimal control is u∗ (t) = Rτ 0

x0 − ax0 (1 − Fσ (s))ds

(1 − Fσ (t)) =

x0 (1 − a)λ1 exp(−λ1 t). (1 − exp(−λ1 τ ))

The optimal control on the second interval is ax0 ax0 λ2 u∗ (t) = R∞ (1 − Fσ (t)) = exp(−λ2 t). exp(−λ2 τ ) (1 − Fσ (s))ds τ

The remaining step is to determine the value of the optimal switching time τ ∗ , which is equal to λ1 x0 (1−a) ln 1 + λ2 −λ1 +λ1 ln(aλ2 x0 ) λ2 e τ∗ = . λ1 The limit case (α = 0) looks as follows: ln λ ex0 + 1 τs = λ

We compute the optimal switching time for different values of a and for the two different sequences of modes. We remind that the parameter a determines the state, and implicitly the time instant, at which the switching between two modes occur. The resulting dependencies are shown in Fig. 3, 4. 5.

Conclusions

A new class of differential games with random duration and a composite cumulative distribution function has been introduced. It has been shown that these games can be well defined within the hybrid systems framework and that the problem of finding the optimal strategy can be posed and solved with the methods of hybrid optimal control theory. An illustrative example along with a qualitative analysis of the results have been presented. The further work on the topic will be devoted to the analysis of the cooperative behaviour in this class of differential games. In particular, we will study the impact which the change of mode may have of the coalition agreement of the players. Appendix Proof of Proposition 1 Property C1 is satisfied since it is satisfied for the function F1 (t): Fσ (t0 ) = F1 (t0 ) = 0


117

Fig. 3: Dependence of the switching time τs on the parameter a for λ1 = 0.1; λ2 = 0.01

Property C2 follows from lim FN (t) = 1 and from the definition of αi (τi ) and t→∞

βi (τi ): lim Fσ (t) = αN −1 (τN −1 ) lim FN (t) + βN −1 (τN −1 ) =

t→∞

t→∞

=

− Fσ (τN −1 )−1 FN (τN −1 )−1

·1+1−

− Fσ (τN −1 )−1 FN (τN −1 )−1

= 1,

where τN −1 is a fixed switching time. To show that Property C3 holds true for Fσ , we first show that Fσ is continuous. This follows from the equality of left and right limits at t = τi : Fσ (τi− )−1 Fσ (τi− )−1 lim Fσ = lim Fi+1 (τi )−1 Fi+1 (t) + 1 − Fi+1 (τi )−1 = t→τi +

=

t→τi +

Fσ (τi− )−1

Fσ (τi− )−1 Fi+1 (τi )−1 Fi+1 (τi )+1− Fi+1 (τi )−1

= Fσ (τi− ) = lim Fσ t→τi −

Next, to demonstrate that the function Fσ (t) is non-decreasing we consider two cases: i) t1 , t2 ∈ [τi , τi+1 ), i = 0, . . . , N − 1. Then, Fσ (t1 ) 6 Fσ (t2 ) as Fσ (t) is proportional to Fi+1 (t) on [τi , τi+1 ) and Fi+1 (t) is non-decreasing. ii) t1 ∈ [τi , τi+1 ), t2 ∈ [τj , τj+1 ), i, j = 0, . . . , N − 1, i < j. Taking into account the continuity property, we have Fσ (t1 ) 6 Fσ (τi+1 ) 6 . . . 6 Fσ (τj ) 6 Fσ (t2 ).

118


Fig. 4: Dependence of the switching time τs on the parameter a for λ1 = 0.01; λ2 = 0.1

Thus, the function Fσ (t) is non-decreasing. Finally, we show that Fσ (t) is absolutely continuous. This is equivalent to the following requirement (Royden, 1988): ∀ε > 0, ∃δ > 0 such that for P any finite set of non-intersecting intervals (x , y ) from [t , ∞), the inequality |yk − xk | 6 δ k k 0 P implies |Fσ (yk ) − Fσ (xk )| 6 ε. We use the fact that the functions Fi (t), i = 1, . . . , N are absolutely continuous. ε Then, for any i = 1, . . . , N and for any εi = 2N > 0, there exists δi > 0 such that for any finite set of non-intersecting intervals (x (i,k) , y(i,k) ) from [τi−1 , τi ], satisfying P |y(i,k) − x(i,k) | 6 δi , holds k

X k

|Fi (y(i,k) ) − Fi (x(i,k) )| 6 εi .

(27)

Let δ = min(δi , (τj − τj−1 )), i, j = 1, .P . . , N . For an arbitrary finite set of nonintersecting intervals (xk , yk ), satisfying |yk − xk | 6 δ there are two possible k

variants:

i) Intervals (xk , yk ) are proper subsets of the partition intervals [τi , τi+1 ]. Then, using the absolute continuity property of Fi and summing over all partition intervals we get X k

|Fσ (yk ) − Fσ (xk )| =

N X X i=1

k

|Fi (yk ) − Fi (xk )| < N εi = ε,


119

whereas the following convention is employed: |Fi (a) − Fi (b)| = 0, if (a, b) ∩ [τi , τi+1 ] = ∅. ii) Some intervals of the finite set (xk , yk ) may include switching instants τi(k) . According to the definition of δ, an interval (xk , yk ) can intersect with at most two partition intervals. Therefore, one can represent (xk , yk ) as a union of two intervals (xk , τi(k) ) ⊂ (τi(k)−1 ) ⊂ (τi(k) , τi(k)+1 P, τi(k) ), and (τi(k) , ykP P). In this way, we can subdivide the sum |y − x | into two: |y − x | = |yk − τi(k) | + k k k k P |xk − τi(k) | < δ. Summing over all partition intervals and using the triangle inequality we get P k

6

|Fσ (yk ) − Fσ (xk )| = N P P

i=1 k

N P P

i=1 k

|Fi (yk ) − Fi (xk )| 6

|Fi (yk )−Fi (τi(k) )| + |Fi (τi(k) )−Fi (xk )|
I2 , I1 , I2 ≥ 0, Ij , j = 1, 2 are agents’ incames. There are three equilibI2 riums in the game (H, H), (B, B), (e x, x e) x e= . Two pure strategies B and I1 + I2 H are evolutionary stable, H, B ∈ ∆ESS .

124 3.

Gubar Elena Extended games

Consider an extension of basic models. Suppose the small share of agents, which can recognize opponents’ behavior during their meeting, invades on the stock market. This type of agents we will call rational agents or rationalist. We add new strategy E to each basic game, which describes new type of behavior of the market agents. Rationality of the player means that agents, which use strategy E can recognize actions of his opponents and adjust their behaviors in compliance with actions of the opponents. In each basic game we suppose that rational agents use their best responses as strategies and if rational agent meets another rational agent, then both play Nash equilibrium strategies. For each situation on the stock market construct new payoff matrices. Situation A: In basic game we have three equilibriums, hence extended game will have three variants, which describe agents preferences. We present payoff matrix which depends on the equilibriums profiles: H B E H (I, I) (3I/2, 0) (I, I) B (0, 3I/2) (2I, 2I) (2I, 2I) E (I, I) (2I, 2I) (u(e x, x e), u(e x, x e))

where values (u(e x, x e), u(e x, x e)) are players payoffs in the Nash equilibrium situation. In basic game we have three different equilibriums hence we get three variations of the extended game. Situation B: In situation B after invasion of rational agents,we get following payoff matrix. Basic game in situation B has three equilibriums hence extended game will have three modifications: B S E B ((I − C)/2, (I − C)/2) (I, 1) (1, 1) S (1,I) (I/2, I/2) (I, I) E (1, 1) (I, I) (u(e x, x e), u(e x, x e))

where as in previous case values (u(e x, x e), u(e x, x e)) is agents payoff on the equilibrium strategies. Strategy E is the strategy of rational agents, which can identify behavioral type of their opponents. Situation C: In situation C we also have three modifications of payoff matrix, depending on the various agent’s payoffs in Nash equilibriums profiles. B H E B (I1 , I1 ) (0, 0) (I1 , I1 ) H (0, 0) (I2 , I2 ) (I2 , I2 ) E (I1 , I1 ) (I2 , I2 ) (u(e x, x e), u(e x, x e))

where u(e x, x e) is equal to agent’s payoff on corresponding equilibrium strategies.

Differential Simulations of Evolutionary Models of a Stock Market 4.

125

Replication by Imitation

For all situations A, B, C we analyze, which behavioral type will prevail on the market during long-rum period with suggestion that in each situation at initial time moment small share of rational players are invaded. For all these models we will consider selection dynamics arising from adaptation by imitation. In all situations we suppose that all agents in the large group are infinitely live and interact forever. This assumption can be interpreted in following way, if one agent physically exits from the market, then he is replaced by another one. Each agent has some pure strategy for some time interval and then reviews his strategy and sometimes changes the strategy. There are two basic elements in this model (Weibull, 1995). The first element is time rate at which agents in the group review their strategy choice. The second element is choice probability at which agents change their strategies. Both elements depend on the current group state and on the performance of the agent’s pure strategy. Let K = {H, B, S, E} is the set of agents pure strategies. In each of the previously described situations agents match at random in total group and each agent use one of pure strategy from the set K. Player with pure strategy i will be called as i-strategist. Denote as ri (x) an average review rate of the agent, who uses pure strategy i in the group state x = (xH , xB , xE ). Variable pij (x) is probability at which i-strategist switches to some pure strategy j, i, j ∈ K. Here pi (x) = (p1i (x), p2i (x), p3i (x)), i = H, B, E, S is the resulting probability distribution over the set of pure strategies and depends on the population state. Value pii (x) is probability that a reviewing i-strategist does not change his strategy. Consider imitation process generally in finite large group of agents. Suppose that each reviewing agent samples another agent at random from the group with equal probability for all agents and observes with some noise the average payoff to his own and to the sampled agent’s payoff. If payoff of the samples agents is better then his own he can switch to the sampled agent’s strategy. In general case the imitation dynamics is described by the formula: X x˙ i = xj rj (x)pij (x) − ri (x)xi , i ∈ K. (1) j∈K

In this paper we use special variation of the imitation dynamics of successful agents. 5.

Imitation of Pairwise Comparison

Suppose that each agent samples another stock agent from the total group with equal probability for all agents and observes the average payoff to his own and the sampled agent’s strategy. When both players show their strategies then player who uses strategy i gets payoff u(ei , x) + ε and player, who uses strategy j gets u(ej , x) + ε′ , where ε, ε′ is random variables with continuously probability distribution function φ. The random variables ε and ε′ can be interpreted as individual preference differences between agents in the market. Each agent can get various preferences, i.e. agent, which receive the large block of shares or the control of target company, can be more satisfied, because he can influence to the company or have additional profit. Other agents, which hold own assets and receive only fixed payoff

126

Gubar Elena

can be less satisfied of their profit. Use as distribution function φ(z) = exp(αz), α ∈ R. Players compare their payoffs: if the payoff of the sampled agent is better than of the reviewing agent, he switches to the strategy of the sampled agent. In other words, if this inequality u(ej , x) + ε′ > u(ei , x) + ε is justify for player with pure strategy i then he switches to the strategy j. For the general case the following formula describes the imitation dynamics of pairwise comparison:   X x˙ i =  xj (φ[u(ei − ej , x)] − φ[u(ej − ei , x)]) xi , i ∈ K. (2) j∈K

To simplify calculations use certain numerical values for the models parameters and construct dynamics for each extended game. Situation A: Using following values for incomes, which are I = 2, α = 1 and values of parameter u are u = 4, 2, 8/3 then we get three different systems of differential equations, corresponding to various cases of the extended games: x˙ H = (xB (e(4xH +xB −2) − e(−4xH −xB +2) )+ xE (e(−xB +(2−u)xE ) − e(xB +(−2+u)xE ) ))xH ; x˙ B = (xH (e(−4xH −xB +2) − e(4xH +xB −2) )+ xE (e(−2xH +(4−u)xE ) − e(2xH +(−4+u)xE ) ))xB ; x˙ E = (xH (e(xB +(−2+u)xE ) − e(−xB +(2−u)xE ) )+ xB (e(2xH +(−4+u)xE ) − e(−2xH +(4−u)xE ) ))xE . Situation B: Let values for income and costs are: I = 2, C = 4, α = 1 then we get three systems of differential equations with values of parameter u: u = 4, 3, 1 in various cases: x˙ H = (xB (e(xH +5xS −3) − e(−xH −5xS +3) )+ xE (e(−2xH +(1−u)xE ) − e(2xH +(−1+u)(1−xH −xS )) ))xH ; x˙ S = (xH (e(−xH −5xS +3) − e(xH +5xS −3) )+ xE (e(−2xB +(4−u)xE ) − e(2xS +(−4+u)xE ) ))xB ; x˙ E = (xH (e(2xH +(−1+u)xE ) − e(−2xH +(1−u)xE ) )+ xS (e(2xS +(−4+u)xE ) − e(−2xS +(4−u)xE ) ))xE .

Differential Simulations of Evolutionary Models of a Stock Market

127

Situation C: Let incomes are I1 = 2, I2 = 1, α = 1, then systems of differential equations that define pairwise comparison dynamics are following: x˙ B = (xB (e(xH −2xB +1) − e(−xH +2xB −1) )+ xE (e(−xB +(2−u)xE ) − e(xB +(−2+u)xE ) ))xB ; x˙ H = (xH (e(−xH +2xB −1) − e( xH − 2xB + 1))+ xE (e(−2xH +(1−u)xE ) − e(2xH +(−1+u)xE ) ))xH ; x˙ E = (xH (e(xB +(−2+u)xE ) − e(−xB +(2−u)xE ) )+ xB (e(2xH +(−1+u)xE ) − e(−2xH +(1−u)xE ) ))xE ; Values of parameter u: u = 2, 1, 4/3. For each system we get numerical solution using next initial states: xE = 0.1, xH = 0.01, 0.02, . . . , 0.98, xB = 0.98, 0.97, . . . , 0.01 and xE = 0.1, xB = 0.01, 0.02, . . . , 0.98, xS = 0.98, 0.97, . . . , 0.01 solution trajectories are presented in Table 1, where rows represent different situations on the stock market and columns correspond to various modifications of extended games. We get that in situation A for all cases of extended games behavioral type "invest" and behavioral type of rationalist are preferable and that strategies will survive in the long-run period, however in case 2 we can see that behavior "to hold" also can be preserved. In situation B in case 1 only rational agents prevail on the market, in case 2 and 3 mixture of agents, who sell their blocks of shares and rationalists will survive and situation (xB , xS , xE ) = (0, 1/3, 2/3) will be the rest point of the system. In situation C we get different variants of prevailed behaviors. In case 1 solutions trajectories aspire to states xB and xE and on the stock market "investors" and "rationalists" will be survived in long-run period. In case 2 behaviors "invest" and partly "hold" will be conserved and and in case 3 only state xB is stable. Table 1: Imitation dynamics of pairwise comparison. Case 1

Case 2

Case 3

u=4

u=2

u = 8/3

A

128

Gubar Elena

B u=4

u=3

u=1

u=2

u=1

u = 4/3

C

6.

Imitation of successful agents

Suppose that the choice probabilities pji (x) are proportional to popularity of j’s strategy xj , and the proportionality factor is described by the currently payoff to strategy j. It is thus as if agent observes other agents choices with some small noise and would imitate another agent from the population with a higher probability for relatively more successful agents. Denote the weight factor that a reviewing agent with strategy i attaches to pure strategy j as ω[u(ei , x), x] > 0, where ω is a continuous function. Then ω[u(ej , x), x]xj pji = P ω[u(ej , x), x]xp p∈K

Selection dynamics for that model is described by following equations: 



  X Pω[u(ej , x), x]xj x˙i =  − 1 xi . ω[u(ej , x), x]xp j∈K

(3)

p∈K

As in earlier case we have some additional assumptions for choice probability such as these is not that a reviewing agent necessarily knows the current average payoff to all pure strategies. It is sufficient that some agent have some possibly noisy empirical information about payoff to some pure strategies in current use and on average more agents prefer to imitate an agent with higher payoff than one with lower average payoff. In this paper accept as weight function ω = exp(α−1 u(ei , x)), where α is small noise of observation and get following expression (Sandholm, 2008, 2010):

Differential Simulations of Evolutionary Models of a Stock Market xi exp(α−1 u(ei , x)) x˙i = P , i, k ∈ K. xk exp(α−1 u(ek , x))

129

(4)

k∈K

To simplify calculations, as in previous section, we use certain numerical values for the models’ parameters and construct dynamics for each extended game. Situation A: Let agents’ income is I = 2 then we get three different systems of differential equations, corresponding to various cases of the extended games: −1

x˙ H =

xH e(α

(xB +2))

−1 (2x +4x +x u)) ; H )) + x e(α H B E + xB e E −1 (α (4−4xH )) xB e ; x˙ B = (4−4xH )) + x e(α−1 (2xH +4xB +xE u)) xH e(α−1 (xB +2)) + xB e(α−1 E −1 xE e(α (2xH +4xB +xE u)) x˙ E = ; −1 xH e(α (xB +2)) + xB e(α−1 (4−4xH )) + xE e(α−1 (2xH +4xB +xE u))

xH e

(α−1 (x

B +2))

(α−1 (4−4x

Values of parameter u: u = 4, 2, 8/3. Situation B: Let agents’ income is I = 2 and costs are C = 4 then we get three systems of differential equations describe imitation dynamics of successful agents with values of parameter u: u = 4, 2, 1. −1

x˙ H =

xH e(α (−2xH +3xS +1)) ; xH e(α−1 (−2xH +3xS +1)) + xS e(α−1 (−3xH −2xS +4)) + xE e(α−1 (xH +4xS +xE u))

x˙ S =

xS e(α (−3xH −2xS +4)) ; −1 (−2x +3x +1)) (α H B xH e + xB e(α−1 (−3xH −2xS +4)) + xE e(α−1 (xH +4xS +xE u))

−1

−1

xE e(α (xH +4xS +xE u)) x˙ E = ; −1 xH e(α (−2xH +3xS +1)) + xS e(α−1 (−3xH −2xS +4)) + xE e(α−1 (xH +4xS +xE u)) Situation C: Let incomes are I1 = 2 and I2 = 1 then for different values of parameter u: u = 2, 1, 4/3. we have following systems of differential equations: −1

x˙ H =

xH e

(α−1 (2−2x

xH e(α (2−2xB )) ; −1 (1−x )) B )) + x e(α H + xE e(α−1 (2xH +xB +xE u)) B −1

x˙ B =

xB e(α (1−xH )) ; −1 (2−2x )) (α B xH e + xB e(α−1 1−xH ) + xE e(α−1 (2xH +xB +xE u))

x˙ E =

xE e(α (2xH +xB +xE u)) ; −1 xH e(α (2−2xB )) + xB e(α−1 (1−xH )) + xE e(α−1 (2xH +xB +xE u))

−1

Table 2 contains pictures with solution trajectories for each extended game, as in Table 1 rows represent different situations on the stock market and columns correspond to various modifications of extended games.

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Gubar Elena

We get that in situation A in case 1 all trajectories converge to stationary state xE and in cases 2 and 3 the state xH is stable and according behavioral type "hold" the blocks of shares will prevail on the market. In situation B, in case 1 stable point is (xB , xS , xE ) = (0, 0, 1), in case 2 system has only one stable rest point (xB , xS , xE ) = (0, 1/3, 2/3), in case 3 stable point is (xB , xS , xE ) = (0, 0.6, 0.4), hence we can say that during long-run period behaviors of rationalists and "sellers" will survive. In situation C in case 1 solutions trajectories converge to border between xE and xB and in case 2 and 3 the stable point is (xB , xS , xE ) = (1, 0, 0). Table 2: Imitation dynamics of successful agents. Case 1 Case 2 Case 3

A u=4

u=2

u = 8/3

u=4 Case 1

u=3 Case 2

u=1 Case 3

B

XI

C u=2

7.

XR

XH

u=1

u = 4/3

Conclusion

This paper’s main contribution is in using general results from evolutionary game theory to simulation of agents’ interaction on the stock market and analysis the behavior stability over time. Applying numerical simulation, we get that in some

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situation behavior of agents with bounded rationality, which can not recognize the actions of the opponent, will survive in long-run period. And the behavior of rational players will preserve in some other situations. In future research we are planning to use other probability distributions for agent’s revision profiles. References Weibull, J. (1995). Evolutionary Game Theory. — Cambridge, MA: The M.I.T.Press. Subramanian, N.(2005). En evolutionary model of speculative attacks. Journal of Evolutionary Economics, 15, 225–250. Springer-Verlag. Gintis, H. (2000). Game theory evolving. Princeton University Press. Gubar, E. A.(2008). An Evolutionary models of Trades in the Stock Market. Proceeding of Workshop on Dynamic Games in Management Science. Montreal, Canada, p.20. Evstigneev, I. V., Hens, T. and Schenk-Hopp?e, K. R. (2006). Evolutionary stable stock markets. Economic Theory, 27, 449–468 Sandholm, W. H. (2009).Pairwise Comparison Dynamics and Evolutionary Foundations for Nash Equilibrium. Econometrica, 769(3), 749–764. Sandholm, W. H., Durlauf, S. N. and L. E. Blume (2008). Deterministic Evolutionary Dynamics The New Palgrave Dictionary of Economics, 2nd edition, eds., Palgrave Macmillan. Sandholm, W. H., E. Dokumaci, and F. Franchetti (2010). Dynamo: Diagrams for Evolutionary Game Dynamics, version 0.2.5. http://www.ssc.wisc.edu/ whs/dynamo.

Equilibrium in Secure Strategies in the Bertrand-Edgeworth Duopoly Model⋆ Mikhail Iskakov1 and Alexey Iskakov2 V.A.Trapeznikov Institute of Control Sciences, Ul.Profsoyuznaya 65, Moscow 117997, Russia E-mail: [email protected] 2 V.A.Trapeznikov Institute of Control Sciences, Ul.Profsoyuznaya 65, Moscow 117997, Russia E-mail: [email protected], [email protected] 1

Abstract We analyze the Bertrand-Edgeworth duopoly model using a solution concept of Equilibrium in Secure Strategies (EinSS), which provides a model of cautious behavior in non-cooperative games. It is suitable for studying games, in which threats of other players are an important factor in the decision-making. We show that in some cases when Nash-Cournot equilibrium does not exist in the price duopoly of Bertrand-Edgeworth there is an EinSS with equilibrium prices lower than the monopoly price. The corresponding difference in price can be interpreted as an additional cost to maintain security when duopolists behave cautiously and secure themselves against mutual threats of undercutting. We formulate and prove a criterion for the EinSS existence. Keywords: Bertrand-Edgeworth Duopoly, Equilibrium in Secure Strategies, Capacity Constraints.

1.

Introduction

It is well known that the model of Bertrand-Edgeworth may not posses a Nash equilibrium (see e.g. d’Aspremont and Gabszewicz (1985)). Let the receipt function pD(p) be strictly concave and reach its maximum at monopoly price pM . When D(pM ) > S1 + S2 there is a Nash price equilibrium (p∗ , p∗ ) in the BertrandEdgeworth duopoly model such that D(p∗ ) = S1 +S2 . When D(pM ) < S1 +S2 there is no (pure strategy) Nash equilibrium in this game. There were several attempts to restore the concept of equilibrium in this model. For example d’Aspremont and Gabszewicz (1985) proposed the concept of quasi-monopoly, which restores the existence of pseudo equilibrium when one capacity is quite small compared to the other. Dasgupta and Maskin (1986) and Dixon (1984) demonstrated the existence of mixed-strategy equilibrium. However it proved not to be easy to characterize what the equilibrium actually looks like. Allen and Hellwig (1986a) were able to show that in a large market with many firms, the average price set would tend to the competitive price. In this paper we analyze the Bertrand-Edgeworth duopoly model using a solution concept of Equilibrium in Secure Strategies (EinSS), which provides a model of cautious behavior in non-cooperative games (M.Iskakov 2005, 2008). It is suitable for studying games, in which threats of other players are an important factor in ⋆

This work was supported by the research project No.14-01-00131-a of the Russian Foundation for Basic Research.

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the decision-making (M.Iskakov and A.Iskakov 2012b). This concept proved to be effective and allowed to find equilibrium positions in some well-known economic games that fail to have Nash-Cournot equilibrium. In particular this approach has been successfully applied for the classic Hotelling’s model with the linear transport costs (Hotelling 1929). There is no price Nash-Cournot equilibrium in this game when duopolists choose locations too close to each other (d’Aspremont et al., 1979). However, there is a unique EinSS price solution for all location pairs under the assumption that duopolists secure themselves against being driven out of the market by undercutting (M.Iskakov and A.Iskakov 2012a). Equilibria in secure strategies have been also successfully characterized for the contest described by Tullock (1969, 1980) as a rent-seeking contest. It is well known that a Nash-Cournot equilibrium does not exist in a two-player contest when the contest success function parameter is greater than two. However an EinSS always exists in the Tullock contest (M.Iskakov et al., 2013). Moreover, when the success function parameter is greater than two, this equilibrium is unique up to a permutation of players, and has a lower rent dissipation than in a mixed-strategy Nash equilibrium. Our aim is to analyze the original Bertrand-Edgeworth duopoly model with capacity constraints, which may not possess a Nash-Cournot equilibrium. We show that in some cases when Nash-Cournot equilibrium does not exist there is an EinSS with equilibrium prices lower than the monopoly price. The corresponding difference in price can be interpreted as an additional cost to maintain security when duopolists are cautious and avoid mutual threats. We formulate and prove a criterion for the EinSS existence. The remaining paper is organized as follows. In Section 2 we remind the BertrandEdgeworth model. In Section 3 the solution concept that we are going to use for analyzing the price duopoly game is presented. In Section 4 the equilibria in secure strategies are characterized for the linear demand function. In Section 5 the obtained results are generalized for the strictly concave receipt functions. Finally we provide an interpretation and summarize our results in the Conclusion. 2.

Bertrand-Edgeworth duopoly model

In this section we consider a model of price setting duopolists with capacity constraints originated in papers of Bertrand (1883) and Edgeworth (1925). We consider the market for some homogeneous product with a continuous strictly decreasing consumer’s demand function D(p). There are two firms in the industry i = 1, 2, each with a limited amount of productive capacity Si such that D(0) > S1 + S2 . As in Edgeworth’s work we assume these limits as physical capacity constraints, which are the same at all prices. Firms choose prices pi and play non-cooperatively. The firm quoting the lower price serves the entire market up to its capacity and the residual demand is met by the other firm. All consumers are identical and choose the lower available price on a first-comefirst-serve basis. Following Shubik (1959) and Beckmann (1965) we assume in our analysis that the residual demand to the firm quoting the higher price is a proportion of total demand at that price. If duopolists set the same prices firms share the market in proportion to their capacities. Formally we define the payoff functions of players

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Mikhail Iskakov, Alexey Iskakov

to be:   p1 min{S1 , D(p1 )}, p1 < p2 1 D(p1 )}, p1 = p2 u1 (p1 , p2 ) = p1 min{S1 , S1S+S 2  p min{S , D(p1 ) max{0, D(p ) − S }}, p > p 2 2 1 2 1 1 D(p2 )   p2 min{S2 , D(p2 )}, p2 < p1 2 D(p2 )}, p2 = p1 u2 (p1 , p2 ) = p2 min{S2 , S1S+S 2  p min{S , D(p2 ) max{0, D(p ) − S }}, p > p 2 2 D(p1 ) 1 1 2 1

(1)

In the particular case of a linear demand function D(p) = 1 − p, which we consider below, the payoff functions can be written as:   p1 min{S1 , 1 − p1 }, p1 < p2 1 (1 − p1 )}, p1 = p2 u1 (p1 , p2 ) = p1 min{S1 , S1S+S 2  p min{S , (1−p1 ) max{0, 1 − p − S }}, p > p 1 1 (1−p2 ) 2 2 1 2 (2)   p2 min{S2 , 1 − p2 }, p2 < p1 2 (1 − p2 )}, p2 = p1 u2 (p1 , p2 ) = p2 min{S2 , S1S+S 2  p min{S , (1−p2 ) max{0, 1 − p − S }}, p > p 1 1 2 1 2 2 (1−p1 )

The study of this case will allow us relatively easy to prove the main results in Section 4. However, these results will be generalized to arbitrary strictly concave receipt functions in Section 5. 3.

Equilibrium in Secure Strategies

We now proceed to define the solution concept that we are going to use to analyze the Bertrand-Edgeworth duopoly model (1). Below we provide definitions of Equilibrium in Secure Strategies from (M.Iskakov and A.Iskakov 2012b). Consider n-person non-cooperative game in the normal form G = (i ∈ N, si ∈ Si , ui ∈ R). The concept of equilibria is based on the notion of threat and on the notion of secure strategy. Definition 1. A threat of player j to player i at strategy profile s is a pair of strategy profiles {s, (s′j , s−j )} such that uj (s′j , s−j ) > uj (s) and ui (s′j , s−j ) < ui (s). The strategy profile s is said to pose a threat from player j to player i. Definition 2. A strategy si of player i is a secure strategy for player i at given strategies s−i of all other players if profile s poses no threats to player i. A strategy profile s is a secure profile if all strategies are secure. In other words a threat means that it is profitable for one player to worsen the situation of another. A secure profile is one where no one gains from worsening the situation of other players. Definition 3. A secure deviation of player i with respect to s is a strategy s′i such that ui (s′i , s−i ) > ui (s) and ui (s′i , s′j , s−ij ) ≥ ui (s) for any threat {(s′i , s−i ), (s′i , s′j , s−ij )} of player j 6= i to player i.


135

There are two conditions in the definition. In the first place a secure deviation increases the profit of the player. In the second place his gain at a secure deviation covers losses which could appear from retaliatory threats of other players. It is important to note that secure deviation does not necessarily mean deviation into secure profile. After the deviation the profile (s′i , s−i ) can pose threats to player i. However these threats can not make his or her profit less than in the initial profile s. We assume that the player with incentive to maximize his or her profit securely will look for secure deviations. Definition 4. A secure strategy profile is an Equilibrium in Secure Strategies (EinSS) if no player has a secure deviation. There are two conditions in the definition of EiSS. There are no threats in the profile and there are no profitable secure deviations. The second condition implicitly implies maximization over the set of secure strategies. Any Nash-Cournot equilibrium poses no threats to players so it is a secure profile. And no player in Nash-Cournot equilibrium can improve his or her profit using whatever deviation. Both conditions of the EinSS are fulfilled. Therefore we obtain Proposition 1. Any Nash-Cournot equilibrium is an Equilibrium in Secure Strategies. The Nash equilibrium is the profile, in which the strategy of each player is the best response to strategies of other players. In a similar way, the strategy of each player in the EinSS turns out to be the best secure response. Definition 5. A secure strategy si of player i is a best secure response to strategies s−i of all other players if player i has no more profitable secure strategy at s−i . A profile s∗ is the Best Secure Response profile (BSR-profile) if strategies of all players are best secure responses. The EinSS is a secure profile by definition. And it must be the best secure response for each player since otherwise there is a player who can increase the payoff by secure deviation. Therefore we get: Proposition 2. Any Equilibrium in Secure Strategies is a BSR-profile. This property provides a practical method for finding EinSS in three steps: (i) to analyze threats and determine conditions for secure profile, (ii) to find all BSRprofiles as a solution of the corresponding maximization problem in the set of secure profiles, and (iii) to verify the definition of EinSS for the found BSR-profiles. 4.

Solution in secure prices for the linear demand function

In this Section we illustrate how to find an EinSS in the simplest case of the linear demand function D(p) = 1 − p. First of all, let us analyze the threats between players and define secure profiles in the Bertrand-Edgeworth duopoly model. Proposition 3. The profile (p1 , p2 ) is a secure profile in the duopoly price game of Bertrand-Edgeworth with the linear demand function D(p) = 1 − p and payoff functions (2) if and only if it lies in the set M = {(p1 , p2 ) : 0 < pi 6 p∗ , i = 1, 2}, where p∗ = 1 − S1 − S2 .

136


Proof. (1). Consider the case p∗ < p1 < p2 . If 1 − p1 > S1 player 1 always threatens player 2 by slight increasing his price p1 . If 1 − p1 6 S1 then player 2 can decrease his price till p˜2 < p1 . In this case, according to (2): u˜1 (p1 , p˜2 ) = p1 (1 − 2 −S2 } p1 ) max{0,1−p < p1 (1 − p1 ) = u1 (p1 , p2 ) and u˜2 (p1 , p˜2 ) = p˜2 min{S2 , 1 − p2 } > 1−p2 0 = u2 (p1 , p2 ), and therefore there is always a threat of player 2 to player 1. Symmetrically, if p∗ < p2 < p1 either player 2 threatens player 1 or vice versa. If p∗ < p2 = p1 there is always a bilateral threat of undercutting by slight decreasing of price. (2). If p1 6 p∗ < p2 player 1 always threatens player 2 by increasing his price till p∗ + 0 < p2 which exceeds p∗ by an arbitrarily small amount. Indeed, in this case 1 − p1 > 1 − (p∗ + 0) > S1 and according to (2) u1 (p1 , p2 ) = p1 S1 < (p∗ + 2 0)S1 = u1 (p∗ + 0, p2 ). On the other hand, u2 (p∗ + 0, p2 ) = p2 S1−p S2 < p2 S2 and 1 +S2 S1 S1 ∗ u2 (p + 0, p2 ) = p2 (1 − p2 ) 1 − 1−(p∗ +0) < p2 (1 − p2 ) 1 − 1−p1 => u2 (p∗ +

0, p2 ) < u2 (p1 , p2 ). Symmetrically, if p2 6 p∗ < p1 player 2 always threatens player 1. (3). From the above it follows that all secure profiles must lie in the set M = {(p1 , p2 ) : 0 < pi 6 p∗ , i = 1, 2}. From the other hand if p1 6 p∗ : u1 (p1 , p2 ) = S1 p1 linearly increases in p1 and does not depend on p2 . Hence there are no threats for player 1. Symmetrically, if p2 6 p∗ there are no threats for player 2. Therefore (p1 , p2 ) is a secure profile in the game (2) if and only if it lies in the set M = {(p1 , p2 ) : 0 < pi 6 p∗ , i = 1, 2}.

Let us now find all Best Secure Response profiles in the set M of secure profiles.

Proposition 4. In the duopoly price game of Bertrand-Edgeworth with the linear demand function D(p) = 1−p and payoff functions (2) there is a unique Best Secure Response profile (p∗ , p∗ ), where p∗ = 1 − S1 − S2 .

Proof. According to Definition 5 and Proposition 3 any Best Secure Response profile must lie in the set M = {(p1 , p2 ) : 0 < pi 6 p∗ , i = 1, 2} of secure profiles found in Proposition 3. According to (2) the payoff functions u1 = S1 p1 and u2 = S2 p2 increase in the set M linearly in p1 and in p2 respectively. Therefore there can be only one BSR-profile (p∗ , p∗ ) in the set M (otherwise at least one player can securely slightly increase his price and get a more profitable secure strategy in M ). Let us now prove that it is indeed a BSR-profile. Any decrease in price in the profile (p∗ , p∗ ) is not profitable for players. On the other hand, as shown in paragraph 2 of the proof of Proposition 3 any increase in price in the profile (p∗ , p∗ ) is not secure for players. Therefore no player has a more profitable secure strategy in (p∗ , p∗ ) and therefore this profile by definition is a Best Secure Response profile in the game. According to Proposition 2 there can not be other EinSS in the BertrandEdgeworth duopoly game except the BSR-profile (p∗ , p∗ ) found in Proposition 4. Below we formulate and prove a necessary and sufficient condition for the BSRprofile (p∗ , p∗ ) to be an Equilibrium in Secure Strategies. Proposition 5. In the game of Bertrand-Edgeworth with the linear demand function D(p) = 1 − p and payoff functions (2) there is an Equilibrium in Secure Prices (p∗ , p∗ ) where p∗ = 1 − S1 − S2 if and only if 1 − S1 1 − S2 6 p∗ and 6 p∗ 2 2

(3)

Equilibrium in Secure Strategies in the Bertrand-Edgeworth Duopoly Model If the equilibrium price is not less than the monopoly price p∗ > pM = Nash equilibrium. There are no other EinSS in the game.

1 2

137 it is a

Fig. 1: Equilibria in secure prices in the Bertrand-Edgeworth duopoly model with D(p) = 1 − p in the space of capacity parameters (S1 , S2 ). Dark gray area: EinSS which coincide with Nash equilibria. Light gray area: EinSS which are not Nash equilibria. White area: there are neither Nash equilibria nor EinSS.

Equilibria in secure prices in the space of capacity parameters (S1 , S2 ) are shown in Fig.1. The profile (p∗ , p∗ ) is a Nash equilibrium if S1 + S2 6 12 (dark gray area). Under the weaker conditions (3) it is an EinSS. The area of EinSS which are not Nash equilibria are shaded by light gray in Fig.1. If conditions (3) do not hold this profile is no longer an EinSS and corresponds to an unstable BSR-profile. The found solution can be compared with the price which would maximize the joint profits in the industry pM = max{1 − S1 − S2 , 21 }. If Nash equilibrium exists (i.e. if S1 + S2 6 12 ) then both equilibrium prices coincide. However if EinSS exists and Nash equilibrium does not exist (i.e. if S1 + S2 > 12 and (3) holds) both EinSS prices p∗ = 1 − S1 − S2 are lower than the monopoly price pM = 12 . One can interpret the price difference S1 + S2 − 21 as an additional cost to maintain security in the situation when players take into account mutual threats of undercutting and behave cautiously. Proof. (1) Necessity. Let us consider profile (p∗ , p∗ ) and prove the conditions 2 (3). Suppose for example that p∗ < pˆ(S2 ) ≡ 1−S 2 . Then player 1 can deviate p∗1 → pˆ. His payoff will increase since p∗ < pˆ 6 pM = 12 and u1 (p1 , p2 ) is strictly increasing in p1 if p1 6 pM = 12 according to (2). Any retaliatory threat of player 2 according to (2) can not make the payoff of player 1 less than min u1 (ˆ p, p2 ) = p2

min u1 (ˆ p, p2 ) = u1 (ˆ p, p2 )|p2 =p−0 = pˆ min{S1 , 1 − pˆ − S2 }. The payoff of player 1 in ˆ

p2 p∗ . Player 1 increases ∗the payoff if ∗ −S2 ∗ ∗ ∗ ∗ 1−p −S2 and only if u1 (p , p ) = p S1 = p 1−p∗ (1−p∗ ) < u1 (p1 , p∗ ) = p1 1−p 1−p∗ (1−p1 ), ∗ ∗ i.e. there must be p (1 − p ) < p1 (1 − p1 ). Then there is retaliatory threat of player 2 to deviate from profile (p1 , p∗ ) into profile arbitrarily close to (p1 , p1 − 0). From p∗ S2 < p1 S2 and p∗ (1 − p∗ ) < p1 (1 − p1 ) it follows that player 2 increases the payoff at this deviation. The payoff of player 1 in this profile is arbitrarily close to u1 (p1 , p1 − 0) = p1 min{S1 , 1 − p1 − S2 }|p∗ pˆ(S2 ) = 1−S and p1 > p∗ > pˆ(S2 ) = 1−S then 2 2 ∗ ∗ ∗ ∗ u1 (p , p ) = p (1 − p − S2 ) > p1 (1 − p1 − S2 ) = u1 (p1 , p1 − 0). Therefore the deviation of player 1 into profile (p1 , p∗ ) is not a secure deviation. Symmetrically an arbitrary deviation of player 2 is not a secure deviation either. No player can make secure deviation in the profile (p∗ , p∗ ). By definition it is an EinSS. The sufficiency of (3) is proven. (3) Nash equilibrium condition. One can easily check that pM 6 p∗ (S1 + S2 6 21 ) is the maximum condition of functions u1 (p1 ) = u1 (p1 , p∗ ) and u2 (p2 ) = u2 (p∗ , p2 ) in the points p1 = p∗ and p2 = p∗ respectively. In other words it is a condition of Nash equilibrium for the profile (p∗ , p∗ ). (4) Uniqueness of EinSS. It follows from Proposition 2 and the uniqueness of BSR-profile proven in Proposition 4. 5.

Solution in secure prices for the strictly concave receipt function

Obtained in the previous section results can be generalized to the more general case of the strictly concave receipt function pD(p). Although the proofs become slightly more involved they follow the similar logic. Proposition 6. Let the receipt function pD(p) be strictly concave and reach its maximum at pM . Then in the game of Bertrand-Edgeworth with payoff functions (1) there is an EinSS (p∗ , p∗ ) where D(p∗ ) = S1 + S2 if and only if  arg max{p(D(p) − S1 )} 6 p∗ p>0 (4) arg max{p(D(p) − S2 )} 6 p∗ p>0

If p∗ > pM it is a Nash equilibrium. There are no other EinSS in the game.

Remark. Since the receipt function pD(p) is strictly concave then the function p(D(p) − S) at a given S is also strictly concave in p and reaches the unique maximum at p > 0. Therefore arg max{p(D(p) − S)} can be considered as a function of p>0

S. The proof is given in (M.Iskakov, A.Iskakov, 2012b).

The condition (4) can be easily formulated in differential form.


139

Proposition 7. If function pD(p) is differentiable the condition (4) is equivalent to d pD(p) 6 min{S1 , S2 } (4’) dp p=p∗

d Proof. One can easily check that pˆ = arg max{p(D(p)−S)} dp pD(p) = p>0 p=pˆ d S. Besides dp pD(p) is strictly decreasing. Therefore pˆ 6 p∗ d d 6 dp pD(p) = S. Hence the equivalence of (4) and (4’). dp pD(p) ∗

p=p

p=pˆ

The obtained results are generally similar to the results obtained for the linear demand function. In the EinSS all firms set equal prices such that market demand equals total supply. If the equilibrium price exceeds or equals the monopoly price this solution coincides with the well-known Nash equilibrium solution. However, in the EinSS which is not Nash equilibrium the prices are lower than the monopoly price. The corresponding difference in price can be interpreted as an additional cost to maintain security in the situation when players behave cautiously and secure themselves against mutual threats of undercutting. 6.

Conclusion

In this paper we considered the Bertrand-Edgeworth duopoly model, with capacity constraints which may not possess a Nash-Cournot equilibrium. Whilst the existence of mixed-strategy equilibrium was demonstrated by Dixon (1984), it has not proven easy to characterize what the equilibrium actually looks like. It has been argued that non-pure strategies are not plausible in the context of the Bertrand-Edgeworth model. Therefore several alternative approaches have been proposed as a response to the non-existence of pure-strategy equilibrium. Allen and Hellwig (1986b) proposed a modification of the game, in which firms choose the quantity they are willing to sell up to at each price. Dastigar (1995) proposed that firms have to meet all demand at the price they set. Benassy (1989) introduced in the Bertrand-Edgeworth model the product differentiation. Dixon (1993) explored the Bertrand-Edgeworth model with ”integer pricing” when firms cannot undercut each other by an arbitrarily small amount. All these approaches in some sense or another change the setting of the original game and introduce specific ad hoc modification to the Bertrand-Edgeworth model. As an alternative approach to analyze the Bertrand-Edgeworth duopoly model we propose in this paper the concept of Equilibrium in Secure Strategies (EinSS). We present a new intuitive formulation of EinSS concept from (M.Iskakov and A.Iskakov, 2012b). An existence condition of the EinSS in the Bertrand-Edgeworth price duopoly is formulated and proved, which allowed us to extend the set of NashCournot price equilibria in the game. If the equilibrium price exceeds or equals the monopoly price this solution coincides with the Nash-Cournot equilibrium solution. However, in the EinSS, which is not Nash-Cournot equilibrium the prices are lower than the monopoly price. The corresponding difference in price can be interpreted as an additional cost to maintain security when duopolists secure themselves against the threats of being undercut.

140


Although the proposed approach does not completely solve the problem of the non-existence of Nash-Cournot equilibria in the Bertrand-Edgeworth model, it nevertheless provides some advantages. In contrast to the above mentioned ad hoc equilibrium concepts developed in the context of the Bertrand-Edgeworth model, the EinSS is a general equilibrium concept that proved to be effective and allowed to find equilibrium positions in several well-known economic games that fail to have Nash-Cournot equilibrium (M.Iskakov and A.Iskakov, 2012b). On the other hand unlike equilibria in mixed strategies it offers a solution in an explicit form and can be easily interpreted in terms of cautious behavior. Acknowlegments. We are deeply indebted to C.d’Aspremont who helped us to find an elegant and intuitively clear formulation of the EinSS concept and inspired us to apply it to the Bertrand-Edgeworth duopoly model. We are thankful to F.Aleskerov and D.Novikov for regular discussions on the subject of EinSS concept during seminars at Moscow High School of Economics and at V.A.Trapeznikov Institute of Control Sciences. This work was supported by the research project No.14-01-00131-a of the Russian Foundation for Basic Research. References Allen, B. and Hellwig, M. (1986a). Bertrand-Edgeworth oligopoly in large markets. Rev. Econ. Stud., 53, 175–204. Allen, B. and Hellwig, M. (1986b). Price-Setting Firms and the Oligopolistic Foundations of Perfect Competition. The American Economic Review, 76(2), 387–392. d’Aspremont, C., Gabszewicz, J. and Thisse, J.-F. (1979). On Hotelling’s ’Stability in Competition’. Econometrica, 47(5), 1145–1150. d’Aspremont, C. and Gabszewicz, J. (1985). Quasi-Monopolies. Economica, 52(206), 141– 151. Benassy, J.-P. (1989). Market Size and Substitutability in Imperfect Competition: A Bertrand-Edgeworth-Chamberlin Model? Review of Economic Studies, 56(2), 217–234. Bertrand, J. (1883). Review of Cournot’s ’Rechercher sur la theoric mathematique de la richesse’. Journal des Savants, 499–508. Dastidar, K. G. (1995). On the existence of pure strategy Bertrand equilibrium. Econ. Theory, 5, 19–32. Dasgupta, P. and Maskin, E. (1986). The existence of equilibrium in discontinuous economic games, II: Applications. Rev. Econ. Stud., LIII, 27–41. Dixon, H. D. (1984). The existence of mixed-strategy equilibria in a price-setting oligopoly with convex costs. Econ. Lett., 16, 205–212. Dixon, H. D. (1993). Integer Pricing and Bertrand-Edgeworth Oligopoly with Strictly Convex Costs: Is It Worth More Than a Penny? Bulletin of Economic Research, 45(3), 257-268. Edgeworth, F. M. (1925). Papers Relating to Political Economy I. Macmillan: London. Hotelling, H. (1929). Stability in Competition. Econ. J., 39(153), 41–57. Iskakov, M. B. (2005). Equilibrium in Safe Strategies. Automation and Remote Control, 66(3), 465–478. Iskakov, M. B. (2008). Equilibrium in Safety Strategies and equilibriums in objections and counter objections in noncooperative games. Automation and Remote Control, 69(2), 278–298. Iskakov, M. and Iskakov, A. (2012a). Solution of the Hotelling’s game in secure strategies. Econ. Lett., 117, 115–118. Iskakov, M. and Iskakov, A. (2012b). Equilibrium in secure strategies. CORE Discussion Paper 2012/61, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).


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Iskakov, M., Iskakov, A. and Zakharov, A. (2013). Tullock Rent-Seeking Contest and its Solution in Secure Strategies. HSE Working Paper WP7/2013/01, Publishing House of the Higher School of Economics: Moscow. Tullock, G. (1967). The welfare costs of tariffs, monopoly and theft. Western Econ. J., 5, 224–232. Tullock, G. (1980). Efficient rent seeking. In: Toward a theory of the rent-seeking society (Buchanan, J. M., Tollison, R. D. and Tullock, G., eds.), pp.97–112, A&M University Press: College Station, TX.

Equilibrium Strategies in Two-Sided Mate Choice Problem with Age Preferences⋆ Anna A. Ivashko1 and Elena N. Konovalchikova2 Institute of Applied Mathematical Research, Karelian Research Center of RAS, Pushkinskaya str. 11, Petrozavodsk, 185910, Russia E-mail: [email protected] URL: http://mathem.krc.karelia.ru 2 Transbaikal State University, Alekzavodskaya str. 30, Chita, 672039, Russia E-mail: [email protected] 1

Abstract In the paper the two-sided mate choice model of Alpern, Katrantzi and Ramsey (2010) is considered. In the model the individuals from two groups (males and females) want to form a couple. It is assumed that the total number of unmated males is greater than the total number of unmated females and the maximum age of males (m) is greater than the maximum age of females (n). There is steady state distribution for the age of individuals. The aim of each individual is to form a couple with individual of minimum age. We derive analytically the equilibrium threshold strategies and investigate players’ payoffs for the case n = 3 and large m. Keywords: mutual mate choice, equilibrium, threshold strategy.

1.

Introduction

In the paper the two-sided mate choice model of Alpern, Katrantzi and Ramsey (2010) (Alpern et al., 2010) is considered. The problem is following. The individuals from two groups (males and females) want to form a long-term relationship with a member of the other group, i.e. to form a couple. Each group has steady state distribution for the age of individuals. In the model males and females can form a couple during m and n periods respectively. It is assumed that the total number of unmated males is greater than the total number of unmated females and m ≥ n. The discrete time game is considered. In the game unmated individuals from different groups randomly meet each other in each period. If they accept each other, they form a couple and leave the game, otherwise they go into the next period unmated and older. It is assumed that individuals of both sexes enter the game at age 1 and stay until they are mated or males (females) pass the age m (n). The initial ratio of age 1 males to age 1 females is given. The payoff of mated player is the number of future joint periods with selected partner. Payoff of a male age i and a female age j if they accept each other is equal to min{m − i + 1, n − j + 1}. The aim of each player is to maximize his/her expected payoff. In each period players use threshold strategies: to accept exactly those partners who give them at least the same payoff as the expected payoff from the next period. ⋆

This research is supported by Russian Fund for Basic Research (project 13-01-91158Γ ΦEH_ a, project 13-01-00033-a) and the Division of Mathematical Sciences of RAS (the program "Algebraic and Combinatorial Methods of Mathematical Cybernetics and New Information System").

Equilibrium Strategies in Two-Sided Mate Choice Problem with Age Preferences 143 In the literature such problems are called also marriage problems or job search problems. We use here the terminology of ”mate choice problem”. In papers (Alpern and Reyniers, 1999; Alpern and Reyniers, 2005; Mazalov and Falko, 2008) the mutual mate choice problems with homotypic and common preferences are investigated. In (Alpern et al., 2013) a continuous time model with age preferences is considered. Other two-sided mate choice models were considered in papers (Gale and Shapley, 1962; Kalick and Hamilton, 1986; Roth and Sotomayor, 1992). Alpern, Katrantzi and Ramsey (Alpern et al., 2010) derive properties of equilibrium threshold strategies and analyse the model for small m and n. The case n = 2 was considered in paper (Konovalchikova, 2012). In this paper using dynamic programming method we derive analytically the equilibrium threshold strategies and investigate players’ payoffs for the case n = 3 and large m. 2.

Two-Sided Mate Choice Model with Age Preferences

Denote ai — the number of unmated males of age i relative to the number of females of age 1 and bj — the number of unmated females of age j relative to the number of females of age 1 (b1 = 1). The vectors of the relative numbers of unmated males and females of each age a = (a1 , ..., am ), b = (b1 , ..., bn ) remain constant over time. Denote the ratio of the rates at which males and females enter the adult popua1 lation by R, R = = a1 . b1 n m P P The total groups of unmated males and females are A = ai , B = bj . A Denote the total ratio by r and assume that r > 1. B Consider the following probabilities:

i=1

i=1

ai — the probability a female is matched with a male of age i, A B – — the probability a male is matched. A bj – — the probability a male is matched with a female of age j, given that a B male is mated. bj bj B – = · — the probability a male is matched with a female of age j. A B A –

Denote Ui , i = 1, ..., m — the expected payoff of male of age i and Vj , j = 1, ..., n — the expected payoff of female of age j. Players use the threshold strategies F = [f1 , ..., fm ] for males and G = [g1 , ..., gn ] for females, where fi = k, k = 1, ..., n — to accept a female of age 1, ..., k, gj = l, l = 1, ..., m — to accept a male of age 1, ..., l. 3.

Model for n = 3 and m ≥ 3

Consider the two-sided mate choice model with age preferences for the case n = 3 and m ≥ 3. The expected payoffs of male have the following form

144

Anna A. Ivashko, Elena N. Konovalchikova

V3 =

m−1 P i=1

V2 =

m−2 P i=1

V1 =

m−2 P i=1

ai am I{fi = 3} + ≤ 1, A A ai am−1 am 2I{fi ≥ 2} + 2+ 1 ≤ 2, A A A ai am−1 am 3+ 2+ max{1, V2 }, A A A

where I{C} is indicator of event C. From this it follows that g3 = g2 = m. Also g1 = m, if V2 < 1 or g1 = m − 1, if V2 ≥ 1. There are three forms of strategies which are presented in the table:

G1 = [m − 1, m, m]

G2 = [m, m, m] I. F1 = [1, ..., 1, 2, ..., 2, 3, ..., 3] | {z } | {z } | {z } k

l

II. F3 = [2, ..., 2, 3, ..., 3] | {z } | {z }

m−k−l

k = 1, ..., m − 3, l = 1, ..., m − 3

k

m−k

k = 1, ..., m − 2

III. F2 = [1, ..., 1, 2, ..., 2, 3, ..., 3] | {z } | {z } | {z } k

l

m−k−l

k = 1, ..., m − 3, l = 1, ..., m − 3 Note that for female strategy G2 = [m, m, m] in the equilibrium male strategy it should be f1∗ = 1. Consider these strategies consequently. I. Players use strategy profile (F1 , G2 ), where G2 = [m, m, m] (to accept any partner), F1 = [f1 , ..., fm ] = [1, ..., 1, 2, ..., 2, 3, ..., 3], k = 1, ..., m− 3, l = 1, ..., m− 3. | {z } | {z } | {z } k

l

m−k−l

For equilibrium strategies the male’s payoff V2 =

am 1 must be less than 1. It is equivalent to A m−1 X i=1

1−

1 r

i−1

+

m−2 P i=1

ai am−1 2I{fi ≥ 2} + 2+ A A

i−1 k X 1 2 1− I{fi ≥ 2} < 0. r i=1

Consider the expected payoffs of males

Equilibrium Strategies in Two-Sided Mate Choice Problem with Age Preferences 145

Um =

b1 b2 b3 B 1 1+ 1+ 1+ 1− 0 = < 1, A A A A r

Um−1 =

Um−2 =

b2 b3 1 2 b3 1 b1 2+ 2+ 1+ 1− Um = − + 1− Um < 2, A A A r r A r b1 b2 b3 1 3+ 2+ max{1, Um−1 } + 1 − Um−1 < 3, A A A r

(1)

b1 b2 b3 1 3+ max{2, Um−i+1 }+ max{1, Um−i+1 }+ 1− Um−i+1 < 3, A A A r i = 3, ..., m − 1.

Um−i =

From these expressions it follows that equilibrium strategies are equal to ∗ ∗ = 3, 1) fm−1 = fm ∗ ∗ 2)fm−2 = 3, if Um−1 < 1; fm−2 = 2, if Um−1 ≥ 1, ∗ ∗ ∗ 3) fm−i = 3, if Um−i+1 < 1; fm−i = 2, if 1 ≤ Um−i+1 < 2; fm−i = 1, if 2 ≤ Um−i+1 , i = 3, ..., m − 1. The equilibrium strategies and the optimal payoffs are presented in the theorem. Theorem 1. If players use the equilibrium strategy profile (F1∗ , G∗2 ), where G∗2 = [m, m, m], F1∗ = [1, ..., 1, 2, ..., 2, 3, ..., 3], for certain values of k and l (k = 1, ..., m − | {z } | {z } | {z } k

l

m−k−l

3, l = 1, ..., m − 3) then the males’ optimal payoffs are equal to  Um = 1 − z,      Um−1 = 2 − z 2 − z,      Um−i = 3 − z i+1 − z i − z i−1 , i = 2, ..., m − 1,

(2)

the equilibrium age distributions are equal to

where z = 1 − 1/r.

a = (R, Rz, Rz 2, ..., Rz m−1 ); b = (1, 0, 0), 1 R= , (1 − z)(1 + z + z 2 + ... + z m−1 ) A = r = 1/(1 − z),

Proof. Let the players use strategy profile (F1 , G2 ), where G2 = [m, m, m], F1 = [1, ..., 1, 2, ..., 2, 3, ..., 3], k = 1, ..., m − 3, l = 1, ..., m − 3. | {z } | {z } | {z } k

l

m−k−l

Then the age distributions are equal to b = (1, 0, 0); a = (R, a1 (1− 1r ), ..., am−1 (1− r1 )) or a = (R, R(1− r1 ), R(1− 1r )2 , ..., R(1− r1 )m−1 ). m n P P Taking into account that Br = A, where A = ai , B = bj we get i=1

R=

i=1

r . 1 + (1 − 1/r) + (1 − 1/r)2 + ... + (1 − 1/r)m−1

146


Then we can rewrite the expected male’s payoffs (1) in the following recurrent form (z = 1 − 1/r): Um = Um−1

1 = 1 − z, r 2 1 = + 1− Um = 2(1 − z) + zUm , r r

Um−i =

3 1 + 1− Um−i+1 = 3(1 − z) + zUm−i+1 , i = 2, ..., m − 1. r r

Substituting each expression into the next one we get Um = 1 − z, Um−1 = 2(1 − z) + zUm = (1 − z)(2 + z),

Um−2 = 3(1 − z) + zUi+1 = (1 − z) 3 + z 2 + 2z , ! i−1 P j Um−i = 3(1 − z) + zUm−i+1 = (1 − z) 3 z − z i−1 + z i , i = 3, ..., m − 1. j=0

Simplifying the payoffs we obtain (2). For the equilibrium females’ strategy G∗2 = [m, m, m] the equilibrium males’ strategy F1∗ can be obtained from the system  V2 < 1,     Um−1 < 1,     ...     Uk+l+2 < 1,    1 ≤ Uk+l+1 < 2, ...     1 ≤ Uk+2 < 2,     Uk+1 ≥ 2,     ...    U2 ≥ 2 for different value of r.

Example 1. For m = 4 and r = 2, we obtain a = F1∗ = [1, 2, 3, 3], G∗2 = [4, 4, 4].

16 8 4 2 , , , 15 15 15 15

, b = (1, 0, 0),

Example 2. F1∗ = [1, ..., 1, 2, 3, 3] for r ∈ (1; 2.191] and m ≥ 4, where r∗ = 2.191 is the solution of the equation 2r3 − 6r2 + 4r − 1 = 0 F1∗ = [1, ..., 1, 2, 2, 3, 3] for r ∈ [2.191; 2.618] and m ≥ 6, where ãäå r1∗ = 2.191 is the solution of the equation 2r3 − 6r2 + 4r − 1 = 0, and r2∗ = 2.618 is the solution of the equation r2 − 3r + 1 = 0, F1∗ = [1, ..., 1, 2, 3, 3, 3] for r ∈ [2.618; 3.14] and m ≥ 6, F1∗ = [1, ..., 1, 2, 2, 3, 3, 3] for r ∈ [3.14; 4.079] and m ≥ 7.

Equilibrium Strategies in Two-Sided Mate Choice Problem with Age Preferences 147 II. Consider the case when the female’s strategy is G1 = [m − 1, m, m] (V2 ≥ 1). The expected males’ payoffs are equal to b1 b2 b3 B 1 b1 0= − < 1, Um = 0 + 1 + 1 + 1 − A A A A r A b1 b2 b3 1 2 b3 1 Um−1 = 2 + 2 + 1 + 1 − Um = − + 1− Um < 2 A A A r r A r b1 b2 b3 1 Um−2 = 3 + 2 + max{1, Um−1} + 1 − Um−1 < 3, A A A r b2 b3 1 b1 max{2, Um−i+1 } + max{1, Um−i+1 } + 1 − Um−i+1 < 3, Um−i = 3 + A A A r i = 3, ..., m − 1.

∗ ∗ It follows that fm−1 = fm = 3, and fi∗ = {1, 2, 3}, i = 1, ..., m − 2 depending on the values of r. Consider the case when the male’s strategy is F3 = [2, ..., 2, 3, ..., 3], k = 1, ..., m− | {z } | {z } k

m−k

2.

Theorem 2. If players use the equilibrium strategy profile (F3∗ , G∗1 ), where G∗1 = [m − 1, m, m], F3∗ = [2, ..., 2, 3, ..., 3], for certain values of k (k = 1, ..., m − 2) then | {z } | {z } k

m−k

the males’ optimal payoffs are equal to

Um = 1 − z −

1 , A

Um−1 = 2(1 − z) + zUm ,

am am 1 i−1 Um−i = 3 − 2 − 1− 2 z − 1+ z i − z i+1 , A (1 − z) A (1 − z) A i = 2, ..., m − 2, the equilibrium age distributions are equal to 



 z m−1  a = R, Rz, Rz 2, ..., Rz m−1 , b =  , 0 . 1, m−1 P i  z i=0

1 + z + z 2 + ... + z m−2 + 2z m−1 R= , (1 − z)(1 + z + z 2 + ... + z m−1 )2 m−1 P i A=R z, i=0

where z = 1 − 1/r.

Proof. The distributions for the age of males and females have form

148

Anna A. Ivashko, Elena N. Konovalchikova a m a = R, Rz, Rz 2, ..., Rz m−1 , b = 1, , 0 , where z = 1 − 1/r. A The ratio of the rates at which males and females enter the adult population has form 1 + z + z 2 + ... + z m−2 + 2z m−1 , where z = 1 − 1/r. (1 − z)(1 + z + z 2 + ... + z m−1 )2 am We have that V2 = 2 − ≥ 1. A The expected payoffs have form R=

Um = Um−1

1 1 − , r A 2 1 = + 1− Um , r r

Um−i = or Um = 1 − z −

1 am 3 + 1− Um−i+1 − 2 , i = 2, ..., m − 2 r r A 1 , A

Um−1 = 2(1 − z) + zUm , am Um−i = 3(1 − z) + zUm−i+1 − 2 = A am 1 am i−1 =3− 2 − 1− 2 z − 1+ z i − z i+1 , A (1 − z) A (1 − z) A i = 2, ..., m − 2, where z = 1 − 1/r

For the equilibrium female’s strategy G∗1 = [m − 1, m, m] the equilibrium males’ strategies F3∗ can be obtained from the system  Um−1 < 1,     ...    Uk+1 < 1, 1 < Uk < 2,     ...    1 < U2 < 2 for different value of r.

III. Finally, consider the case when the female’s strategy is G1 = [m − 1, m, m] (V2 ≥ 1) and the male’s strategy is F2 = [1, ..., 1, 2, ..., 2, 3, ..., 3], k = 1, ..., m − 3, | {z } | {z } | {z } k

l

m−k−l

l = 1, ..., m − 3. Then the expected payoff of female of age 2 is equal to k a P am i V2 = 2 − −2 and it must be greater then or equal to 1. A i=1 A Then the distributions for the age of males and females have forms

Equilibrium Strategies in Two-Sided Mate Choice Problem with Age Preferences 149 k a am am P i a = (a1 , ..., am ); b = 1, , , A A i=1 A where a1 = R, ai= ai−1 (1 − 1/A), i = 2, ..., k + 1, 1 b3 +1− , j = k + 2, ..., k + l + 1, aj = aj−1 A r 1 as = as−1 1 − , s = k + l + 2, ..., m. r The expected payoffs of males are equal to 1 1 − , r A 2 b3 1 + 1− Um , Um−1 = − r A r 3 b2 b3 1 Ui = − −2 + 1− Ui+1 , i = k + l + 1, ..., m − 2, r A A r 3 b2 b3 1 b3 Uj = − −3 + 1− + Uj+1 , j = k + 1, ..., k + l, r A A A r 3 b2 b3 1 b2 b3 Us = − 3 − 3 + 1 − + + Us+1 , s = 2, ..., k + 1. r A A r A A Um =

For the equilibrium females’ strategy G∗1 = [m − 1, m, m] the equilibrium males’ strategy F2∗ can be obtained from the system  V2 ≥ 1,     Um−1 < 1,     ...     Uk+l+2 < 1,    1 ≤ Uk+l+1 < 2, ...     1 ≤ Uk+2 < 2,     Uk+1 ≥ 2,     ...    U2 ≥ 2.

In Table 1. the numerical results for the optimal threshold strategies are given for different values of r. Table 1: Equilibrium strategy for m = 5 for different values of r. Equilibrium ([1, 1, 2, 3, 3], [5, 5, 5]) ([1, 2, 3, 3, 3], [4, 5, 5]) ([2, 2, 3, 3, 3], [4, 5, 5]) ([2, 3, 3, 3, 3], [4, 5, 5]) ([3, 3, 3, 3, 3], [4, 5, 5])

A B (1, 2.191] [2.016, 2.79] [2.85, 4.517] [4.517, 6.87] [6.87, +∞)

r=

R (1, 1.049] [1.081, 1.191] [1.209, 1.560] [1.560, 2.097] [2.097, +∞)

Acknowlegments. The authors express their gratitude to Prof. V. V. Mazalov for useful discussions on the subjects.

150


References Alpern, S. and Reyniers, D. (1999). Strategic mating with homotypic preferences. Journal of Theoretical Biology, 198, 71–88. Alpern, S. and Reyniers, D. (2005). Strategic mating with common preferences. Journal of Theoretical Biology, 237, 337–354. Alpern, S. Katrantzi, I. and Ramsey, D. (2010). Strategic mating with age dependent preferences. The London School of Economics and Political Science. Alpern, S. Katrantzi, I. and Ramsey, D. (2013). Partnership formation with age-dependent preferences. European Journal of Operational Research, 225, 91–99. Gale, D. and Shapley, L. S. (1962). College Admissions and the Stability of Marriage. The American Mathematical Monthly, 69(1), 9–15. Kalick, S.M. and Hamilton, T. E. (1986). The matching hypothesis reexamined. J. Personality Soc. Psychol., 51, 673–682. Mazalov, V. and Falko, A. (2008). Nash equilibrium in two-sided mate choice problem. International Game Theory Review. 10(4), 421–435. Roth, A. and Sotomayor, M. (1992). Two-sided matching: A study in game-theoretic modeling and analysis. Cambridge University Press. Konovalchikova, E. (2012). Model of mutual choice with age preferences. Mathematical Analysis and Applications. Transbaikal State University, 10–25 (in Russian).

Stationary State in a Multistage Auction Model⋆ Aleksei Y. Kondratev Institute of Applied Mathematical Research, Karelian Research Center of RAS, Pushkinskaya str. 11, Petrozavodsk, 185910, Russia E-mail: [email protected]

Abstract We consider a game-theoretic multistage bargaining model with incomplete information related with deals between buyers and sellers. A player (buyer or seller) has private information about his reserved price. Reserved prices are random variables with known probability distributions. Each player declares a price which depends on his reserved price. If the bid price is above the ask price, the good is sold for the average of two prices. Otherwise, there is no deal. We investigate model with infinite time horizon and permanent distribution of reserved prices on each stage. Two types of Nash-Bayes equilibrium are derived. One of them is a threshold form, another one is a solution of a system of integro-differential equations. Keywords: multistage auction model, Nash equilibrium, integro-differential equations for equilibrium, threshold strategies.

1.

Introduction

In (Mazalov and Kondratyev, 2012; Mazalov and Kondratyev, 2013) there was considered bargaining model with incomplete information, where a buyer and a seller have an opportunity to make a deal at only one stage. In (Mazalov et al., 2012) there was proposed auction model with finite number of steps. We fix time horizon n. A seller and a buyer meet each other at random. Reserved prices s and b are independent random variables on interval [0, 1] with density functions f (s) and g(b) accordingly. Seller asks Sk = Sk (s) ≥ s, buyer bids Bk = Bk (b) ≤ b on step k = 1, 2, . . . n We have a deal on the average of the two prices (Sk (s) + Bk (b))/2 if Bk ≥ Sk . If there is no deal then agents go to the next step k + 1. Differential equations for equilibrium strategies were found. In this paper we generalize and research this auction model for case of infinite time horizon. Let δ be discount factor. Consider a game with infinite time horizon. Let reserved prices of sellers and buyers s and b at the stage i = 1, 2, . . . have density functions fi (s), s ∈ [0, 1] and gi (b), b ∈ [0, 1] accordingly. At the stage i players use strategies Si (s) and Bi (b). If there was a deal then the buyer b and the seller s get outcome δ i−1 (b − B(b)+S(s) ) and δ i−1 ( B(b)+S(s) − s) accordingly and in this case they do 2 2 not move to the next stage. Additionally let fix count of new agents appears in the market at the each stage. We will study this model assuming that when i → ∞ and if agents act optimal then fi (s) and gi (b) tend to the limit density distribution f (s) and g(b). Hence we research stationary state on the market, when distributions ⋆

This research is supported by Russian Fund for Basic Research (project 13-01-91158Γ ΦEH_ a, project 13-01-00033-a), the Division of Mathematical Sciences of RAS (the program "Algebraic and Combinatorial Methods of Mathematical Cybernetics and New Information System") and Strategic Development Program of PetrSU.

152

Aleksei Y. Kondratev

f (s) and g(b) are the same at each stage, i.e. new agents replace making a bargain players. 2.

Integro-differential equations for Nash equilibrium

To find optimal strategies we count them as functions of reserved prices accordingly S = S(s) and B = B(b). Let they are differentiable and strictly increasing. Then inverse functions (differentiable and strictly increasing too) are U = B −1 and V = S −1 , i.e. accordingly s = V (S) and b = U (B). There is a deal, if B > S. If there is a deal then we have a deal price (S(s) + B(b))/2. Pay functions are (1) and (2). Fix buyer’s strategy B(b) and derive the best response of the seller s. Condition B(b) > S is equivalent to b > U (S). Outcome of the seller equals Z1

Hs (S) =

U(S)

B(b) + S − s g(b)db + δG(U (S))Hs (S), 2

1 Hs (S) = 1 − δG(U (S))

Z1

U(S)

B(b) + S − s g(b)db. 2

(1)

Differentiating (1) with respect to S, we have first order condition h 1 − G(U (S)) ∂Hs (S) 1 ′ = − (S − s)g(U (S))U (S) · ∂S (1 − δG(U (S)))2 2 Z1 i B(b) + S · (1 − δG(U (S))) + − s g(b)db · δ · g(U (S))U ′ (S) , 2 U(S)

and so we get integro-differential equation for equilibrium strategies (inverse functions) U (B), V (S) 1 − G(U (S)) 2

− (S − V (S))g(U (S))U ′ (S) (1 − δG(U (S)))+

Z1 S 1 δ · g(U (S))U (S) − V (S) (1 − G(U (S))) + B(b)g(b)db = 0. 2 2 ′

U(S)

The same way let S(s) be seller’s strategy. We find the best response of the buyer b. His outcome is

Hb (B) =

VZ(B) 0

b−

S(s) + B 2

1 Hb (B) = 1 − δ + δF (V (B))

f (s)ds + δ(1 − F (V (B)))Hb (B), VZ(B) 0

S(s) + B b− 2

f (s)ds.

(2)

153

Stationary State in a Multistage Auction Model Differentiating (2) with respect to B, we have first order condition h ∂Hb (B) 1 F (V (B)) ′ = (b−B)f (V (B))V (B)− · ∂B (1−δ+δF (V (B)))2 2 · (1−δ+δF (V (B)))−

VZ(B)

b−

0

S(s)+B 2

i f (s)ds · δ · f (V (B))V ′ (B) ,

and so we get the second integro-differential equation for equilibrium strategies (inverse functions) U (B), V (S)

F (V (B)) · (1 − δ + δF (V (B)))− 2 VZ(B) B 1 ′ δ · f (V (B))V (B) · (U (B) − )F (V (B)) − S(s)f (s)ds = 0. 2 2

(U (B) − B)f (V (B))V ′ (B) −

0

Now we have the system of equations for Nash equilibrium ∂U = ∂t

(1−G(U ))(1−δG(U )) , R1 2g(U ) (t−V )(1−δG(U ))−δ( 2t −V )(1−G(U ))− δ2 B(b)g(b)db

(3)

F (V )(1−δ+δF (V ))

(4)

U

∂V = ∂t

"

2f (V ) (U −t)(1−δ+δF (V

))−δ(U − 2t )F (V

)+ 2δ

RV

S(s)f (s)ds

0

#.

Functions U and V must satisfy U (a) = a, U (c) = 1, V (a) = 0, V (c) = c. From (3) and (4) it is easy to find U ′ (a) =

(1 − G(a))(1 − δG(a)) , R1 δ δ 2g(a) a(1 − δG(a)) − 2 a(1 − G(a) − 2 B(b)g(b)db

(5)

a

V ′ (c)=

F (c)(1−δ+δF (c)) . Rc 2f (c) (1−c)(1−δ+δF (c))−δ(1− 2c )F (c)+ 2δ S(s)f (s)ds

(6)

0

To figure out marginal prices a and c assume that there exist finite derivative V ′ (a) > 0 and density f (0) > 0. Using L’Hopital’s rule we derive f (V )V ′ (1 − δ + δF (V )) + δF (V )f (V )V ′ t→a 2f (V )[(U ′ −1)(1−δ+δF (V ))+(U −t)δf (V )V ′ −δ(U ′ − 1 )F (V ) 2 f (0)V ′ (a)(1 − δ) V ′ (a) = = , 2f (0)(U ′ (a) − 1)(1 − δ) 2(U ′ (a) − 1) −δ(U − 2t )f (V )V ′ + 21 δtf (V )V ′ ]

V ′ (a)= lim

and so U ′ (a) = 1.5.

154


The same way assume that there exist finite derivative U ′ (c) > 0 and density g(1) > 0. Using L’Hopital’s rule we derive −g(U )U ′ (1 − δG(U )) − δ(1 − G(U ))g(U )U ′ t→c 2g(U )[(1−V ′ )(1−δG(U ))−(t−V )δg(U )U ′ −δ( 1 −V ′ )(1−G(U )) 2

U ′ (c)= lim +δ( 2t

−V

)g(U )U ′

+

1 ′ 2 δtg(U )U ]

=

−g(1)U ′ (c)(1 − δ) −U ′ (c) = , ′ 2g(1)(1 − V (c))(1 − δ) 2(1 − V ′ (c))

and so V ′ (c) = 1.5. So we find necessary condition for differentiable strictly increasing strategies to be Nash equilibrium. The case of δ = 0 leads to single-stage auction model researched in (Mazalov and Kondratyev, 2012).

Fig. 1: Equilibrium strategies Fig. 2: Deal area (for theorem 1)

Theorem 1. If density functions g(b) and f (s) are continuous on [0, 1], 0 < f (0) < +∞, 0 < g(1) < +∞. Derivatives 0 < S ′ (0), B ′ (1) < +∞ exist. Then differentiable and strictly increasing strategies S(s) on [0, c] and B(b) on [a, 1] are Nash equilibrium, if they satisfy (3),(4) on interval (a, c), with respect to boundary conditions U (a) = a, U (c) = 1, V (a) = 0, V (c) = c. Marginal prices a and c must be derived from equations U ′ (a) = 1.5, V ′ (c) = 1.5, using (5), (6). 3.

Nash equilibrium with threshold strategies

We derive necessary and sufficient condition for threshold strategies to be Nash equilibrium in the underlying

155

Stationary State in a Multistage Auction Model

Fig. 3: Threshold strategies Fig. 4: Deal area (for theorem 2)

Theorem 2. If strategies S(s), B(b) are threshold with price of a deal a ∈ [0, 1], i.e. S(s) = max{a, s}, B(b) = min{a, b}. Then it is Nash equilibrium if and only if (∗) Hs=0 (S) on [0, a] has a maximum for S = a, (∗∗) Hb=1 (B) on [a, 1] has a maximum for B = a. Proof. The deal is made if seller’s reserved price s ∈ [0, a] and ask price S ∈ [s, a], S ≤ B(b). Outcome of the seller (1) equals  a  Z Z1 1 b + S a + S  Hs (S) = − s dG(b) + − s dG(b) = 1 − δG(S) 2 2 a S   Za 1 S 1 a  = − s (1 − G(S)) + bdG(b) + (1 − G(a)) . (7) 1 − δG(S) 2 2 2 S

It is easy to check that Hs (S) = Hs=0 (S) +

(1 − δ)s s − , δ(1 − δG(S)) δ

and in respect that G(S) is increasing, from (∗) we have a result that for any s ∈ [0, a] seller’s outcome has maximum point S = a.

156


We can hold the same reasoning for buyers. The deal is made if buyer’s reserved price b ∈ [a, 1], and a bid price B ∈ [a, b], B > S(s). From (2) we find his outcome Hb (B)



 ZB 1  b − B F (B) − 1 sdF (s) − a F (a) . 1 − δ + δF (B) 2 2 2

=

Note that

Hb (B) = Hb=1 (B) −

(8)

a

(1 − b) (1 − b)(1 − δ) + , δ δ(1 − δ + δF (B))

and in respect that F (B) is increasing, from (∗∗) we get that for any b ∈ [a, 1] buyer’s outcome has a maximal value for B = a. Theorem 3. If distribution functions F (s), G(b) have piecewise-continuous and limited density functions f (s) ≤ L on [a, 1] and g(b) ≤ M on [0, a], then in theorem 2 for (∗) it is sufficient that 2

δ ≥1−

(1 − G(a)) , 2aM

(9)

F 2 (a) . 2(1 − a)L

(10)

and condition (∗∗) is true if δ ≥1−

Proof. At the points of continuity for g(b), differentiating the (7), we derive that ′ Hs=0 (S) =

1 2(1 − δG(S))

2

h 1 − G(S) − 2Sg(S) − δG(S) + δG2 (S)+

+ δSg(S)G(S) + δg(S)

Za S

i bg(b)db + δag(S) − δaG(a)g(S) + δSg(S) . (11)

Using that g(S)

Za S

bg(b)db ≥ g(S)

Za

Sg(b)db = Sg(S)(G(a) − G(S)),

S

substituting δ = 1 − (1 − δ), it is easy to check that in (11) expression in square brackets is not less than 2

(1 − G(S)) + g(S)(a − S)(1 − G(a)) − (1 − δ)(−G(S) + G2 (S)+

+ (S − a)g(S)G(a) + (a + S)g(S)) ≥

further as S ≤ a and (9) it results that 2

2

≥ (1 − G(S)) − (1 − δ)(a + S)g(S)) ≥ (1 − G(a)) − (1 − δ)2aM ≥ 0. ′ Hence we prove that derivative Hs=0 (S) is nonnegative on the interval [0, a], so it leads to (∗).

Stationary State in a Multistage Auction Model

157

At the points of continuity for f (s), differentiating the (8), we find that ′ Hb=1 (B) =

1 2(1 − δ + δF (B))

− δBf (B)F (B) + δf (B)

2

h

ZB a

− (1 − δ)F (B) + 2(1 − δ)f (B) − δF 2 (B)+ i sf (s)ds + δaF (a)f (B) − 2(1 − δ)Bf (B) . (12)

Noting that f (B)

ZB a

sf (s)ds ≤ f (B)

ZB a

Bf (s)ds = Bf (B)(F (B) − F (a)),

substituting δ = 1 − (1 − δ), we calculate that in (12) expression in square brackets is not less than 2(1 − δ)(1 − B)f (B) − (1 − δ)(F (B) − F 2 (B)) − F 2 (B) − δ(B − a)f (B)F (a) ≤ further as B ≥ a and (10) we get that ≤ 2(1 − δ)(1 − B)f (B) − F 2 (B) ≤ 2(1 − δ)(1 − a)L − F 2 (a) ≤ 0. ′ We prove that derivative Hb=1 (B) not positive on [a, 1], and this fact implies (∗∗).

Threshold strategies and deal area are on pic. 3 and pic. 4. Theorem 3 shows that for any price a ∈ (0, 1), with limited density functions f (s), g(b) and discount factor δ close to 1 then Nash equilibrium with threshold strategies exists. Example 1. Let consider a situation of uniform distribution for reserved prices on the interval [0, 1], i. e. F (s) = s, G(b) = b. As f (s) = 1, g(b) = 1, so in the theorem 2 a2 3 we can set L = M = 1. By (9), (10) we find that if δ ≥ max{1 − (1−a) 2a , 1 − 2(1−a) } then threshold strategies with price a ∈ (0, 1) are Nash equilibrium. For a = 21 we get sufficient condition (by using theorem 3) δ ≥ 34 . Now we calculate explicit minimal value for discount factor δ when it is Nash equilibrium with threshold at price a = 12 . Derivative of outcome (11) is 1 3 2 3 1 1 2 1 ′ Hs=0 (S) = δS − S + − δa + δa , 2 2 4 2 (1 − δS)2 4 solving appropriate inequality we derive that for √ 3 − 9 − 6δ + 3δ 2 a2 − 6δ 2 a S≤ 3δ

derivative of seller’s outcome is nonnegative. Hence, we have necessary and sufficient condition √ 3 − 9 − 6δ + 3δ 2 a2 − 6δ 2 a . a≤ 3δ From where we find " # √ 3 − δ − δ 2 − 10δ + 9 a ∈ 0, . 2δ

158


The same reasoning for buyers leads to condition " # √ 3δ − 3 + δ 2 − 10δ + 9 ,1 , a∈ 2δ and finally we find that for δ ≥ 23 threshold strategies with price a = equilibrium in multistage auction model. 4.

1 2

are Nash

Conclusion

We offer multistage double closed auction model. Distribution of reserved prices are common knowledge. On every stage pairs of agents with different reserved prices are randomly selected. After then they decide to make a deal or no deal. In classical version (Chatterjee and Samuelson, 1983) it is single-stage process. In our model if there is no deal then agents move to the next stage. Outcome is discounted. We find Nash equilibrium in the model. Strategies are functions of reserved prices. Assuming the existence of stationary state for distribution of reserved prices from stage to stage, we research criteria for strategies to be Nash equilibrium. In theorem 1 we prove criteria for equilibrium in class of strictly increasing strategies, and in theorem 2 in class of threshold strategies. References Chatterjee, K. and Samuelson, W. (1983). Bargaining under incomplete informationg. Operations Research, 31(5), 835–851. Mazalov, V. V. and Kondratyev, A. Y. (2012). Bargaining model with incomplete information. Vestnik St. Petersburg University. Ser. 10, 1, 33-Ű40. Mazalov, V. V. and Kondratev, A. Y. (2013). Threshold strategies equilibrium in bargaining model. Game theory and applications, 5(2), 46–63. Mazalov, V. V., Mentcher, A. E. and Tokareva, J. S. (2012). Negotiations. Mathematical theory. Lan. Saint-Petersburg. 304 P. Mazalov, V. V. and Tokareva, J. S. (2011). Equilibrium in bargaining model with nonuniform distribution for reservation prices. Game theory and applications. 3(2), 37–49. Myerson, R. and Satterthwait, M. A. (1983) Efficient mechanisms for Bilateral Trading. Journal of Economic Theory, 29, 265–281. Myerson, R. (1984) Two-Person Bargaining Problems with Incomplete Information. Econometrica, 52, 461–487.

Phenomenon of Narrow Throats of Level Sets of Value Function in Differential Games⋆ Sergey S. Kumkov and Valerii S. Patsko Institute of Mathematics and Mechanics, S.Kovalevskaya str., 16, Ekaterinburg, 620990, Russia; Institute of Mathemaics and Computer Sciences, Ural Federal University Turgenev str., 4, 620083, Ekaterinburg, Russia E-mail: [email protected] [email protected]

Abstract A number of zero-sum differential games with fixed termination instant are given, in which a level set of the value function has one or more time sections that are almost degenerated (have no interior). Presence of such a peculiarity make very high demands on the accuracy of computational algorithms for constructing value function. Analysis and causes of these degeneration situations are important during study of applied pursuit problems. Keywords: linear differential games, fixed termination instant, level sets of value function, geometric methods, narrow throats

1.

Introduction

During investigating zero-sum differential games, the main topic is constructing and studying the value function of the game. One of the traditional approaches to value function construction is to solve the corresponding Hamilton–Jacobi–Bellman–Isaacs partial differential equation. Another approach is based on the representation of the value function as a collection of its level sets (Lebesgue sets). These sets are built by means of a geometric method. This representation is the most intuitive when the phase vector of the game is two-dimensional or when the game can be reduced to such a situation. In this case, any level set is located in a three-dimensional space time × two-dimensional phase space and can be effectively constructed and visualized to graphic study of its structure and peculiarities. The result of constructions is often a collection of polygons that approximate its time sections (t-sections) on some time grid. A very important thing both from theoretic and numerical points of view is loss of interior by t-sections of a level set at some instant. Further its evolution (in the backward time) can lead to complete degeneration of the set (its t-sections become empty), or can bring back the interior. The last case corresponds to the situation when we say that the level set has a narrow throat. Earlier, the authors have investigated appearance of narrow throats in linear differential game with fixed termination instant and terminal convex payoff function (Kumkov et al., 2005). That game appears during study an interception problem of one weak-maneuvering object by another one. This paper contains a number of ⋆

This work was supported by the Russian Foundation for Basic Research (projects nos. 12-01-00537 and 13-01-96055), by the Program “Dynamic systems and control theory” of the Presidium of the RAS (project no.12-Π-1-1002), and by the Act 211 Government of the Russian Federation 02.A03.21.0006

160

Sergey S. Kumkov, Valerii S. Patsko

examples of another games with convex payoff function, in which there are narrow throats. Also, we consider games having non-convex payoff. They arise from a pursuit game with two pursuers and one evader. The study is made numerically by algorithms and programs worked out by the authors. 2.

Games with Convex Payoff Function

2.1. Problem Formulation Let us consider a zero-sum linear differential game (Krasovskii and Subbotin, 1974; Krasovskii and Subbotin, 1988): z˙ = A(t)z + B(t)u + C(t)v, t ∈ [t0 ; T ], z ∈Rn , u ∈ P ⊂ Rp , v ∈ Q ⊂ Rq , ϕ zi (T ), zj (T ) → min max . u

(1)

v

The first player governs the control u and minimizes the payoff ϕ; the second player choosing its control v maximizes the payoff. The sets P and Q that constrain the players’ controls are convex compacta in their spaces. The payoff function ϕ depends on values of two components of the phase vector at the termination instant and is convex. It is necessary to construct level sets of the value function and study them from the point of view of narrow throat presence. 2.2.

Equivalent Differential Game

A standard approach to study linear differential games with fixed termination instants and payoff function depending on a part of phase coordinates at the termination instant assumes a passage to a new phase vector; see, for example, (Krasovskii and Subbotin, 1974; Krasovskii and Subbotin, 1988). These new variables are regarded as the values of the target components forecasted to the termination instant under zero players’ controls. Often they are called zero effort miss coordinates (Shima and Shinar, 2002; Shinar and Shima, 2002). In our case, we pass to new coordinates x1 and x2 , where x1 (t) is the value of the component zi forecasted from the current instant t to the termination instant T , and x2 is the forecasted value of the component zj . To obtain constructively the forecasted values, one uses a matrix combined of two rows of the fundamental Cauchy matrix X(T, t) for the system z˙ = A(t)z. These rows correspond to the target components of the phase vector. In our case, we use the ith and jth rows of the Cauchy matrix. The change of variables is described by the formula x(t) = Xi,j (T, t)z(t). (The subindices i, j of the matrix X denote taking the corresponding rows of the fundamental Cauchy matrix.) The equivalent game has the following form: x˙ = D(t)u + E(t)v, t ∈ [t0 ; T ], x ∈ R2 , u ∈ P, v ∈ Q, ϕ x1 (T ), x2 (T ) , D(t) = Xi,j (T, t)B(t), E(t) = Xi,j (T, t)E(t).

(2)

Further to analyze the evolution in time of the time sections of the level sets of the value function, it is useful to involve the sets P(t) = D(t)P , Q(t) = E(t)Q, which are called vectograms of the players at the instant t. The sense of the vectograms is the collection of velocities that can be given to the system by the players at the

Phenomenon of Narrow Throats of Level Sets

161

corresponding time instant. If one has that Q(t) ⊂ P(t), then it can be said that at the instant t the first player has (dynamic) advantage. In the case of opposite inclusion, we say about advantage of the second player. 2.3.

Numerical Construction of Level Sets

Fix a value c and describe construction of an approximation of the level set Wc of the value function V of game (2). The set will correspond to the chosen constant c. For a numerical construction, at first, let fix a time grid {tj }, t0 < t1 < . . . < tN = T . The constructions are made in the backward time from the termination instant T . Let at some instant tj+1 we have an approximation Wc (tj+1 ) of the t-section Wc (tj+1 ) of the level set Wc . Then the approximation Wc (tj ) of the t-section Wc (tj ) is described by the following formula (Pschenichnyi and Sagaidak, 1970): ∗ Wc (tj ) = Wc (tj+1 ) + (−∆j )D(tj )P − ∆j E(t)Q. (3)

Here, ∆j = tj+1 − tj ; D(tj ) and E(tj ) are the matrices from dynamics (2) computed at the instant tj ; P and Q are the sets constraining the controls of the first and second players. The sign “+” denotes the operation of algebraic sum (Minkowski ∗ denotes the geometric difference (Minkowski difference). sum), and “ −” The initial set Wc (T ) for the procedure is taken asa convex polygon Mc close in the Hausdorff metrics to the convex level set Mc = (x1 , x2 ) : ϕ(x1 , x2 ) ≤ c of the payoff function. Convexity of the set Mc is due to the convexity of the payoff function. It is known that in linear differential games with fixed termination instant, convexity of the target set provides convexity of all t-sections Wc (tj ) of the corresponding solvability set (the maximal stable bridge). Therefore, in procedure (3) we can apply algorithms for processing convex sets. In iteration procedures suggested by one of the authors (Isakova et al., 1984; Kumkov et al., 2005), convex sets in the plane are described by their support functions. (There is a one-to-one correspondence between a convex compact non-empty set S and its support function ρ(l; S) = max hl, si : s ∈ S , which is positively-homogeneous; here, h·, ·i denotes a dot product.) With that, to construct the support function of Minkowski sum of two sets we should just construct the sum of the support functions of the summands. To obtain the support function of Minkowski difference of two sets, it is necessary to build convex hull of difference of support functions of the initial sets. Also there is a very helpful fact that the support function of a convex polygon is piecewise-linear with areas of linearity in the cones between outer normals to its neighbor edges. Due to all these properties, it is possible to suggest effective procedures for addition of sets, subtraction, and convex hull construction. During the backward constructions, the current section Wc (tj+1 ) is summed with the dynamic capabilities (−∆j )D(tj )P = (−∆j )P(tj ) of the first player and further subtracted by dynamic capabilities ∆j E(tj )Q = ∆j Q(tj ) of the second player. Thus, the change of the t-section is connected to the correlation of the players’ vectograms. If the first player’s vectogram is “greater” than the vectogram of the second one (that is, if the first player has advantage), then the t-section grows in the backward time. In the opposite situation when the second player has advantage, vice versa, the section contracts in the backward time. If neither P(t) ⊂ Q(t), nor Q(t) ⊂ P(t), then the first player has advantage in some directions and disadvantage in others. Studying the situation of advantage of one or other player

162


allows to estimate qualitatively the evolution of the level set in time without its exact construction. 2.4.

Examples

One-to-One Interception Problem. In the works (Shinar et al., 1984; Shinar and Zarkh, 1996; Melikyan and Shinar, 2000), a three-dimensional problem of interception in near space or upper atmosphere is considered. The pursuer P is an intercept-missile; the evader E is a weak maneuverable target (for example, another missile or a large aircraft). The geometry of the interception is drawn in Fig. 1. The three-dimensional problem reduces naturally to a two-dimensional one. The longitudinal velocities of the objects are rather large, and the approach time is small. Thus, the control accelerations aP and aE that are orthogonal to the current velocity of the corresponding object cannot turn significantly the velocity vectors. Due to this, the longitudinal motion of the objects can be considered as uniform. Also, the minimal approach distance, which is the natural payoff in this game, can be changed by the lateral distance at the instant of nominal longitudinal passage of the objects. This instant is fixed as the termination one. The three-dimensional geometric coordinates can be introduced as it is shown in Fig. 1. The origin O is put at the position of the pursuer P . The axis OX coincides with the nominal line-of-sight. The axis OY is orthogonal to OX and is located in the plane defined by the vectors of the nominal velocities of the objects. The axis OZ is orthogonal to OX and OY . After excluding the longitudinal motion along the axis OX from consideration, we pass to a two-dimensional problem of lateral motion in the plane OY Z. The control of the evader defines its acceleration directly; the pursuer has a more complicated dynamics. Its control affects the acceleration through a link of the first order: r¨P = F, t ∈ [0; T ], rP , rE ∈ R2 , u ∈ P, v ∈ Q,

(4) F˙ = (u − F )/lP ,

rP (T ) − rE (T ) . ϕ x(T ), y(T ) = rË = v,

Here, rP and rE are the radius-vectors of the positions of the pursuer and evader in the plane OY Z; lP is the time constant that describes the inertiality of servomechanisms transferring the control command signal u to the acceleration F ; v is the

Fig. 1: The geometry of the three-dimensional interception. The actual realizations of the velocity vectors VP (t) and VE (t) are close to the nominal values (VP )col and (VE )col

163


evader’s control; T is the termination instant coinciding with the instant of longitudinal passage of objects along the nominal motions. The sets P and Q constraining the controls of the players are ellipses. These ellipses are obtained after projection of the original round vectograms on accelerations (that are orthogonal to the nominal velocities (VP )col and (VE )col ) into the plane OY Z. The parameters of the ellipse (the semiaxes) are defined by the maximal acceleration of the corresponding object (aP or aE ) and by the angle between the vector of its velocity and the line-of-sight ((χP )col or (χE )col ). To pass to a standard game with the payoff depending on two components of the phase vector, we use the following change of variables: z1 z3 z5 z7

= (rP )Y − (rE )Y , = (r˙P )Y , = (r˙P )Z , = (¨ rP )Y ,

z2 z4 z6 z8

= (rE )Z − (rE )Z , = (r˙E )Y , = (r˙E )Z , = (¨ rP )Z .

(5)

In this case, the payoff function (which is the lateral miss) depends on the values of z1 and z2 at the instant T : q ϕ z1 (T ), z2 (T ) = z12 (T ) + z22 (T ). Proceeding to a two-dimensional equivalent game, we obtain the dynamics x˙ = D(t)u + E(t)v, t ∈ [0; T], x ∈ R2 , u ∈pP, v ∈ Q, ϕ x(T ) = x(T ) = x21 (T ) + x22 (T ),

where

D(t) = ζ(t) · I2 , E(t) = η(t) · I2 ,

ζ(t) = (T − t) + lP e−(T −t)/lP − lP ,

(6)

η(t) = −(T − t),

and I2 is a unit 2 × 2 matrix. The sets P and Q are 1/ cos2 (χP )col 0 u ∈ P = u : u′ u ≤ a2P , 0 1 1/ cos2 (χE )col 0 v ∈ Q = v : v′ v ≤ a2E . 0 1

Example 1. Below, we give the results (Kumkov et al., 2005) of numerical study of problem (4). The following parameters have been used: lP = 1.0, u21 u22 v12 v22 2 2 2 + ≤ 1.30 , Q = v ∈ R : + ≤1 . P = u∈R : 0.672 1.002 0.712 1.002 Using notations of the original formulation, we have |VE | = 1.054, |VP |

aP = 1.3, aE

cos χP = 0.67,

cos χE = 0.71,

lP = 1.

164


Fig. 2: Example 1. A general view of the level set of the value function with a narrow throat

Fig. 3: A large view of the narrow throat

In Fig. 2, one can see a general view of the level set Wc computed for c = 2.391, which is a bit greater than the critical one (that is, to the one corresponding to the level set, which t-section has no interior at some instant). The main interesting properties of this tube is that it has the narrow throat and that the direction of elongation of t-sections changes near the throat. A large view of the narrow throat is given in Fig. 3. Such a complicated shape of the throat is conditioned by the process of passage of the advantage from the second player to the first one in this time interval.

165


This example was computed in the time interval τ ∈ [0; 7]. Here and below, τ = T − t denotes the backward time. The time step ∆ equals 0.01. The level sets of the payoff function (that are rounds) and the ellipses of the constraints for the players’ controls are approximated by 100-gons. Generalized L.S.Pontryagin’s Test Example. In work (Pontryagin, 1964), the following differential game ¨ + αx˙ = u, x

(7)

¨ + β y˙ = v. y

was taken as an illustration to the theoretic results. Here, α and β are some positive constants; x, y ∈ Rn ; kuk ≤ µ, kvk ≤ ν. The termination of the game happens when the coordinates x, y of the objects coincide. The first player tries to minimize the duration of the game, the second one hinders this. Later, differential games with dynamics (7) and termination conditions depending only on the geometric coordinates of the objects were called in Russian mathematical literature as “L.S.Pontryagin’s test example”. Another well-known example with the dynamics ¨ = u, x

(8)

y˙ = v

and constraints for the player’s controls kuk ≤ µ, kvk ≤ ν was called by L.S.Pontryagin (Pontryagin and Mischenko, 1969) as game “boy and crocodile”. The first player (the “crocodile”) controls its acceleration and tries to catch the second one (the “boy”) to some neighborhood. The second player is more maneuverable because it controls its velocity. Game (8) is a particular case of the game “isotropic rockets” (Isaacs, 1965), which dynamics is ¨ + k x˙ = u, x y˙ = v. (9) In works (Pontryagin, 1972; Mezentsev, 1972; ?; Nikol’skii, 1984; Grigorenko, 1990; Chikrii, 1997), problems with dynamics more complicated than (7), (8), (9) were studied: x(k) + ak−1 x(k−1) + · · · + a1 x˙ + a0 x = u, y

(s)

+ bs−1 y

(s−1)

+ · · · + b1 y˙ + b0 y = v,

u ∈ P,

v ∈ Q.

(10) (11)

Games having dynamics (10), (11) and termination conditions depending only on the geometric coordinates x, y, are often called “generalized L.S.Pontryagin’s test example”. In this paper, let us assume that the payoff function is defined by the formula ϕ x(T ), y(T ) = x(T ) − y(T ) . Also, let us count that x, y ∈ R2 . A variable change similar to (5) z1 = x1 − y1 , z3 = x˙ 1 , ............ (k−1) z2k−1 = x1 , z2k+1 = y˙ 1 , ............ (s−1) z2(k+s)−3 = y1 ,

z2 = x2 − y2 , z4 = x˙ 2 , ............ (k−1) z2k = x2 , z2k+2 = y˙ 2 , ............ (s−1) z2(k+s)−2 = y2 ,

166


transforms system (10), (11) to standard form (1): z˙ = Az + Bu + Cv,

z ∈ R2(k+s)−2 , u ∈ P, v ∈ Q,

with the matrices A, B, and C that do not p depend on the time. The payoff function is terminal and convex: ϕ z1 (T ), z2 (T ) = z12 (T ) + z22 (T ). There can be other variants of the change, which are more convenient in particular situations, but all of them assume introduction of relative geometric coordinates (z1 , z2 in our case). When some experience had been accumulated in numerical study of level sets with narrow throats in the case of problem from works (Shinar et al., 1984; Shinar and Zarkh, 1996; Melikyan and Shinar, 2000), the author decided to construct another examples with narrow throats in the framework of games with the dynamics of the generalized L.S.Pontryagin’s test example. The most interesting results of constructing level sets of the value function for the generalized L.S.Pontryagin’s test example are when at least one of the sets P and Q is not a round (since the level sets of the payoff, which is distance between objects at the termination instant, are rounds, we need something that destroys uniformity of the sets). So, let us take the sets P and Q as ellipses with center at the origin and main axes parallel to the coordinate axes. Then the players’ vectograms P(t) and Q(t) for all instants are ellipses homothetic to the ellipses P and Q respectively. As it becomes clear from the previous example, a narrow throat appears when there is a change of advantage of players. Namely, at the initial period of the backward time the second player should be stronger to contract t-section of level sets. Then the advantage should pass to the first player to allow him to expand the sections. The easiest way to obtain such a change of advantage is to assign an oscillating dynamics to one or both players. The most illustrative way to study the passages of the advantage is to investigate tubes of vectograms, that is the sets P = (t, u) : u ∈ P(t) , Q = (t, v) : v ∈ Q(t) . If one of the tubes includes the other in some period of time, then in this period the corresponding player has complete advantage. Example 2. The dynamics is the following: ¨ + 2 x˙ = u, x ¨ + 0.2 y˙ + y = v, y

x, y ∈ R2 ,

u ∈ P,

v ∈ Q.

Here, the first player controls an inertial point in the plane. The second object is a two-dimensional oscillator. Both objects have a friction proportional to their velocities. The controls are constrained by the ellipses u21 u22 v12 v22 2 2 P = u∈R : + ≤1 , Q= v∈R : + ≤1 . 0.82 0.42 1.52 1.052 The tubes of vectograms appearing in this example are shown in Fig. 4a. Since the dynamics of the second player describes an oscillating system, the advantage passes from one player to another several times. At the beginning of the backward time, the second player has the advantage, but later after a number of passes, the advantage comes to the first player. An enlarged fragment of the tubes can be seen in Fig. 4 b.


167

a)

b) Fig. 4: Example 2. Two views of the vectogram tubes. Number 1 denotes the tube of the first player (the set P), number 2 corresponds to the second player’s vectogram tube (the set Q)

Fig. 5 shows a level set Wc for c = 2.45098. This level set breaks (that is, is finite in time and has empty t-sections from some instant of the backward time). Before the break, orientation of elongation of the t-sections of Wc (t) changes. Namely, before the last contraction of the tube, the sections are elongated vertically, and after it the elongation is horizontal. As in the example in the previous subsection, this change is due to delicate interaction of the vectogram tubes P(t) and Q(t) in the time interval of the narrow throat. If to increase the value of c, the length in time of the level sets grows jump-like. The level set for c = 2.45100 can be seen in Fig. 6. In Fig. 7, its enlarged fragment is given, which is near the narrow throat at τ = 11.95. This value of c can be regarded as critical: for c < 2.45100 level sets break, for c ≥ 2.45100 they are infinite in time. More exact reconstruction of the level sets corresponding to values c close to the critical one needs a very accurate computations. This example was computed in the time interval τ ∈ [0; 20]. The time step is ∆ = 0.05. The round level sets Mc of the payoff function and the ellipses P and Q were approximated by 100-gons.

168


Fig. 5: Example 2. A broken level set close to the critical one, c = 2.45098

Fig. 6: Example 2. A general view of the level set with a narrow throat, c = 2.45100

Fig. 7: A large view of the narrow throat

169


Example 3. To get an example with a level set of the value function with more than one narrow throat, we should choose players’ dynamics to provide multiple passage of advantage in such a way that each of players has it for a quite long time (to allow the second player contract t-section almost to nothing). The most reasonable way to get such a situation is to put an oscillating dynamics to both players. Let the dynamics be the following: ¨ − 0.025 x˙ + 1.3 x = u, x ¨ + y = v, y x, y ∈ R2 ,

u ∈ P,

v ∈ Q.

Constraints for the players’ controls are equal ellipses: v12 v22 2 P =Q= v∈R : + ≤1 . 1.52 1.052 Since the sets P and Q constraining the players’ controls are equal, then at any instant the players’ vectograms P(t) and Q(t) are homothetic. In Figs. 8a and 8b, the tubes of players’ vectograms are shown. The difference of these figures is that in Fig. 8b the second player’s tube is transparent. Fig. 9 contains a general view of the level set Wc for c = 1.2. In Fig. 10, there is an enlarged fragment of the set near the first (in the backward time) narrow throat. The instants of the backward time of the most thin parts of the set are τ1 = 5.65 and τ2 = 8.50. The level set has been computed in the time interval τ ∈ [0; 16]. The time step is ∆ = 0.05. Near the narrow throats, the time step was ten times smaller: ∆′ = 0.005. Again, the approximating polygons for the constraints for the controls and for the payoff level set have 100 vertices. Note again that despite the players’ vectograms are homothetic, the t-sections of the level set and the vectograms are not. Absence of this homothety leads to complicated shape of the t-sections of the level set of the value function and, therefore, complicated shape of narrow throats. Example 4. The dynamics of this example is described by the relations ¨ + 0.025 x˙ + 1.2 x = u, x ¨ + 0.01 y˙ + 0.85 y = v, y The constraints are taken as follows: u2 u2 P = u ∈ R2 : 12 + 22 ≤ 1 , 2.0 1.3

x, y ∈ R2 ,

Q=

u ∈ P,

v ∈ R2 :

v ∈ Q.

v12 v22 + ≤ 1 . 1.52 1.052

The vectograms appearing in this game are given in Fig. 11. The level set Wc corresponding to c = 0.397 is shown in Fig. 12a. One can see three narrow throats. An enlarged view of the middle one (which is the narrowest among them) can be seen in Fig. 12b. 3.

Games with Non-Convex Payoff Function

In the previous section, we demonstrate examples where number of narrow throats is more than one. One can think that examples of this kind are artificial and, therefore,

170


a)

b) Fig. 8: Example 3. A general view of the vectogram tubes. Number 1 denotes the tube of the first player (the set P), number 2 corresponds to the second player’s vectogram tube (the set Q). In subfigure b), the tube of the second player is transparent


171

Fig. 9: Example 3. A general view of the level set with two narrow throats, c = 1.2

Fig. 10: An enlarged view of the first (in the backward time) narrow throat

172


rare. During last few years, the authors investigate differential games arising from consideration of pursuit problems in near space or in upper atmosphere. Descriptions of dynamics of the objects involved in the pursuit were taken from works by J. Shinar and his pupils. Games of this type also bring examples having level sets with, at least, two narrow throats. 3.1.

Problem Formulation

Consider a game z˙ P1 = AP1 (t)zP1 + BP1 (t)u1 , z˙ P2 = AP2 (t)zP2 + BP2 (t)u2 , z˙ E = AE (t)zE + BE (t)v, zP1 ∈ Rn1 , zP2 ∈ Rn2 , zE ∈ Rm , |ui | ≤ µi , |v| ≤ ν

(12)

with three objects moving in a straight line. The objects P1 and P2 described by the phase vectors zP1 and zP2 are the pursuers. The object E with the phase vector zE is the evader. The first components zP1 , zP2 , and zE of the vectors zP1 , zP2 , and zE respectively are the one-dimensional geometric coordinates of the objects. Two instants T1 and T2 are prescribed. At the instant T1 , the pursuer P1 terminates its pursuit, and the distance between him and the evader E is measured: r1 (T1 ) = |zP1 (T1 ) − zE (T1 ) . Similarly, the second pursuer P2 stops to pursue at the instant T2 , when the distance r2 (T2 ) = |zP2 (T2 ) − zE (T2 ) is measured. The payoff is the minimum of these distances: ϕ = min r1 (T1 ), r2 (T2 ) . The first player that consists of the pursuers and governs the controls u1 , u2 minimizes the value of payoff ϕ. The second player, which is identified with the evader E, maximizes the payoff. All controls are scalar and have bounded absolute value.

Fig. 11: Example 4. A large view of the vectogram tubes. Number 1 denotes the tube of the first player (the set P), number 2 corresponds to the second player’s vectogram tube (the set Q)


173

a)

b)

Fig. 12: Example 4. a) A general view of a level set with three narrow throats, c = 0.397; b) An enlarged view of the narrowest of the throats (the middle one)

3.2. Equivalent Differential Game Let us pass from system (12) with separated objects to two relative dynamics. To do this, introduce new phase vectors y (1) ∈ Rn1 +nE −1 and y (2) ∈ Rn2 +nE −1 such that (1) (2) y1 = zP1 − zE , y1 = zP2 − zE . (1)

The rest components yi , i = 2, . . . , n1 + nE − 1, of the vector y (1) equal components of the vectors zP1 and zE other than zP1 and zE . In the same way, the rest (2) components yi , i = 2, . . . , n2 + nE − 1, of the vector y (2) are the components of the vectors zP2 and zE other than zP2 and zE . Due to linearity of dynamics (12), the new dynamics consisting of the two relative ones, is linear too: y˙ (1) = A1 (t)y (1) + B1 (t)u1 + C1 (t)v, t ∈ [t0 ; T1 ], y˙ (2) = A2 (t)y (2) + B2 (t)u2 + C2 (t)v, t ∈ [t0 ; T2 ], y (1) ∈ Rn1 +nE −1 , y (2) ∈ Rn2 +nE −1 , (1) (2) |ui | ≤ µi , |v| ≤ ν, ϕ = min y1 (T1 ) , y1 (T2 ) .

(13)

The payoff function depends now on the first components of the phase vectors of the individual games. An individual game of the pursuer Pi against the evader E is the

174


(i) game with the dynamics of the vector y (i) and the payoff y1 (Ti ) . The dynamics of the individual games are linked only through the control of the evader. In each individual game, let us pass to the forecasted geometric coordinates in the same way as it is done from game (1) to (2). In the game of the pursuer Pi , (i) i = 1, 2, against the evader E, the passage is provided by the matrix X1 (Ti , t) constructed from the first row of the fundamental Cauchy matrix X (i) (Ti , t) that corresponds to the matrix Ai . The variable changes are defined by the formulas (1) (2) (1) x1 (t) = X1 (T1 , t)y (1) , x2 (t) = X1 (T2 , t)y (2) . Note that x1 (T1 ) = y1 (T1 ) and (2) x2 (T2 ) = y1 (T2 ). Dynamics of the individual games is the following: x˙ 1 = d1 (t)u1 + e1 (t)v, t ∈ [t0 ; T1 ], x˙ 2 = d2 (t)u2 + e2 (t)v, t ∈ [t0 ; T2 ], x1 , x2 ∈ R, |ui | ≤ µi , |v| ≤ ν.

(14)

Here, di (t) and ei (t) are scalar functions: (i)

(i)

di (t) = X1 (Ti , t)Bi (t), ei (t) = X1 (Ti , t)Ci (t),

i = 1, 2.

In the joint game of the pursuers against the evader, the payoff function is ϕ = min x1 (T1 ) , x2 (T2 ) .

3.3. Numerical Construction of Level Sets Numerical constructions for the taken formulation are more complicated due to the following circumstances. At first, for problem (2), any level set of the payoff function is plunged into the phase space at the instant T . But for the new formulation, level sets of the payoff can consist of two parts at two different (generally speaking) instants T1 and T2 . At second, level sets of the payoff in problem (2) compact. But now the components of (1) level sets corresponding to a constant c are infinite strips Mc = {x : |x1 | ≤ c} at (2) the instant T1 (that is an infinite strip along the axis x2 ) and Mc = {x : |x2 | ≤ c} at the instant T2 (an infinite strip along the axis x1 ). Presentation of infinite objects in a computational program is a quite difficult problem. At third, a realization of procedure (3) for problem (2) is oriented on work with convex sets. In problem (14), we need to proceed non-convex time sections. An algorithm taking into account these considerations and also based on procedure (3) can be formulated as follows. For definiteness, let us assume that T2 ≤ T1 . The opposite case is considered in the same way. For numerical constructions, fix a time grid in the interval [t0 ; T1 ]. It should (1) include the instant T2 . At the instant T1 , the set Mc is taken as the start for the procedure (3). In the case of numerical constructions, the infinite strip is cut becoming a rectangle with a quite large size along the axis x2 . Then, in the interval (T2 , T1 ], the procedure (3) is applied with the set D(t)P taken as a segment [−|d1 (t)|µ1 ; |d1 (t)|µ1 ] × {0} (by this, we ignore the action of the second pursuer). When the construction are made up to the instant T2 , we unite the obtained t(2) section Wc (T2 + 0) with the set Mc (also cut to a finite size in the case of numerical constructions). Thus, we get the set Wc (T2 ), which is the start value

175


for the further iterations. In the time interval [t0 ; T2 ], the rectangle vectogram [−|d1 (t)|µ1 ; |d1 (t)|µ1 ]×[−|d2 (t)|µ2 ; |d2 (t)|µ2 ] of the first player is taken; now, actions of both players are involved. The vectogram of the second player (of the evader) equals −|e1 (t)|ν, −|e2 (t)|ν ; |e1 (t)|ν, |e2 (t)|ν for all time instants from the grid. (1) If T2 = T1 = T , then the start set Mc at the instant T is union of the strips Mc (2) and Mc (possibly, cut). As it was mentioned above, the necessary realization of procedure (3) should be able to process non-convex sets. Namely, we need operations of Minkowski sum and difference, which first operand is not convex (the second one is convex because it is computed as a convex vectogram multiplied by the time step). A helpful fact is that both operations, sum and difference, can be fulfilled by one operation, namely, sum. Indeed, it is true that ∗ B = (A′ − B)′ . A−

Here, the prime denotes set complement. The authors worked out an algorithm for construction Minkowski sum when the first operand is a union of a number of simple-connected closed polygonal sets (possibly, non-convex), or is a complement to such a polygon (in other words, is an infinite closed set with a number of polygonal holes). 3.4.

Variants of Servomechanism Dynamics

1. A First Order Link. In work (Le Ménec, 2011), a pursuit problem is formulated that includes two pursuers and one evader. Each object has a three-dimensional phase variable: one-dimensional coordinate, velocity, and acceleration. The acceleration is affected by the control through a link of the first order: z˙1 = z2 ,

z˙2 = z3 ,

z˙3 = (u − z3 )/l.

(15)

Dynamics of the pursuer in problem (4) is similar, but now the geometric coordinate is one-dimensional. As above, l is the time constant describing the inertiality of the servomechanisms. 2. Damped Oscillating Control Contour. In work (Shinar et al., 2013), a game is considered, in which one of the objects has a damped oscillating control contour: z˙1 = z2 ,

z˙2 = z3 ,

z˙3 = z4 ,

z˙4 = −ω 2 z3 − ζz4 + u.

(16)

Here, ω is the natural frequency of the contour, ζ is the damping coefficient. 3. Tail/Canard Air Rudders. When considering an objects moving in the atmosphere, it is important to take into account position of its rudders with respect to the center of mass. A corresponding model is set forth in (Shima, 2005): z¨ = a + du, a˙ = (1 − d)u − a /l. (17)

The parameter d is defined by the position of the rudder. A positive (negative) value corresponds to the situation when the rudder is located in front of (behind) the center of mass. The first situation is called canard control scheme, the second one is called tail control scheme. The absolute value of d describes now far from the center of mass the rudder is. The parameter l again is the time constant.

176


4. Dual Tail/Canard Scheme. As a development of model (17), work (Shima and Golan, 2006) suggests a dynamics of an objects, which has both canard and tail rudders: z˙1 = z2 , z˙2 = z3 + dc uc + dt ut , z˙3 = (1 − dc )uc + (1 − dt )ut − z3 /l. (18)

Here, the constant dc > 0 describes the capabilities of the canard rudder (then index c means “canard” here), the constant dt < 0 corresponds to the tail rudder (the index t means “tail”). The time constant l regarded to be common for inertiality of both rudders. In this model, one can see two scalar controls uc and ut (or one vector control u = (uc , ut )⊤ taken from a rectangle). Therefore, formally this model does not belong to class (12). But the procedures for construction level sets of the value function suggested by the authors can be applied to such a dynamics with double scalar control. The difference is that formula (3) includes now two summands connected to two controls of the first player. 3.5.

Examples

In this subsection, we assume that the evader has dynamics of type (15). Example 5. Let both pursuers have the same dynamics of type (15). The parameters of the game are µ1 = µ2 = 1.5, ν = 1.0, lP1 = lP2 = 1/0.25, lE = 1/1.0, T1 = T2 = 15. The level set of the value function corresponding to c = 1.32 is shown in Fig. 13. In similar problems studied in detail by the authors (Ganebny et al., 2012; Kumkov et al., 2013), the advantage of a player in an individual game, can be detected analytically by analyzing the parameters ηi = µi /ν and εi = lE /lPi . When ηi > 1, ηi εi > 1, the ith pursuer has advantage over the evader. Vice versa, if ηi < 1, ηi εi < 1, the advantage belongs to the evader. If the parameters do not obey one of these conditions, then there is a situation of changing advantage of the ith pursuer over the evader in time. For the example, the data are such that both pursuers are weaker then the evader at the beginning of the backward time (near the target set, which is located in the plane t = T1 = T2 ). Due to this, at the beginning of the backward time, the

Fig. 13: Example 5. Narrow throats in a problem with three objects having dynamics (15)


177

Fig. 14: Example 6. A level set for example 2 with two narrow throats

t-sections start to contract. In Fig. 13, an instant can be distinguished when the infinite strips (the rectangles elongated along the corresponding axes) degenerate to a line due to this contraction and disappear. After this instant, the t-sections consist of two finite disconnected parts that correspond to zones of joint capture. If the position of the system is in such a zone, then the evader escaping from one pursuer is captured (with the given miss) by the another one. These parts continue to contract until an instant when the pursuers get the advantage. Further, the contraction turns to expansion, and at some instant growing parts joins into one simple-connected set that continue to grow infinitely. Since the parameters of both pursuers coincide and the time lengths of both individual games are equal, the dynamics of the coordinates x1 and x2 are the same. Therefore, the evolution of t-sections is the same along both coordinate axes. As it will be seen from the following examples, this is not true if the pursuers’ parameters or game lengths are different. Example 6. Let both pursuers be equal again, but now they have dynamics (16) with oscillating control contour. The parameters are the following: µ1 = µ2 = 0.3, ν = 1.3, ωP1 = ωP2 = 0.5, ζP1 = ζP2 = 0.0025, lE = 1.0, T = T1 = T2 = 30. The level set of the value function corresponding to c = 1.6 can be seen in Fig. 14. In this problem, due to fundamental difference of the pursuers’ and evader’s dynamics, it is difficult to get analytically the conditions of advantage of one or other player. Thus, the example is constructed on the base of the evolution of the players’ vectograms obtained numerically. Presence of two narrow throats is connected to repeat of a period such that at the beginning the advantage belongs to the evader and at the end it comes to pursuers. The repeat is provided by the oscillating type of the pursuers’ dynamics. More throats can be obtained by putting to the evader an oscillating dynamics too.

178


Fig. 15: Example 7. The level set W0.525 for the pursuers’ dynamics of type (18); the pursuers have different parameters of the dynamics

Example 7. Consider now a pair of pursuers both having dynamics (18). Let some dynamics parameters be different: aP1 ,max = 1.05, aP2 ,max = 1.15, lP1 = lP2 = 1/0.18807, dc,1 = dc,2 = 0.605, dt,1 = dt,2 = −0.5, α1 = 0.9, α2 = 0.8, aE,max = 0.95, dE = 0.157980, lE = 1.0, T1 = 32, T2 = 29. The value αi defines distribution of the control resource aPi ,max of the ith pursuer over the rudders: |uc | ≤ α · aP,max , |ut | ≤ β · aP,max ;

α, β ≥ 0, α + β = 1.

The level set Wc that correspond to c = 0.525 is given in Fig. 15. One can see that due to difference of the pursuers’ dynamics the contraction of the set is different along the axes x1 and x2 : degeneration of the infinite strips happens at different instants. Moreover, the finite parts remaining after degeneration of infinite strips have sufficiently different sizes along the two coordinate axes. 4.

Conclusion

A level set (Lebesgue set) of the value function corresponding to some value c can be regarded as a solvability set of a game problem with the payoff equal to c. For differential games with fixed termination instants, a level set of the value function is a tube in the space time × phase space along the time axis. It is very important to establish the law of evolution of time sections of the tubes in time. For example, if a tube corresponding to some c has a small length in time, then it means that the zone of guaranteed capture with the miss not greater than c is small too. If a solvability set has a narrow throat, that is, a period of time where its t-sections are close to degeneration (to loss of interior), then one should analyze accurately the


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possibility of practical application of the control law based on such a tube. In the paper, it is shown that the presence of narrow throats is not rare both in model differential games and practical pursuit problems. References Chernous’ko, F. L. and A. A. Melikyan (1978). Game problems of control and search. Nauka: Moscow. (in Russian) Chikrii, A. A. (1997). Conflict-Controlled Processes. Springer: Berlin. Ganebny, S. A., S. S. Kumkov, S. Le Ménec, and V. S. Patsko (2012). Model problem in a line with two pursuers and one evader. Dyn. Games Appl., 2, 228–257. Grigorenko, N. L. (1990). Mathematical Methods for Control of a Number of Dynamic Processes. Moscow, Moscow State University. (in Russian) Isaacs, R. (1965). Differential Games. John Wiley and Sons: New York. Isakova, E. A., G. V. Logunova and V. S. Patsko (1984). Computation of stable bridges for linear differential games with fixed time of termination. In: Algorithms and Programs for Solving Linear Differential Games (Subbotin, A. I. and V. S. Patsko, eds), pp. 127– 158. Inst. of Math. and Mech.: Sverdlovsk (in Russian) Krasovskii, N. N. and A. I. Subbotin (1974). Positional Differential Games. Nauka: Moscow. (in Russian) Krasovskii, N. N. and A. I. Subbotin (1988). Game-Theoretical Control Problems. SpringerVerlag: New York. Kumkov, S. S., V. S. Patsko and J. Shinar (2005). On level sets with “narrow throats” in linear differential games. Int. Game Theory Rev., 7(3), 285–311. Kumkov, S. S., V. S. Patsko and S. Le Ménec (2013). Game with two pursuers and one evader: case of weak pursuers. In: Annals of the International Society of Dynamic Games, Vol. 13, Advances in Dynamic Games. Theory, Applications, and Numerical Methods, (Krivan, V. and G. Zaccour, eds), pp. 263–293. Switzerland: Birkhauser. Le Ménec, S. (2011). Linear differential game with two pursuers and one evader. In: Annals of the International Society of Dynamic Games, Vol. 11, Advances in Dynamic Games. Theory, Applications, and Numerical Methods for Differential and Stochastic Games, (Breton, M. and K. Szajowski eds), pp. 209–226. Boston: Birkhauser. Melikyan, A. A. and J. Shinar (2000). Identification and construction of singular surface in pursuit-evasion games. In: Annals of the International Society of Dynamic Games, Vol. 5, Advances in Dynamic Games and Applications (Filar, J. A., V. Gaitsgory and K. Mizukami, eds), pp. 151–176. Springer: Berlin. Mezentsev, A. V. (1972). A class of differential games. Engrg. Cybernetics, 9(6), 975–978. Nikol’skii, M. S. (1984). The first direct method of L.S.Pontryagin in differential games. Moscow State Univ: Moscow. (in Russian) Pontryagin, L. S. (1964). On some differential games. Soviet Math. Dokl., 5, 712–716. Pontryagin, L. S. and E. F. Mischenko (1969). Problem of evading by a controlled object from another one. Doklady Akad. Nauk SSSR, 189(4), 721–723. (in Russian) Pontryagin, L. S. (1972). Linear differential games. International Congress of Mathematics, Nice, 1970. Reports of Soviet Mathematicians. Moscow: Nauka, 248–257. (in Russian) Pschenichnyi, B. N. and M. I. Sagaidak (1970). Differential games of prescribed duration. Cybernetics, 6(2), 72–80. Shima, T. and J. Shinar (2002). Time-varying linear pursuit-evasion game models with bounded controls. J. Guid. Contr. Dyn., 25(3), 425–432. Shima, T. (2005). Capture conditions in a pursuit-evasion game between players with biproper dynamics. JOTA, 126(3), 503–528. Shima, T. and O. M. Golan (2006). Bounded differential games guidance law for dualcontrolled missiles. IEEE Trans. on Contr. Sys. Tech., 14(4), 719–724.

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Shinar, J., V. Y. Glizer and V. Turetsky (2013). The effect of pursuer dynamics on the value of linear pursuit-evasion games with bounded controls. In: Annals of the International Society of Dynamic Games, Vol. 13, Advances in Dynamic Games. Theory, Applications, and Numerical Methods, (Krivan, V. and G. Zaccour, eds), pp. 313–350. Switzerland: Birkhauser. Shinar, J., M. Medinah and M. Biton (1984). Singular surfaces in a linear pursuit-evasion game with elliptical vectograms. JOTA, 43(3), 431–458. Shinar, J. and T. Shima (2002). Non-orthodox guidance law development approach for intercepting maneuvering targets. J. Guid. Contr. Dyn., 25(4), 658–666. Shinar, J. and M. Zarkh (1996). Pursuit of a faster evader — a linear game with elliptical vectograms. Proceedings of the Seventh International Symposium on Dynamic Games, Yokosuka, Japan, 855–868.

Strictly Strong (n − 1)-equilibrium in n-person Multicriteria Games Denis V. Kuzyutin1 , Mariya V. Nikitina2 and Yaroslavna B. Pankratova3 1 St.Petersburg State University, Faculty of Applied Mathematics and Control Processes, Universitetskii pr, 35, St.Petersburg, 198504, Russia E-mail: [email protected] 2 International Banking Institute, Nevski pr, 60, St.Petersburg, 191023, Russia E-mail:[email protected] 3 St.Petersburg State University, Faculty of Applied Mathematics and Control Processes, Universitetskii pr, 35, St.Petersburg, 198504, Russia E-mail: [email protected]

Abstract Using some specific approach to the coalition-consistency analysis in n-person multicriteria games we introduce two refinements of (weak Pareto) equilibria: the strong and strictly strong (n − 1)-equilibriums. Axiomatization of the strictly strong (n − 1)-equilibria (on closed families of multicriteria games) is provided in terms of consistency, strong one-person rationality, suitable variants of Pareto optimality and converse consistency axiom and others. Keywords: multicriteria games; Pareto equilibria; strong equilibrium; consistency; axiomatizations.

1.

Introduction

The concept of strictly strong (n − 1)-equilibria (in n-person strategic games and in multicriteria games) is based on some specific approach to the coalition-consistency analysis, offered in (Kuzyutin, 1995; Kuzyutin, 2000). Namely, we suppose that trying to investigate the coalition-consistency of some acceptable Nash equilibrium x, every player i does not consider the deviations of coalitions S, i ∈ S with her participance (since player i may be sure in her own strategic choice xi ). This approach allows to make the strong Nash equilibria (Aumann, 1959) requirements slightly weaker. We show (in section 2) that the strong and strictly strong (n−1)-equilibrium differs from other closely related solution concepts: coalition-proof equilibrium (Bernheim et al., 1987) and semi-strong Nash equilibrium (Kaplan, 1992). The axiomatization of strong and strictly strong (n − 1)-equilibria in n-person strategic games was given in (Kuzyutin, 2000). In section 3 we explore the same approach to coalition-consistency analysis in n-person multicriteria games (or the games with vector payoffs) and offer two refinements of the weak Pareto equilibria (Shapley, 1959; Voorneveld et al., 1999). The axiomatic characterization of strictly strong (n − 1)- equilibria (on closed families of multicriteria games) is provided in section 4 using the technique offered in

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Denis V. Kuzyutin, Mariya V. Nikitina, Yaroslavna B. Pankratova

(Peleg and Tijs, 1996; Norde et al., 1996; Voorneveld et al., 1999). In this axiomatization the suitable variants of Pareto-optimality and converse consistency axioms play a role to distinguish between the strictly strong (n − 1)-equilibria and other equilibrium solutions in multicriteria games. 2.

Strong and strictly strong (n − 1)-equilibrium in strategic games

Consider a game in strategic form G = (N, (Ai )i∈N , (ui )i∈N , where N is a finite set | = n Ai 6= ∅ is the set of player’s i strategies; and ui : A = Q of players |N 1 A → R is the payoff function of player i ∈ N . A solution (optimality j j∈N principle) ϕ, defined on a class of strategic games Γ , is a function that assigns to each game G = (N, (Ai )i∈N , (ui )i∈N ∈ Γ a subset ϕ(G) of A. We’ll call a strategy profile x the optimal situation, if x ∈ ϕ(G). Q Let S ⊂ N , S 6= ∅, be a coalition; S ⊂ N , S 6= ∅, N Ů proper coalition; AS = j∈S Aj — a set of all possible players’ i ∈ S strategy profiles. The concept of strong (Nash) equilibria was offered by Aumann, 1959. Definition 1. x ∈ A is a strong Nash equilibrium (SNE), if ∀S ⊂ N , S 6= ∅, ∀yS ∈ AS , ∃i ∈ S: ui (x) ≥ ui (yS , xN \S ), where yS = (yj )j∈ S, xN \S = (xj )j∈N \S . Definition 2. x ∈ A is weakly Pareto-optimal (W P O), if ∀y ∈ A, ∃i ∈ N : ui (x) ≥ ui (y). Definition 3. x ∈ A is a strictly strong Nash equilibrium (SSN E), if there do not exist coalition S ⊂ N and yS ∈ AS such that: ui (yS , xN \S ) ≥ ui (x) ∀i ∈ S, ∃j ∈ S : uj (yS , xN \S ) > uj (x). Notice that the concept of SN E (as well as SSN E) deals with a r b i t r a r y deviations of a l l p o s s i b l e coalitions S ⊂ N . We denote by N E(G), SN E(G), SSN E(G) the set of Nash equilibriums (Nash, 1950), strong Nash equilibriums and strictly strong Nash equilibriums of G respectively. The following inclusions hold: N E(G) ⊃ SN E(G) ⊃ SSN E(G). Unfortunately, the sets SN E(G) and SSN E(G) are often empty (see, for instance, Petrosjan and Kuzyutin, 2008) by the reason of "too strong" requirements to the solution used in def. 1, 3. We’ll consider an opportunity to make these requirements slightly weaker that leads to new concept of coalition-stable equilibrium. We guess a game G = (N, (Ai )i∈N , (ui )i∈N ) is "of common knowledge", when every player knows all players’ strategy sets and payoff functions. Moreover, suppose that trying to investigate the coalition stability of some acceptable strategy profile x, every player i does not consider the deviations of coalitions S ∈ i with her participance since player i may be sure in her own strategic choice xi . The related motivation was used early for other purposes in Kuzyutin, 1995 to define the istability property in n-person extensive game.

Strictly Strong (n − 1)-equilibrium in n-person Multicriteria Games

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Definition 4. Let G = (N, (Ai )i∈N , (ui )i∈N ) ∈ Γ , |N | = n. 1. n ≥ 2: x ∈ A is a strong (n−1)-equilibrium (SN E n−1 ), if for every player i ∈ N the following condition holds: ∀S ⊂ N \ {i}, ∀ yS ∈ AS , ∃ j ∈ S : uj (x) ≥ uj (yS , xN \S ); 2. n = 1: xi ∈ Ai is a strong (n − 1)-equilibrium in one-player game G = ({i}, Ai , ui ), if ui (xi ) ≥ ui (yi ) ∀yi ∈ Ai Definition 5. 1. n ≥ 2: x ∈ A is a strictly strong (n−1)-equilibrium (SSN E n−1 ), if for every player i ∈ N there do not exist a coalition S ⊂ N \ {i} and yS ∈ AS such that: uj (yS , xN \S ) ≥ uj (x) ∀ j ∈ S, 2. n = 1: SSN E

n−1

∃ k ∈ S : uk (yS , xN \S ) > uk (x).

coincides with SN E n−1 .

Remark 1. Another possible definition of SN E n−1 (n ≥ 2) is as follows: x is a SN E n−1 if ∀ S ⊂ N, |S| ≤ n − 1, ∀ yS ∈ AS ∃ j ∈ S: uj (x) ≥ uj (yS , xN \S ). However, we guess the def. 4 is more useful to clarify the offered approach every player (independently of others) holds on to check the coalition stability of x. The optimally principles SN E n−1 and SSN E n−1 deal with a r b i t r a r y deviations of c e r t a i n (c r e d i b 1 e) coalitions. It is clear that

Further we have:

N E(G) ⊃ SN E n−1 (G) ⊃ SSN E n−1 (G). N E(G) ⊃ SN E n−1 (G) ⊃ SN E(G),

N E(G) ⊃ SSN E n−1 (G) ⊃ SSN E(G).

The example shows that these inclusions may be strict.

Example 1. Let the three-person game G = (N = {1, 2, 3}, A1 = {x1 , y1 }, A2 = {x2 , y2 }, A3 = {x3 , y3 }, (ui )i∈N ), be given by the following normal form: x3 y3 x2 y2 x2 y2 x1 (7, 7, 0) (0, 0, 5) x1 (4, 4, 9) (0, 0, 5) y1 (0, 0, 5) (5, 5, 15) y1 (0, 0, 5) (0, 0, 0) For the convenience we’ll restrict ourselves to the players’ pure strategies (player 1 chooses a row, player 2 a column and player 3 a block of the table). Here: N E(G) = {(y1 , y2 , x3 ), (x1 , x2 , y3 )}; SN E(G) = SSN E(G) = ∅; SN E n−1 (G) = SSN E n−1 (G) = {x1 , x2 , y3 )}; W P O(G) = {(y1 , y2 , x3 ), (x1 , x2 , x3 )} is the set of weak Pareto-optimal strategy profiles in G.

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Certainly, player 3 can not accept the situation (y1 , y2 , x3 ) ∈ N E(G)∩W P O(G) since the best player 1 and player 2 joint response to the player 3 strategy x3 is (x1 , x2 ) that leads to the least possible payoff of player 3. At the same time the strategy profile (x1 , x2 , y3 ) is free from such danger, and satisfies the requirements of coalition stability from def. 4, 5 although does not satisfy weak Pareto-optimality. Thus one can use the strong (n − 1)-equilibrium concept to obtain a unique optimal outcome in strategic game G. Notice, that the extended analysis of closely related example is offered in Bernheim et al., 1987 in connection with the coalition-proof Nash equilibrium concept. Definition 6. Let G = (N, (Ai )i∈N , (ui )i∈N ) be a game, let x ∈ A and let ∅ 6= S ⊂ N . An internally consistent improvement (ICI) of S upon x is defined by induction on |S|. If |S| = 1, that is S = {i} for some i ∈ N , then yi ∈ Ai is an ICI of i upon x if it is an improvement upon x, that is, ui (yi , xN \{i} ) > ui (x). If |S| > 1 then yS ∈ AS is an ICI of S upon x if: 1. ui (ys , xN \S ) > ui (x) for all i ∈ S, 2. no T ⊂ S, T 6= ∅, S has an ICI upon (yS , xN \S ). x is a coalition-proof Nash equilibrium (CP N E) if no T ⊂ N , T 6= ∅, has an ICI upon x. The reader is refereed to Bernheim et al., 1987 for discussion and motivation. Definition 7 (Kaplan, 1992). Let G = (N, (Ai )i∈N , (ui )i∈N ) be a game. x ∈ A is a semi-strong Nash equilibrium (SM SN E), if for every ∅ 6= S ⊂ N and every yS ∈ N E(GS,x ) there exists i ∈ S such that ui (x) ≥ ui (yS , xN \S ). Notice that the concept of CP N E (as well as SM SN E) deals only with c e r t a i n deviations of a l l p o s s i b l e coalitions S ⊂ N . To clarify the difference between CP N E and SM SN E from the one hand and the strong (n − 1)-equilibrium concept from the other we consider the following example. Example 2. G = (N = {1, 2, 3}, A1 = {x1 , y1 }, A2 = {x2 , y2 }, A3 = {x3 , y3 }, (ui )i∈N ), is the three-person strategic game: x3 y3 x2 y2 x2 y2 x1 (9, 9, 0) (4, 10, 0) x1 (4, 4, 9) (0, 0, 5) y1 (0, 0, 5) (5, 5, 10) y1 (0, 0, 5) (0, 0, 0) Here: N E(G) = {(y1 , y2 , x3 ), (x1 , x2 , y3 )}; W P O(G) = {(x1 , x2 , x3 ), (x1 , y2 , x3 ), (y1 , y2 , x3 )}; SN E(G) = SSN E(G) = ∅; CP N E(G) = SM SN E(G) = N E(G) ∩ W P O(G) = {(y1 , y2 , x3 )}, but SSN E n−1 (G) = {(x1 , x2 , y3 )}. Notice that (as in example 1) player 3 can reject the strategy profile (y1 , y2 , x3 ) by the reason of other players have the profitable joint deviation (x1 , x2 ) from (y1 , y2 , x3 ) that is still possible (although (x1 , x2 ) is not ICI of S = {1, 2} upon (y1 , y2 , x3 )). If such deviation takes place player 3 will receive the least feasible payoff (independently of further possible deviation y2 of player 2).


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Remark 2. SSN E n−l does not coincide with CP N E (as soon as with SM SN E) in general case. Remark 3. Let ϕ be one of the optimality principles: CP N E or SM SN E. SSN E n−1 is not a refinement of ϕ, and ϕ is not a refinement of SSN E n−1 . 3.

Coalition-stable equilibriums in multicriteria games

Now let us turn to so-called multicriteria games (or the games with vector payoffs) when every player may take several criteria into account. Formally, let G = (N, (Ai )i∈N , (ui )i∈N ) be a finite multicriteria game, were N is a finite set of players, |N | = n, Ai 6= ∅ is the finite set of player i ∈ N , Q of pure strategies r(i) and for each player i ∈ N the function ui : A → R maps each strategy j j∈N profile to a point in r(i)-dimensional Euclidean space. Note that player i in multicriteria game G tries to maximaize r(i) scalar criteria (i.e. all the components of her vector valued payoff function ui (xi , x−i )). The concept of equilibrium point for multicriteria games was proposed by Aumann, 1959 as a natural generalization of the Nash equilibrium concept for unicriterium games. Let a, b ∈ Rt , and a > b means that ai > bi for all i = 1, . . . , t; a ≥ b means that ai ≥ bi for all i = 1, . . . , t, and a 6= b. The vector a ∈ M ⊆ Rt is weak Pareto efficient (or undominated) in M iff {b ∈ t R : b > a} ∩ M = ∅. In this case we’ll use the following notation: a ∈ W P O(M ). Given strategy profile x = (xi , x−i ) in the finite multicriteria game G denote by Mi (G, x−i ) = {ui (yi , x−i ), yi ∈ Ai } the set of all player’s i attainable vector payoffs (due to arbitrary choice of his strategy yi ∈ Ai ). Q Definition 8. The strategy profile x = (x1 , . . . , xn ) ∈ j∈N Aj is called (weak Pareto) equilibrium in multicriteria game G iff for each player i ∈ N there does not exist a strategy yi ∈ Ai such that: ui (yi , x−i ) > ui (xi , x−i )

(1)

Note that (1) is equivalent to the following condition: ui (xi , x−i ) ∈ W P O(Mi (G, x−i )) ∀i ∈ N.

(2)

Let E(G) be the set off all (weak Pareto) equilibriums in multicriteria game G. Definition 9. The strategy profile x = (x1 , . . . , xn ) is called strong equilibrium (in a sense of Aumann, 1959 ) in multicriteria game G iff Y ui (yS , x−S ) > ui (xS , x−S ) ∀S ⊂ N, S 6= ∅ ∄ yS ∈ Aj : i∈S . j∈S

Definition 10. The strategy profile x = (x1 , . . . , xn ) is called strictly strong equilibrium in multicriteria game G iff Y ui (yS , x−S ) ≥ ui (xS , x−S ) ∀S ⊂ N, S 6= ∅ ∄ yS ∈ Aj : i∈S . j∈S

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Definition 11. Let G = (N, (Ai )i∈N , (ui )i∈N ) be a finite multicriteria game with n players, |N | = n. Q 1. n ≥ 2: x = (x1 , . . . , xn ) ∈ j∈N Aj is a strong (n − 1)-equilibrium if for each player i ∈ N the following condition holds: uj (yS , x−S ) > ui (xS , x−S ) ∀S ⊂ N \{i}, ∄ yS ∈ AS : (3) j∈S . 2. n = 1: xi ∈ Ai is a strong (n − 1)-equilibrium in one player multicriteria game G = ({i}, Ai , ui ) if ∄ yi ∈ Ai : ui (yi ) > ui (xi ). Let SE(G), SSE(G) and SE n−1 (G) be the sets S of all strong equilibriums, strictly strong equilibriums and strong (n − 1)-equilibriums in multicriteria game G correspondly. Definition 12. 1. n ≥ 2: x = (x1 , . . . , xn ) ∈ A is a strictly strong (n − 1)equilibrium if for every player i ∈ N the following condition holds: uj (yS , x−S ) ≥ uj (xS , x−S ) ∀S ⊂ N \{i}, ∄ yS ∈ AS : (4) j∈S . 2. n = 1: xi ∈ Ai is a strictly strong (n − 1)-equilibrium in one-person multicriteria game G = ({i}, Ai , ui ), if ∄ yi ∈ Ai : ui (yi ) ≥ ui (xi ). The set of all strictly strong (n − 1)-equilibriums in G denote by SSE n−1 (G). Q Definition 13. The strategy profile x = (x1 , . . . , xn ) ∈ j∈N Aj is called Pareto efficient in a multicriteria game G iff Y ui (y) ≥ ui (x) ∄y∈ Aj : (5) i∈N . j∈N

The set of all Pareto efficient strategy profiles in G denote by P OMG (G). It is clear that E(G) ⊃ SE n−1 (G) ⊃ SSE n−1 (G), SSE n−1 (G) ⊃ SSE(G), P OMG (G) ⊃ SSE(G).

4.

Axiomatization of strictly strong (n − 1)- equilibria in multicriteria games

In this section we give axiomatization of SSE n−1 correspondence on closed classes of multicriteria games in terms on consistency, strong one-person rationality, suitable variants of converse consistency and Pareto-optimality axiom and others. Let Γ be a set of muliticriteria games G and let ϕ be a solution on Γ . Definition 14. ϕ satisfies strong one-person rationality (SOP R) if for every oneperson game G = ({i}, Ai , ui ) ∈ Γ ϕ(G) = {xi ∈ Ai | ∄ yi ∈ Ai : ui (yi ) ≥ ui (xi )}


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Let G = (N, (Ai )i∈N , (ui )i∈N ) be a game, n =| N |≥ 2, let S ⊂ N be a proper coalition, i.e. S 6= ∅, N . Definition 15. The proper reduced game GS,x of G (with respect to S and x) is the multicriteria game GS,x = (S, (Ai )i∈S , (uxi )i∈S ), where uxi (yS ) = ui (yS , xN \S ) ∀ yS ∈ AS , ∀ i ∈ S. A family Γ of multicriteria games is r-closed, if G = (N, (Ai )i∈N , (ui )i∈N ) ∈ Γ , S ⊂ N , S 6= ∅, N and x ∈ A imply that GS,x ∈ Γ . Definition 16. Let Γ be a r-closed family of strategic games. A solution ϕ on Γ satisfies consistency (CON S), if for every G = (N, (Ai )i∈N , (ui )i∈N ) ∈ Γ , ∀ S ⊂ N , S 6= ∅, N , ∀ x ∈ ϕ(G) the following condition holds: xS ∈ ϕ(GS,x ). The CON S property means the restriction xS of the optimal strategy profile x ∈ ϕ(G) still satisfies the optimality principle ϕ in every reduced game GS,x . If G = (N, (Ai )i∈N , (ui )i∈N ) ∈ Γ , and n ≥ 2, then we denote: ϕ(G) e = {x ∈ A | ∀ S ⊂ N, S 6= ∅, N, xS ∈ ϕ(GS,x )}

(6)

Taking (6) into account one can notice that CON S property means ϕ(G) ⊂ ϕ(G) e for every G ∈ Γ .

Definition 17. A solution ϕ on Γ satisfies (n − 1)-Pareto optimality for multicrin−1 teria games (P OMG ), if for every G = (N, (Ai )i∈N , (ui )i∈N ) ∈ Γ with at least two players (n ≥ 2), for every x ∈ ϕ(G) the following conditions holds: uj (y−i , xi ) ≥ uj (x−i , xi ) ∀ i ∈ N ∄ y−i ∈ A−i : (7) j ∈ N \ {i}. . n−1 Notice that ϕ satisfies P OMG iff ∀ x ∈ ϕ(G), ∀i ∈ N

xN \{i} ∈ P OMG (GN \{i},x ). n−1 Let P OMG (G) be the set of all strategy profiles x ∈ Πi∈N Ai in G, satisfying (7).

Definition 18. Let Γ be a r-closed family of strategic games. A solution ϕ satisfies COCON S∗n−1 (the appropriate version of converse for SSE n−1 ), if for every G = (N, (Ai )i∈N , (ui )i∈N ) ∈ Γ , n ≥ 2, it is true that: n−1 x ∈ ϕ(G) e and x ∈ P OMG (G) ⇒ x ∈ ϕ(G) (8)

In accordance with COCON S∗n−1 property if the restrictions xS of some strategy profile x ∈ A satisfy optimality principle ϕ in every reduced game GS,x , and x ∈ n−1 P OMG (G) then x is the optimal strategy profile in the original game G. Theorem 1. A solution ϕ on a r-closed family of multicriteria games Γ satisn−1 fies CON S, SOP R, P OMG and COCON S∗n−1 , if and only if ϕ = SSE n−1 (i.e. ϕ(G) = SSE n−1 (G) for every G ∈ Γ ).

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n−1 Proof. 1. It is not difficult to verify that SSE n−1 satisfies CON S, SOP R, P OMG n−1 n−1 and COCON S∗ . Let us check here that SSE satisfies CON S (for instance). If x ∈ SSE n−1 then ∀ S ⊂ N \{i} ∄ yS ∈ AS : uj (yS , x−S ) ≥ uj (xS , x−S ) = uj (x) (9) j ∈ S. .

Consider an arbitrary coalition S1 ⊂ N , S1 6= ∅, N and reduced game GS1 ,x . Let S ⊂ S1 \{i} ⊂ S1 ⊂ N . Using (9) we have that uj (yS , xS1 \S , xN \S1 ) = uj (yS , xN \S ) ≥ uj (x) ∀ S ⊂ S1 \{i} ∄ yS ∈ AS : j ∈ S. . This means that xS1 ∈ SSE n−1 (GS1 ,x ), i.e. SSE n−1 satisfies CON S. 2. Now let ϕ be a solution on Γ that satisfies the foregoing four axioms. We prove by induction (on the number of players n) that ϕ(G) = SSE n−1 (G) for every G ∈ Γ. By SOP R ϕ(G) = SSE n−1 (G) for every one-person multicriteria game G ∈ Γ . Now assume that ϕ(G) = SSE n−1 (G) ∀ G = (N, (Ai )i∈N , (ui )i∈N ) ∈ Γ,

(10)

where 1 ≤ |N | ≤ k, k ≥ 1, and consider an arbitrary (k + 1)-person multicriteria game G ∈ Γ . Let x ∈ ϕ(G). From the CON S of ϕ it follows that (11)

x ∈ ϕ(G). e

Using the induction hypothesis and the notation (6) we obtain:

n−1 Moreover, by P OMG of ϕ,

] ϕ(G) e = SSE

n−1

(G).

n−1 x ∈ P OMG (G).

(12)

COCON S∗n−1 property of n−1

Taking into account (10), (11), and (12), and SSE n−1 , n−1 we obtain that x ∈ SSE (G), and, hence, ϕ(G) ⊂ SSE (G). Similarly, we may prove that SSE n−1 (G) ⊂ ϕ(G) for every (k + 1)-person multicriteria game G ∈ Γ . The inductive conclusion completes the proof. Corollary 1. Let ϕ be a solution on r-closed family of games Γ , that satisfies n−1 CON S and P OMG . Then ϕ(G) ⊂ SSE n−1 (G) ∀ G ∈ Γ, | N |= n ≥ 2. Proof. Let x ∈ ϕ(G), n ≥ 2. To prove that x ∈ SSE n−1 (G) we need to verify that for every possible proper coalition uj (yS , x−S ) ≥ uj (x), S ⊂ N, S 6= N, ∅, ∄ yS ∈ AS : j ∈ S. i.e. xS ∈ P OMG (GS,x ) ∀ S ⊂ N : s =| S |= 1, 2, . . . , n − 1

(13)

Strictly Strong (n − 1)-equilibrium in n-person Multicriteria Games n−1 By P OMG of ϕ

189

xS ∈ P OMG (GS,x ) ∀ S ⊂ N : s = n − 1.

If n = 2 we have already established (13) for all possible proper coalitions. Otherwise (if n ≥ 3), consider a proper reduced game GS,x , where s = n − 1. By CON S of ϕ n−1 s−1 xS ∈ ϕ(GS,x ), and by P OMG xS ∈ P OMG (GS,x ), i.e. xT ∈ P OMG (GT,x ) ∀ T ⊂ S, t = |T | = s − 1 = n − 2. Using the same approach we can establish (13) for every proper coalition S ⊂ N , s =| S |= n − 1, n − 2, . . . , 1. Another axiomatic characterization of the SSE n−1 correspondence involves the following axioms (Peleg and Tijs, 1996). Definition 19. Let Γ be a set of multicriteria games, and ϕ be a solution Γ . ϕ satisfies independence of irrelevant strategies (IIS) if the following condition holds: if G = (N, (Ai )i∈N , (ui )i∈N ) ∈ Γ , x ∈ ϕ(G), xi ∈ Bi ⊂ Ai for all i ∈ N , and G∗ = (N, (Bi )i∈N , (ui )i∈N ) ∈ Γ , then x ∈ ϕ(G∗ ). A family of games Γ is called s-closed, if for every game G = (N, (Ai )i∈N , (ui )i∈N ) ∈ Γ , and Bi ⊂ Ai , Bi 6= ∅, i ∈ N , the game G∗ = (N, (Bi )i∈N , (ui )i∈N ) ∈ Γ . Further, Γ is called closed, if it is both r-closed and s-closed. For example, the set of all finite multicriteria games is closed. Definition 20. A solution ϕ on r-closed family of games Γ satisfies the dummy axiom (DU M ), if for every game G = (N, (Ai )i∈N , (ui )i∈N ) ∈ Γ and every "dummy player" d in G (i.e. player d ∈ N such that | Ad |= 1), the following condition holds: ϕ(G) = Ad × ϕ(GN \{d} , x), where x is an arbitrary strategy profile from A. Note, that SSE n−1 satisfies IIS and DU M . Proposition 1. (Peleg B., Tijs S. [1996]) If a solution ϕ on closed family of games Γ satisfies IIS and DU M , then ϕ also satisfies CON S. The next axiomatic characterization of SSE n−1 correspondence follows from the theorem 1 and proposition 1. Theorem 2. Let Γ be a closed family of multicriteria games. The SSE n−1 n−1 correspondence is the unique solution on Γ that satisfies SOP R, P OMG , n−1 COCON S∗ , IIS and DU M . References Aumann, R. J. (1959). Acceptable points in general cooperative n-person games. Contributions to the theory of games. Prinston. 4, 287–324. Bernheim, B., Peleg, B. and Winston, M. (1987). Coalition-proof Nash equilibria I. Concept. Journal of Economics Theory, 42, 1–12. Borm, P., Van Megen, F. and Tijs, S. (1999). A perfectness concept for multicriteria games. Mathematical Methods of Operation Research, 49, 401–412. Kaplan, G. (1992). Sophisticated outcomes and coalitional stability. MSc Thesis. Department of Statistics, Tel-Aviv University. Kuzyutin, D. (1995). On the problem of solutions’ stability in extensive games. Vestnik of St.Petersburg Univ., Ser. 1, Iss. 4, 18–23 (in Russian).

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Kuzyutin, D. (2000). Strong (N −1)-equilibrium. Concepts and axiomatic characterization. International Journal of Mathematics, Game Theory and Algebra, Vol. 10, No. 3, 217– 226. Kuzyutin, D. (2012). On the consistency of weak equilibria in multicriteria extensive games. Contributions to game theory and management, Collected papers, Vol. V . Editors L.A. Petrosjan, N. A. Zenkevich. St.Petersburg Univ. Press, 168-177. Nash, J. F. (1950). Equilibrium points in n-person games. Proc. Nat. Acad. Sci. USA, Vol. 36, 48–49. Norde, H., Potters, J., Reijnierse, H. and Vermeulen, D. (1996). Equilibrium selection and consistency. Games and Economic Behavior, 12, 219–225. Peleg, B., Potters, J. and Tijs, S. (1996). Minimality of consistent solutions for strategic games, in particular for potential games. Economic Theory, 7, 81–93. Peleg, B. and Tijs, S. (1996). The consistency principle for games in strategic form. International Journal of Game Theory, 25, 13–34. Petrosjan, L. and Kuzyutin, D. (2008). Consistent solutions of positional games. St.Petersburg Univ. Press. 330 p. (in Russian) Petrosjan, L. and Puerto, J. (2002). Folk theorems in multicriteria repeated n- person games. Sociedad de Estatistica e Investigation Operativa Top, Vol. 10, No. 2, 275–287. Shapley, L. (1959). Equilibrium points in games with vector payoffs. Naval Research Logistics Quarterly, 1, 57–61. Voorneveld, M., Vermeulen, D. and Borm, P. (1999). Axiomatization of Pareto equilibria in multicriteria games. Games and Economic Behavior, 28, 146–154.

The Nash Equilibrium in Multy-Product Inventory Model Elena A. Lezhnina, Victor V. Zakharov St.Petersburg State University, Faculty of Applied Mathematics and Control Processes, Universitetskii pr. 35, St.Petersburg, 198504, Russia E-mail: [email protected]

Abstract In this paper game theory model of inventory control of a set of products is treated. We consider model of price competition. We assume that each retailer can use single-product and multi-product ordering . Demand for goods which are in stock is constant and uniformly distributed for the period of planning. Retailers are considered as players in a game with two-level decision making process. At the higher level optimal solutions of retailers about selling prices for the non-substituted goods forming Nash equilibrium are based on optimal inventory solution (order quantity or cycle duration) as a reaction to chosen prices of the players. We describe the price competition in context of modified model of Bertrand. Thus at the lower level of the game each player chooses internal strategy as an optimal reaction to competitive player’s strategies which are called external. Optimal internal strategies are represented in analytical form. Theorems about conditions for existences of the Nash equilibrium in the game of price competition are proved. Keywords:game theory, non-coalition game, Bertrand oligopoly, Nash equilibrium, logistics.

1.

Introduction

Inventory management of physical goods and other products or elements is an integral part of logistic systems common to all sectors of the economy including industry, agriculture, and defense. Since the logistic costs account for up to 20% of the costs of Russian companies under the modern conditions, the issue of reducing costs for optimization of logistics systems is particularly relevant. The first paper on mathematical modeling in inventory management was written by Harris (Harris ,1915) in 1915. We may also note famous book by Hadley and Whitin (Hadley and Whitin, 1963), as well as books by Hax and Candea (Hax and Candea, 1963) and Tersine (Tersine, 1994). Inventory control systems with single decision maker capture many important aspects of inventory management. On the other hand they usually don’t take into account decisions of other competitors on the market. Game theory is a mathematical theory of decision making by participants in conflicting or cooperating situations. Its goal is to explain, or to provide a normative guide for, rational behavior of individuals confronted with strategic decision or involved in social interaction. The theory is concerned with optimal strategic behavior, equilibrium situations, stable outcomes, bargaining, coalition formation, equitable allocations, and similar concepts related to resolving group differences. The field of game theory may be divided roughly in two parts, namely non-cooperative game theory and cooperative game theory. Models in non-cooperative game theory assume that each player

192

Elena A. Lezhnina, Victor V. Zakharov

in the game optimizes its own objective and does not care about the effect of its decisions on others. The focus is on finding optimal strategies for each player. Binding agreements among the players are not allowed. Up to date, many researchers use non-cooperative game theory to analyse supply chain problems. Non-cooperative game theory uses the notion of a strategic equilibrium or simply equilibrium to determine rational outcomes of a game. Numerous equilibrium concepts have been proposed in the literature (van Damme, 1991). A lot of researches are devoted to analytical design of contracting arrangements to eliminate inefficiency of decision making in supply chain with several players like echelon inventory game and local inventory game (see Cachon review, 2003). Two-level strategic decision making in Bertrand type competitive inventory model for the first time was treated in (Mansur Gastratov, Victor Zakharov, 2011). Some widely used concepts are dominant strategy, Nash equilibrium, and sub-game perfect equilibrium. Nash Equilibrium says that strategies chosen by all players are said to be in Nash equilibrium if no player can benefit by unilaterally changing their strategy. Nash (Nash, 1951) proved that every finite game has at least one Nash equilibrium. Historically, most researchers establish the existence of an equilibrium based on the study of the concavity or quasi-concavity of profit function. Dasgupta and Maskin (Dasgupta, Maskin, 1986), Parlar (Parlar, 1988), Mahajan and van Ryzin (Mahajan, van Ryzin, 2001), Netessine et al. (Mahajan, van Ryzin, 2001), among others establish the existence of a Nash equilibrium based on the two above-mentioned properties of the profit function A starting paper on mathematical models of inventory management was Harris (1915). Parlar (Parlar, 1988)was the first to analyse the inventory problem in game theory frameworks. There are two main classes of models depending on whether price or quantity is regarded as the decision variable. The static models of Cournot (Cournot, 1838) and Bertrand (Bertrand, 1883) were developed long before modern game theoretic methods. Hence to control the prices is faster and easier than to good’s quantities, we use the Bertrand oligopoly model of price competition (Friedman, 1983). The Harris – Wilson formula is a traditional method for determining the order or production quantity if you know the total uniform consumption during a period of time. The formula tries to find an optimal balance between the two costs to minimize the total cost , which is known as the economic order quantity (EOQ). The classical EOQ formula is essentially a trade-off between the ordering cost, assumed to be a flat fee per order, and inventory holding cost. A lager order quantity reduces ordering frequency, and, hence ordering cost. On the other hend, a smaller order quantity reduces average inventory but requires more frequent ordering and higher ordering cost. This formula dating for 1913 is extremely well-known (Harris, 1915). 2.

Preliminaries

2.1. Non-cooperative games Let’s consider a system n

n

Γ = hN, {Ωi }i=1 , {Πi }i=1 i . This system is called a non-cooperative game, where N = {1, 2, . . . , n} – set of players, Ωi – set of strategies of player i, Πi – payoff function of player i.

(1)

The Nash Equilibrium in Multy-Product Inventory Model

193

Players make an interactive decisions simultaneously choosing their strategies xi from strategy sets Ωi . The agreements and coalition formations are forbidden. Vector x = (x1 , . . . , xn ) is called situation in the game. As a result players are paid payoff Πi = Πi (x). A game is a formal representation of a situation in which a number of decision makers (players) interact in a setting of strategic interdependence. By that, we mean that the welfare of each decision maker depends not only on his own actions, but also on the actions of the other players. Moreover, the actions that are best for him to take may depend on what he expects the other players to do. We say that game theory analyzes interactions between rational, decision-making individuals who may not be able to predict fully the outcomes of their actions. We call x⋆ = (x⋆1 , . . . , x⋆n ) a Nash equilibrium if for all admissible strategies xi ∈ Ωi , i = 1, . . . , n the following inequalities hold Πi (x⋆ ) ≥ Πi x⋆1 , x⋆2 . . . , x⋆i−1 , xi , x⋆i+1 , . . . , x⋆n . Theorem 1. (Kukushkin, Morozov, 1984) In game (1) there exists Nash equilibrium in pure strategies if for each i ∈ N strategy set Ωi is compact and convex, and payoff function Πi (x) is concave with respect to xi and continuous on Ω = Ω1 × Ω2 × . . . × Ωn .

Assume for any i ∈ N the function Πi (x) is continuously differentiable with respect to xi . From ( Tirol, 2000) we can see that first-order necessary condition for Nash equilibrium is the following ∂Πi (x⋆ ) = 0, i ∈ N. (2) ∂xi Suppose the payoff function Πi (x), i = 1, . . . , n is concave for all xi ∈ Ωi . In this case solution of system (2) appears to be a Nash equilibrium in pure strategies in non-cooperative game n

n

Γ = hN, {Ωi }i=1 , {Πi }i=1 i . 2.2. Oligopoly There are two most notable models in oligopoly theory: Cournot oligopoly, and Bertrand oligopoly. In the Cournot model, firms control their production level, which influences the market price. In the Bertrand model, firms choose the price to charge for a unit of product, which affects the market demand. Definition 1. Non-cooperative oligopoly is a market where a small number of firms act independently but are aware of each other’s actions. In the oligopoly model we suppose that: 1. firms are rational; 2. firms reason strategically. Firms or players meet only once in a single period model. The market then clears one and for all. There is no repetition of the interaction and hence, no opportunity for the firms to learn about each other over time. Such models are appropriate for markets that last only a brief period of time. Cournot and Bertrand oligopolies are modeled as strategic games, with continuous action sets (either production levels or prices). We study competitive markets in which firms use price as their strategic variable.

194 3.

Elena A. Lezhnina, Victor V. Zakharov The Price Competition

Let’s consider a market with n retailers sell m products: i = 1, . . . , n, j = 1, . . . , m. In this model each supplier forms for the planning period T single-product orders to supplier. Let qij be the quantity of product j in the order; qi = (qi1 , . . . , qim ) – the order vector of supplier i. After the order was received retailer assigns the prices for every good for selling: pi = (pi1 , . . . , pim ), i = 1, . . . , n – price vector of player i, where pij – the price assigned by retailer i for product j. Assume the demand for items of the product is known and uniform during a period of planning. Dij (p1j , . . . , pnj ) – demand function for good j with price appointed by player i and other players. This is the inverse function of pi . In the price competition the demand on the good depends on prices appointed by other players. Due to the single-product orders the total inventory cost function of retailer i could be expressed as T Ci (p1 , . . . , pn , qi1 , . . . , qim ) = m X O Dij (p1j , . . . , pnj ) H qij = cj Dij (p1j , . . . , pnj ) + cij + cij , qij 2 j=1 where pi = (pi1 , . . . , pim ), cO ij – order cost per unit of good j for player i, cH ij – holding cost per unit of product j for retailer i during period T , cj – procurement price of product j fixed by supplier. We assume that prices satisfy the conditions: pij > cj ,

i = 1, . . . , n,

j = 1, . . . , m.

The payoff function is expressed as Πi (p1 , . . . , pn , qi ) = =

m X j=1

(3)

pij Dij (p1j , . . . , pnj , qi ) − T Ci (p1 , . . . , pn , qi ).

Following single-product inventory game theory model (Mansur Gastratov, Victor Zakharov, 2011) we consider expanding this model for multi-product one under the condition of single-product ordering strategies of the players (retailers). As in singleproduct model retailer has to calculate inventory decision on order quantity of a product as an optimal reaction for the products prices assigned by all players. That is we also can introduce internal and external strategies of the players as follows. Definition 2. We define qi = (qi1 , . . . , qim ) as internal strategy, and pi = (pi1 , . . . , pim ) as external strategy of player i. To take into account influence of external strategy of player to internal one we could find optimal reaction of the retailer to prices assigned by all players. We would realize two-stage procedure.

195

The Nash Equilibrium in Multy-Product Inventory Model To find optimal internal strategy retailer has to solve the following problem min

(qi1 ,...,qim )

T Ci (pi1 , . . . , pim , qi ) =

m X Dij (p1j , . . . , pnj ) H qij cj Dij (p1j , . . . , pnj ) + cO + c . ij ij (qi1 ,...,qim ) qij 2 j=1

=

min

The value of the economic order quantity is defined as internal player strategy qi = (qi1 , . . . , qim ). In this case it is possible to use the Harris-Wilson formula because of demand function has additive form. Now we have s 2cO ij Dij (p1j , . . . , pnj ) ∗ qij = . (4) cH ij ∗ Now it is possible to substitute the optimal qij in (3). We get the new payoff function: m X e i (pi1 , . . . , pim ) = Π pij Dij (p1j , . . . , pnj , qi ) − T Ci∗ (pi1 , . . . , pim ), j=1

where

=

m X j=1



T Ci∗ (pi1 , . . . , pim ) =

 Dij (p1j , . . . , pnj ) cj Dij (p1j , . . . , pnj ) + cO r + cH ij ij  2cO D (p ,...,p ) ij

1j

ij

r

nj

2cO ij Dij (p1j ,...,pnj ) cH ij

2

cH ij



 . 

On the next stage the modified price competition of Bertrand oligopoly is considered. Retailers choose the goods’ prices according to the price competition with other players in non-cooperative game: E D n on n Γ = N, Π˜i , {Ωi }i=1 , (5) i=1

where N = 1, . . . , n – set of players, Ωi – strategy set of player i, where Ωi = Ωi1 × Ωi2 × . . . , Ωim , Ωij = {pij | pij > cj }, i = 1, . . . , n, j = 1, . . . , m. fi (pi1 , . . . , pim ) – payoff function of player i. Π This function depends on external player strategies (pi1 , . . . , pim ) ∈ Ω1 × Ω2 × . . . × Ωn . Every player i chooses external strategy pi ∈ Ωi , which gives the decision of problem  m X max Πi (pi1 , . . . , pim ) = max  pij Dij (p1j , . . . , pnj )− (p1 ,...,pn )

−

(p1 ,...,pn )



j=1

m X  Dij (p1j , . . . , pnj ) cj Dij (p1j , . . . , pnj ) + cO r + cH ij ij  2cO Dij (p1j ,...,pnj ) j=1

ij

cH ij

r

2cO ij Dij (p1j ,...,pnj ) cH ij

2



  . 

196


The optimal strategy is achieved by finding Nash equilibrium (Nash strategies), which is the most commonly used solution concept in game theory. The convexity, fi (pi1 , . . . , pim ) is necessary and continuity and differentiability for payoff function Π sufficient for existence of Nash equilibrium. In terms of convenience we denote q q cH ij √ Cf = + 1 + cO ij ij . 2 Now we can rewrite the cost function as T Ci∗ (pi1 , . . . , pim ) =

m h i X eij Dij (p1j , . . . , pnj )1/2 cj Dij (p1j , . . . , pnj ) + C j=1

and payoff function is expressed as

fi (pi1 , . . . , pim ) = Π m m h i X X fij Dij (p1j , . . . , pnj )1/2 . = pij Dij (p1j , . . . , pnj ) − cj Dij (p1j , . . . , pnj ) + C j=1

j=1

The existence of Nash equilibrium depends on demand function. There exist two cases for demand function form. First case: the case when demand function is in linear form. Proposition 1. If for every i ∈ N strategy set Ωi is compact and convex and demand function has a linear form then there exists the Nash equilibrium in the game (5). Proof. The demand function is linear, continuous and differentiable. The square 1 root function (Dij (p1j , . . . , pnj )) 2 is concave, continuous and differentiable with fi (p1 , . . . , pn ) is represented as difference respect to pij on Ωi . The payoff function Π of linear function and square root function. From the properties of this function we get that the payoff function is concave, continua, and differentiable with respect to pij on Ωi . Which leads to existence of unique Nash equilibrium. ⊔ ⊓ Second case: the case of non-linear demand function. In general case for existence of Nash equilibrium we need the special conditions. Proposition 2. Let the following conditions be satisfied: 1. for every i ∈ N strategy set Ωi is compact and convex; 2. the demand function Dij (p1j , . . . , pnj ) is continuous and differentiable with respect to pij on Ωi ; fi (pi1 , . . . , pim ) is concave with respect to pij on Ωi , i = 1, . . . , n, 3. the function Π j = 1, . . . , m. D n on E fi Under these conditions the Nash equilibrium in game Γ = N, Π , {Ωi }ni=1 i=1 exists.

197


Proof. If function Dij (p1j , . . . , pnj ) is continuous and differentiable function on Ωi fi (pi1 , . . . , pim ) is continuous and differentiable with respect to pij , then function Π fi (pi1 , . . . , pim ) with respect to pij on Ωi , i = 1, . . . , n, j = 1, . . . , m. And if function Π is concave, then all conditions of Theorem 1 are satisfied and there exists the Nash equilibrium. ⊔ ⊓ According to the Teorem 1, there exist the Nash equilibrium in this game. And, from (Tirol, 2000), the equilibrium point (p⋆1 , . . . , p⋆n ) is found by solving set of equations:

=

m X j=1

fi (p1 , . . . , pn ) ∂Π = ∂pij Dij (pi1 , . . . , pim ) + (pij − cj )

# eij C ∂Dij (p1j , . . . , pnj ) − p , ∂pij 2 Dij (p1j , . . . , pnj )

∂Dij (p1j , . . . , pnj ) − ∂pij i = 1, . . . , n,

j = 1, . . . , m.

The issue appears when Dij (p1j , . . . , pnj ) = 0. In this case prices set by player are too high and demand is zero. The player does not participate in price competition. Thus, we suppose that Dij (p1j , . . . , pnj ) > 0. Substituting the solutions of this ∗ – the optimal goods equation set (p⋆1 , . . . , p⋆n ) to (4) we will finally get the qij quantity for player’s order. 4.

The Multi-Product Orders.

As in the first case, we have the market with m products and n retailers: i = 1, . . . , n, j = 1, . . . , m. Let τi T be duration of planning period. Suppose that all retailers set multi-product orders assigning equal duration of cycles between deliveries as a part of period T . Let τi T be duration of planning period. Due to uniform demand the following equality takes place for quantity qij in multi-product order: qij = τi Dij (p1j , . . . , pnj ). Assume that ordering cost for multi-product order for player i is equal to cMO . i Then total inventory cost function for player i can be expressed as follows: m X cMO H τi Dij (p1j , . . . , pnj ) i T Ci (pi1 , . . . , pim , τi ) = + cj Dij (p1j , . . . , pnj ) + cij . τi 2 j=1 Payoff function is described as Πi (pi1 , . . . , pim , τi ) =

m X j=1

pij Dij (p1j , . . . , pnj )−

m X cMO i H τi Dij (p1j , . . . , pnj ) − − cj Dij (p1j , . . . , pnj ) + cij . τi 2 j=1

198


As in the case of single-product ordering we use the two stage decision making procedure To find optimal iternal strategy of player i we have to solve the following problem: min T Ci (pi1 , . . . , pim , τi ) = = min τi

cMO i τi

+

m X

τi

cj Dij (p1j , . . . , pnj ) +

τi Dij cH ij

i=1

(p1j , . . . , pnj ) 2

!

.

This is the problem EQO (optimal economic order). Now it is possible to use Harris – Wilson formula because of cost function additive form. Using the dependence form qij = τi Dij (p1j , . . . , pnj ), it’s possible to find the optimal τi⋆ : s τi∗ =

2cMO i . H j=1 cij Dij (p1j , ..., pnj )

Pm

(6)

Substituting this optimal τi⋆ into the formula of payoff function we obtain fi (pi1 , . . . , pim ) = Π  1/2 m m X X 2  = (pij − cj )Dij (p1j , . . . , pnj ) − √ (cMO )1/2  cH i ij Dij (p1j , . . . , pnj ) 2 j=1 j=1

As a result we get that the payoff function depends on external strategies only. On the next stage Bertrand oligopoly with price competition is considered. On the second stage player finds optimal prices according to the competition with other players. Let’s consider non-cooperative game D n on E n Γ = N, Π˜i , {Ωi }i=1 , i=1

Ωi – strategy set of player i, where Ωi = Ωi1 × Ωi2 × . . . , Ωim , Ωij = {pij | pij > cj }, i = 1, . . . , n, j = 1, . . . , m. fi (pi1 , . . . , pim ) – payoff function of player i. This function depends on external Π player strategies (pi1 , . . . , pim ) ∈ Ω1 × Ω2 × . . . × Ωn . Every player i chooses external strategy pi ∈ Ωi , which gives the decision of problem. The aim of each player is to maximize their payoff in the price competition: fi (pi1 , . . . , pim ) Π

−→

pi1 ,...,pim

max .

According to the Teorem 1, there exist the Nash equilibrium in this game. The existence of Nash equilibrium depends on the form of payoff function. Proposition 3. Suppose the following conditions for i = 1, . . . , n are satisfied: 1. for every i ∈ N strategy set Ωi is compact and convex;


199

2. demand function has a linear form. Under these conditions Nash equilibrium in pure strategies exists. Proof. If function Dij (p1j , . . . , pnj ) has a linear form, then it is continuous and fi (pi1 , . . . , pim ) is differentiable function on Ωi with respect to pij , and function Π continuous and differentiable with respect to pij on Ωi , i = 1, . . . , n, j = 1, . . . , m. fi (pi1 , . . . , pim ) is represented as difference of linear function and square Function Π root function. From the properties of this functions follows that the payoff function is concave, continua and differentiable with respect to pij on Ωi , i = 1, . . . , n, j = 1, . . . , m. All conditions of Theorem 1 are satisfied and there exists the Nash equilibrium. ⊔ ⊓ Proposition 4. Suppose the following conditions for i = 1, . . . , n are satisfied: 1. for every i ∈ N strategy set Ωi is compact and convex; 2. demand function Dij (p1j , . . . , pnj ) is continuous and differentiable with respect to pij on Ωi ; fi (pi1 , . . . , pim ) is concave with respect to pij on Ωi . 3. payoff function Π

Under these conditions Nash equilibrium in pure strategies exists.

Proof. If function Dij (p1j , . . . , pnj ) is continuous and differentiable function on Ωi fi (pi1 , . . . , pim ) is continuous and differentiable with respect to pij , then function Π fi (pi1 , . . . , pim ) with respect to pij on Ωi , i = 1, . . . , n, j = 1, . . . , m. And if function Π is concave, then all conditions of Theorem 2 are satisfied and there exists the Nash equilibrium. ⊔ ⊓ From (Tirol, 2000), the equilibrium point (p⋆1 , . . . , p⋆n ) is found by solving set of equations: fi (p1 , . . . , pn ) ∂Π = 0, ∂pij i = 1, . . . , n.

After finding optimal strategy value of p⋆ = (p⋆1 , . . . , p⋆n ) it is possible to substitute its to (6) and to calculate the optimal value of period τi⋆ . 5.

Conclusion

In this paper game theory models for multi-product inventory control are treated in case of competition among retailers. The model of price competition in context of modied model of Bertran is considered. Each retailer can use two order types: singleproduct and multi-product ordering. Demand for each product is supposed to be uniform for the period of planning. In game theory model retailers are considered as players using two-level strategies. At the lower level of the game each player chooses internal strategy as an optimal reaction to competitive players strategies which are called external. Optimal internal strategies are represented in analytical form. Necessary and sufficient conditions for existence of Nash equilibrium in pure strategies for the cases of linear and non-linear demand functions are proposed.

200


References Nash, J. F. (1951). Non-Cooperative games. Annals of Mathematics, 54, 286–295. Harris, F. (1915). Operation and Cost. Factory Management Series. A.W. Shaw, Chicago, 48–52. van Damm, E. (1991). Stability and Perfection of Nash Equilibria. Springer-Verlag, Berlin. Nash, J. F. (1950). Equilibrium Points in n-Person Games. Proceedings of the National Academy of Sciences, 36, 48–59. Dasgupta, P., Maskin, E. (1986). The Existence of Equilibrium in Discontinuous Economic Games. Review of Economics Studies 53, 1–26. Mahajan, S., van Ryzin, G. J. (2001). Inventory Competition Under Dynamic Consumer Choice. Operations Research, 49(5), 646–657. Netessine, S., Rudi, N., Wang, Y (2003). Dynamic Inventory Competition and Customer Retention. Working paper. Parlae, M. (1988). Game Theoretic Analysis of the Substitutable Product Inventory Problem with Random Demands. Naval Research Logistics, 35, 397–409. Tirol, Jean (2000). The markets and the market power: The organization and industry theory. Translation from English by J. M. Donc, M. D. Facsirova, under edition A. S. Galperina and N. A. Zenkevich. SPb: Institute Economic school, in 2 Volumes. V.1. 328 p. V.2. 240 p. Kukushkin, N. S., Morozov, V. V. (1984). The nonantagonistic Game Theory. M.: Moscow State University, 103 p. Friedman, James (1983).Oligopoly Theory. Cambridge University Press, 260 p. Bertrand, J. (1883).Theorie mathematique de la richesse sociale. Journal des Savants., 67, 499–508. Cournot, A. A. (1883).Recherches sur les principes mathmatiques de la thorie des richesses. Paris, L. Hachette, 198 p. Hax, A. C. and D. Candea (1984). Production and inventory management. Prentice-Hall. Englewood Cliffs, N.J., 135 p. Haldey, G. and T. M. Whitin (1963). Analysis of inventory. Prentice-Hall, Englewood Cliffs, N.J. Tersine, R. J. (1994). Principles of inventory and materials management. Elsevier North Holland, Amsterdam. Cachon, G. P. (2003). Supply chain coordination with contracts. Handbook in Operations Research and Management Science: Supply Chain Management. Vol. 11, Elsevier B. V., Amsterdam, pp. 229–340. Mansur Gasratov, Victor Zakharov (2011). Games and Inventory Management. In: Dynamic and Sustainability in International Logistics and Supply Chain Management. Cuvillier Verlag, Gottingen.

Nash Equilibria Conditions for Stochastic Positional Games Dmitrii Lozovanu1 and Stefan Pickl2 Institute of Mathematics and Computer Science, Academy of Sciences of Moldova, Academy str., 5, Chisinau, MD–2028, Moldova E-mail: [email protected] http://www.math.md/structure/applied-mathematics/math-modeling-optimization/ 2 Institute for Theoretical Computer Science, Mathematics and Operations Research, Universität der Bundeswehr, München 85577 Neubiberg-München, Germany E-mail: [email protected] 1

Abstract We formulate and study a class of stochastic positional games using a game-theoretical concept to finite state space Markov decision processes with an average and expected total discounted costs optimization criteria. Nash equilibria conditions for the considered class of games are proven and some approaches for determining the optimal strategies of the players are analyzed. The obtained results extend Nash equilibria conditions for deterministic positional games and can be used for studying Shapley stochastic games with average payoffs. Keywords: Markov decision processes, stochastic positional games, Nash equilibria, Shapley stochastic games, optimal stationary strategies.

1.

Introduction

In this paper we consider a class of stochastic positional games that extends deterministic positional games studied by Moulin,1976, Ehrenfeucht and Mycielski, 1979, Gurvich at al., 1988, Condon, 1992, Lozovanu and Pick, 2006, 2009. The considered class of games we formulate and study applying the concept of positional games to finite state space Markov decision processes with average and expected total discounted costs optimization criteria. We assume that the Markov process is controlled by several actors (players) as follows: The set of states of the system is divided into several disjoint subsets which represent the corresponding position sets of the players. Additionally the cost of system’s transition from one state to another is given for each player separately. Each player has to determine which action should be taken in each state of his position set of the Markov process in order to minimize his own average cost per transition or the expected total discounted cost. In these games we are seeking for a Nash equilibrium. The main results of the paper are concerned with the existence of Nash equilibria for the considered class of games and determining the optimal strategies of the players. Necessary and sufficient conditions for the existence of Nash equilibria in stochastic positional games that extend Nash equilibria conditions for deterministic positional games are proven. Based on the constructive proof of these results we propose some approaches for determining the optimal strategies of the players. Additionally we show that the stochastic positional games are tightly connected with Shapley stochastic games (Shapley, 1953) and the obtained results can be used for studying a special class of Shapley stochastic games with average payoffs.

202 2.

Dmitrii Lozovanu, Stefan Pickl Formulation of the Basic Game Models and Some Preliminary Results

We consider two game-theoretic models. We formulate the first game model for Markov decision processes with average cost optimization criterion and call it the stochastic positional game with average payoffs. We formulate the second one for Markov decision processes with discounted cost optimization criterion and call it stochastic positional game with discounted payoffs. Then we show the relationship of these games with Shapley stochastic games. 2.1. Stochastic Positional Games with Average Payoffs To formulate the stochastic positional game with average payoffs we shall use the framework of a Markov decision process (X, A, p, c) with a finite set of states X, a finite set of actions A, a transition probability function p : X × X × A → [0, 1] that satisfies the condition X pax,y = 1, ∀x ∈ X, ∀a ∈ A y∈X

and a transition cost function c : X × X → R which gives the costs cx,y of states transitions of the dynamical system from an arbitrary x ∈ X to another state y ∈ X (see Howard, 1960; Puterman, 2005). For the noncooperative game model with m players we assume that m transition cost functions ci : X × X → R, i = 1, 2, . . . , m are given, where cix,y expresses the cost of the system’s transition from the state x ∈ X to the state y ∈ X for the player i ∈ {1, 2, . . . , m}. In addition we assume that the set of states X is divided into m disjoint subsets X1 , X2 , . . . , Xm X = X1 ∪ X2 ∪ · · · ∪ Xm (Xi ∩ Xj = ∅, ∀i 6= j), where Xi represents the positions set of the player i ∈ {1, 2, . . . , m}. So, the Markov process is controlled by m players, where each player i ∈ {1, 2, . . . , m} fixes actions in his positions x ∈ Xi . We assume that each player fixes actions in the states from his positions set using stationary strategies, i.e. we define the stationary strategies of the players as m maps: si : x → a ∈ Ai (x) for x ∈ Xi , i = 1, 2, . . . , m, where Ai (x) is the set of actions of the player i in the state x ∈ Xi . Without loss of generality we may consider |Ai (x)| = |Ai | = |A|, ∀x ∈ Xi , i = 1, 2, . . . , m. In order to simplify the notation we denote the set of possible actions in a state x ∈ X for an arbitrary player by A(x). A stationary strategy si , i ∈ {1, 2, . . . , m} in the state x ∈ Xi means that at every discrete moment of time t = 0, 1, 2, . . . the player i uses the action a = si (x). Players fix their strategy independently and do not inform each other which strategies they use in the decision process. If the players 1, 2, . . . , m fix their stationary strategies s1 , s2 , . . . , sm , respectively, then we obtain a situation s = (s1 , s2 , . . . , sm ). This situation corresponds si (x)

to a simple Markov process determined by the probability distributions px,y in the states x ∈ Xi for i = 1, 2, . . . , m. We denote by P s = (psx,y ) the matrix of

Nash Equilibria Conditions for Stochastic Positional Games

203

probability transitions of this Markov process. If the starting state x0 is given, then for the Markov process with the matrix of probability transitions P s we can determine the average cost per transition ωxi 0 (s1 , s2 , . . . , sm ) with respect to each player i ∈ {1, 2, . . . , m} taking into account the corresponding matrix of transition costs C i = (cix,y ). So, on the set of situations we can define the payoff functions of the players as follows: Fxi 0 (s1 , s2 , . . . , sm ) = ωxi 0 (s1 , s2 , . . . , sm ),

i = 1, 2, . . . , m.

In such a way we obtain a discrete noncooperative game in normal form which is determined by a finite set of strategies S 1 , S 2 , . . . , S m of m players and the payoff functions defined above. In this game we are seeking for a Nash equilibrium (see Nash, 1951), i.e., we consider the problem of determining the stationary strategies ∗

∗

∗

∗

∗

∗

∗

s1 , s2 , . . . , si−1 , si , si+1 , . . . , sm∗ such that ∗

∗

∗

Fxi 0 (s1 , s2 , . . . , si−1 , si , si+1 , . . . , sm ∗ ) ∗

∗

∗

∗

≤ Fxi 0 (s1 , s2 , . . . , si−1 , si , si+1 , . . . , sm ∗ ), ∀si ∈ S i , i = 1, 2, . . . , m. The game defined above is determined uniquely by the set of states X, the position sets X1 , X2 , . . . , Xm , the set of actions A, the cost functions ci : X × X → R, i = 1, 2, . . . , m, the probability function p : X × X × A → [0, 1] and the starting position x0 . Therefore, we denote this game by (X, A, {Xi }i=1,m , {ci }i=1,m , p, x0 ). In the case m = 2 and c2 = −c1 we obtain an antagonistic stochastic positional game. If pax,y = 0 ∨ 1, ∀x, y ∈ X, ∀a ∈ A the stochastic positional game (X, A, {Xi }i=1,m , {ci }i=1,m , p, x0 ) is transformed into the cyclic game (Ehrenfeucht and Mycielski, 1979, Gurvich at al., 1988, Condon, 1992, Lozovanu and Pick, 2006). Some results concerned with the existence of Nash equilibria for stochastic positional games with average payoffs have been derived by Lozovanu at al., 2011. In particular the following theorem has been proven. Theorem 1. If for an arbitrary situation s = (s1 , s2 , . . . , sm ) of the stochastic positional game with average payoffs the matrix of probability transitions P s = (psx,y ) induces an ergodic Markov chain then for the game there exists a Nash equilibrium. If the matrix P s for some situations do not correspond to an ergodic Markov chain then for the stochastic positional game with average payoffs a Nash equilibrium may not exist. This follow from the constructive proof of this theorem (see Lozovanu at al., 2011). An example of a deterministic positional game with average payoffs for which Nash equilibrium does not exist has been constructed by Gurvich at al., 1988. However, in the case of antagonistic stochastic positional games saddle points always exist (Lozovanu and Pickl, 2014), i.e. in this case the following theorem holds. Theorem 2. For an arbitrary antagonistic positional game there exists a saddle point. The existence of saddle points for deterministic positional games with average payoffs have been proven by Ehrenfeucht and Mycielski, 1979, Gurvich at al., 1988.

204

Dmitrii Lozovanu, Stefan Pickl

2.2. Stochastic Positional Games with Discounted Payoffs We formulate the stochastic positional game with discounted payoffs in a similar way as the game from Section 2.. We assume that for the Markov process m transition cost functions ci : X × X → R, i = 1, 2, . . . , m, are given and the set of states X is divided into m disjoint subsets X1 , X2 , . . . , Xm , where Xi represents the positions set of the player i ∈ {1, 2, . . . , m}. The Markov process is controlled by m players, where each player i ∈ {1, 2, . . . , m} fixes actions in his positions x ∈ Xi using stationary strategies, i.e. the stationary strategies of the players in this game are defined as m maps: si : x → a ∈ A(x)

for

x ∈ Xi ; i = 1, 2, . . . , m.

Let s1 , s2 , . . . , sm be a set of stationary strategies of the players that determine the situation s = (s1 , s2 , . . . , sm ). Consider the matrix of probability transitions P s = (psx,y ) which is induced by the situation s, i.e., each row of this matrix corresponds to si (x)

a probability distribution px,y in the state x where x ∈ Xi . If the starting state x0 is given, then for the Markov process with the matrix of probability transitions P s we can determine the discounted expected total cost σxi 0 (s1 , s2 , . . . , sm ) with respect to each player i ∈ {1, 2, . . . , m} taking into account the corresponding matrix of transition costs C i = (cix,y ). So, on the set of situations we can define the payoff functions of the players as follows: Fbxi 0 (s1 , s2 , . . . , sm ) = σxi 0 (s1 , s2 , . . . , sm ),

i = 1, 2, . . . , m.

In such a way we obtain a new discrete noncooperative game in normal form which is determined by the sets of strategies S 1 , S 1 , . . . , S m of m players and the payoff functions defined above. In this game we are seeking for a Nash equilibrium. We denote the stochastic positional game with discounted payoffs by (X, A, {Xi }i=1,m , {ci }i=1,m , p, γ, x0 ). For this game the following result has been proven (Lozovanu, 2011).

Theorem 3. For an arbitrary stochastic positional game (X, A, {Xi }i=1,m , {ci }i=1,m , p, γ, x0 ) with given discount factor 0 < γ < 1 there exists a Nash equilibrium. Based on a constructive proof of Theorems 1,3 some iterative procedures for determining Nash equilibria in the considered positional games have been proposed (see Lozovanu at al., 2011). 2.3.

The Relationship of Stochastic Positional Games with Shapley Stochastic Games A stochastic game in the sense of Shapley (see Shapley, 1953) is a dynamic game with probabilistic transitions played by several players in a sequence of stages, where the beginning of each stage corresponds to a state of the dynamical system. The game starts at a given state from the set of states of the system. At each stage players select actions from their feasible sets of actions and each player receives a stage payoff that depends on the current state and the chosen actions. The game then moves to a new random state the distribution of which depends on the previous state and the actions chosen by the players. The procedure is repeated at a new


205

state and the play continues for a finite or infinite number of stages. The total payoff of a player is either the limit inferior of the average of the stage payoffs or the discounted sum of the stage payoffs. So, an average Shapley stochastic game with m players consists of the following elements: 1. A state space X (which we assume to be finite); 2. A finite set Ai (x) of actions with respect to each player i ∈ {1, 2, . . . , m} for an arbitrary state x ∈ X;

3. A stage payoff f i (x, a) with respect to each player i ∈ {1, 2, . . .Q , m} for each state x ∈ X and for an arbitrary action vector a ∈ i Ai (x); Q Q 4. A transition probability function p : X × x∈X i Ai (x) × X → [0, 1] that gives the probability transitions pax,y from an arbitrary x ∈ X Q to an arbitrary y ∈ Y for a fixed action vector a ∈ i Ai (x), where P Q a i y∈X px,y = 1, ∀x ∈ X, a ∈ i A (x);

5. A starting state x0 ∈ X.

The stochastic game starts in state x0 . At stage t players observe state xt and simultaneously choose actions ait ∈ Ai (xt ), i = 1, 2, . . . , m. Then nature selects a state xt+1 according to probability transitions paxtt ,y for fixed action vector at = (a1t , a2t , . . . , am t ). A play of the stochastic game x0 , a0 , x1 , a1 , . . . , xt , at , . . . defines a stream of payoffs f0i , f1i , f2i , . . . , where fti = f i (xt , at ), t = 0, 1, 2, . . . . The t-stage average stochastic game is the game where the payoff of player i ∈ {1, 2, . . . , m} is t−1

Fti

1X i f . = t τ =1 τ

The infinite average stochastic game is the game where the payoff of player i ∈ {1, 2, . . . , m} is i F = lim Fti . t→∞

In a similar a Shapley stochastic game with expected discounted payoffs of the players is defined. In such a game along to the elements described above also a discount factor λ (0 < λ < 1) is given and the total payoff of a player represents the expected discounted sum of the stage payoffs. By comparison for Shapley stochastic games with stochastic positional games we can observe the following. The probability transitions from a state to another state as well as the stage payoffs of the players in a Shapley stochastic game depend on the actions chosen by all players, while the probability transitions from a state to another state as well as the stage payoffs (the immediate costs of the players) in a stochastic positional game depend only on the action of the player that controls the state in his position set. This means that a stochastic positional game can be regarded as a special case of the Shapley stochastic game. Nevertheless we can see that stochastic positional games can be used for studying some classes of Shapley stochastic games. The main results concerned with determining Nash equilibria in Shapley stochastic games have been obtained by Gillette, 1957, Mertens and Neyman, 1981, Filar and Vrieze, 1997, Lal and Sinha, 1992, Neyman and Sorin, 2003. Existence

206


of Nash equilibria for such games are proven in the case of stochastic games with a finite set of stages and in the case of the games with infinite stages if the total payoff of each player is the discounted sum of stage payoffs. If the total payoff of a player represents the limit inferior of the average of the stage payoffs then the existence of a Nash equilibrium in Shapley stochastic games is an open question. Based on the results mentioned in previous sections we can show that in the case of the average non-antagonistic stochastic games a Nash equilibrium may not exist. In order to prove this we can use the average stochastic positional game (X, A, {Xi }i=1,m , {ci }i=1,m , p, x0 ) from section 2. It is easy to observe that this game can be regarded as a Shapley stochastic game with average payoff functions of the players, where for a fixed situation s = (s1 , s2 , . . . , sm ) the probability transition psx,y from a state x = x(t) ∈ Xi to a state y = x(t + 1) ∈ X depends only on a strategy si of player i and the P corresponding stage payoff in the state x of player i ∈ {1, 2, . . . , m} is equal to y∈X psx,y cix,y . Taking into account that the cyclic game represents a particular case of the average stochastic positional game and for the cyclic game Nash equilibrium may not exist (see Gurvich at al., 1988) we obtain that for the average non-antagonistic Shapley stochastic game a Nash equilibrium may not exist. However in the case of average payoffs Theorem 1 can be extended for Shapley stochastic games. 3.

Nash Equilibria Conditions for Stochastic Positional Games with Average Payoffs

In this section we formulate Nash equilibria conditions for stochastic positional games in terms of bias equations for Markov decision processes. We can see that Nash equilibria conditions in such terms may be more useful for determining the optimal strategies of the players. Theorem 4. Let (X, A, {Xi }i=1,m , {ci }i=1,m , p, x) be a stochastic positional game with a given starting position x ∈ X and average payoff functions Fx1 (s1 , s2 , . . . , sm ), Fx2 (s1 , s2 , . . . , sm ), . . . , Fxm (s1 , s2 , . . . , sm ) of the players 1, 2, . . . , m, respectively. Assume that for an arbitrary situation s = (s1 , s2 , . . . , sm ) of the game the transition probability matrix P s = (psx,y ) corresponds to an ergodic Markov chain. Then there exist the functions εi : X → R,

i = 1, 2, . . . , m

and the values ω 1 , ω 2 , . . . , ω m that satisfy the following conditions: 1) µix,a + where

P

y∈X µix,a

pax,y εiy − εix − ω i ≥ 0, ∀x ∈ Xi , P a i = px,y cx,y ;

2) min {µix,a + a∈A(x)

∀a ∈ A(x), i = 1, 2, . . . , m,

y∈X

P

y∈X

pax,y εiy − εix − ω i } = 0,

∀x ∈ Xi , i = 1, 2, . . . , m; ∗

3) on each position set Xi , i ∈ {1, 2, . . . , m} there exists a map si : Xi → A such that o n X ∗ si (x) = a∗ ∈ Arg min µix,a + pax,y εiy − εix − ω i a∈A(x)

y∈X

Nash Equilibria Conditions for Stochastic Positional Games and

X

µjx,a∗ +

∗

y∈X ∗

207

pax,y εjy − εjx − ω j = 0, ∀x ∈ Xi , j = 1, 2, . . . , m.

∗

The set of maps s1 , s2 , . . . , sm∗ determines a Nash equilibrium situation s∗ = ∗ ∗ (s1 , s2 , . . . , sm∗ ) for the stochastic positional game (X, A, {Xi }i=1,m , {ci }i=1,m , p, x and ∗

∗

Fxi (s1 , s2 , . . . , sm ∗ ) = ω i , ∀x ∈ X, i = 1, 2, . . . , m. ∗

∗

Moreover, the situation s∗ = (s1 , s2 , . . . , sm∗ ) is a Nash equilibrium for an arbitrary starting position x ∈ X. Proof. Let a stochastic positional game with average payoffs be given and assume that for an arbitrary situation s of the game the transition probability matrix P s = (psx,y ) corresponds to an ergodic Markov chain. Then according to Theorem 1 ∗ ∗ for this game there exists a Nash equilibrium s∗ = (s1 , s2 , . . . , sm∗ ) and we can set ∗ ∗ ω i = Fxi (s1 , s2 , . . . , sm∗ ), ∀x ∈ X, i = 1, 2, . . . , m. ∗

∗

∗

∗

Let us fix the strategies s1 , s2 , . . . , si−1 , si+1 , . . . , sm ∗ of the players 1, 2, . . . , i − 1, i+1, . . . , m and consider the problem of determining the minimal average cost per transition with respect to player i. Obviously, if we solve this decision problem then ∗ we obtain the strategy si . We can determine the optimal strategy of this decision problem with an average cost optimization criterion using the bias equations with respect to player i. This means that there exist the functions ǫi : X → R and the values ω i , i = 1, 2, . . . , m that satisfy the conditions: 1) µix,a + 2) min

a∈A(x)

P

y∈X

n

pax,y εiy − εix − ω i ≥ 0,

µix,a +

P

y∈X

∀x ∈ Xi , ∀a ∈ A(x);

o pax,y εiy − εix − ω i = 0, ∗

∗

∀x ∈ Xi . ∗

∗

Moreover, for fixed strategies s1 , s2 , . . . , si−1 , si+1 , . . . , sm∗ of the corresponding ∗ players 1, 2, . . . , i − 1, i + 1, . . . , m we can select the strategy si of player i where o n X ∗ si (x) ∈ Arg min µix,a + pax,y εiy − εix − ω i a∈A(x)

∗

y∈X

∗

and ω i = Fxi (s1 , s2 , . . . , sm ∗ ), ∀x ∈ X, i = 1, 2, . . . , m. This means that conditions 1)–3) of the theorem hold. Corollary 1. If for a stochastic positional game (X, A, {Xi }i=1,m , {ci }i=1,m , ∗ ∗ p, x) with average payoffs there exist a Nash equilibrium s∗ = (s1 , s2 , . . . , sm∗ ) which is a Nash equilibrium for an arbitrary starting position of the game x ∈ X and ∗ ∗ for arbitrary two different starting positions x, y ∈ X holds Fxi (s1 , s2 , . . . , sm∗ ) = i 1∗ 2∗ m∗ Fy (s , s , . . . , s ) then there exists the functions εi : X → R,

i = 1, 2, . . . , m

208


and the values ω 1 , ω 2 , . . . , ω m that satisfy the conditions 1) − 3) from Theorem 4. ∗ ∗ So, ω i = Fxi (s1 , s2 , . . . , sm∗ ), ∀x ∈ X, i = 1, 2, . . . , m and an arbitrary Nash equilibrium can be found by fixing n o X ∗ si (x) = a∗ ∈ Arg min µix,a + pax,y εiy − εix − ω i . a∈A(x)

y∈X

Using the elementary properties of non ergodic Markov decision processes with average cost optimization criterion the following lemma can be gained. Lemma 1. Let (X, A, {Xi }i=1,m , {ci }i=1,m , p, x) be an average stochastic ∗ ∗ positional game for which there exists a Nash equilibrium s∗ = (s1 , s2 , . . . , sm ∗ ), which is a Nash equilibrium for an arbitrary starting position of the game with ∗ ∗ ∗ ∗ ωxi = Fxi (s1 , s2 , . . . , sm∗ ). Then s∗ = (s1 , s2 , . . . , sm∗ ) is a Nash equilibrium for the average stochastic positional game (X, A, {Xi }i=1,m , {ci }i=1,m , p, x), where cix,y = cix,y − ωxi , ∀x, y ∈ X, i = 1, 2, . . . , m and

i

∗

∗

F x (s1 , s2 , . . . , sm ∗ ) = 0, ∀x ∈ X, i = 1, 2, . . . , m. Now using Corollary 1 and Lemma 1 we can prove the following results. Theorem 5. Let (X, A, {Xi }i=1,m , {ci }i=1,m , p, x) be an average stochastic positional game. Then in this game there exists a Nash equilibrium for an arbitrary starting position x ∈ X if and only if there exist the functions εi : X → R, i = 1, 2, . . . , m and the values ωx1 , ωx2 , . . . , ωxm for x ∈ X that satisfy the following conditions: P a i 1) µix,a + px,y εy − εix − ωxi ≥ 0, ∀x ∈ Xi , ∀a ∈ A(x), i = 1, 2, . . . , m, y∈X P a i where µix,a = px,y cx,y ; y∈X

2)

min {µix,a a∈A(x)

+

P

y∈X

pax,y εiy − εix − ωxi } = 0,

∀x ∈ Xi , i = 1, 2, . . . , m; ∗


and

µjx,a∗ +

X

y∈X

∗

pax,y εjy − εjx − ω j = 0,

y∈X

∀x ∈ Xi , j = 1, 2, . . . , m. ∗

∗

If such conditions hold then the set of maps s1 , s2 , . . . , sm∗ determines a Nash equilibrium of the game for an arbitrary starting position x ∈ X and ∗

∗

Fxi (s1 , s2 , . . . , sm ∗ ) = ωxi , i = 1, 2, . . . , m.

209


Proof. The sufficiency condition of the theorem is evident. Let us prove the necessity one. Assume that for the considered average stochastic positional game there exists ∗ ∗ a Nash equilibrium s∗ = (s1 , s2 , . . . , sm ∗ ) which is a Nash equilibrium for an arbitrary starting position of the game. Denote ∗ ∗ σxi = Fbxi (s1 , s2 , . . . , sm∗ ), ∀x ∈ X, i = 1, 2, . . . , m

and consider the following auxiliary game (X, A, {Xi }i=1,m , {ci }i=1,m , p, x), where cix,y = cix,y − ωxi , ∀x, y ∈ X, i = 1, 2, . . . , m.

Then according to Lemma 1 the auxiliary game has the same Nash equilibrium s∗ = ∗ ∗ (s1 , s2 , . . . , sm ∗ ) as initial one. Moreover, this equilibrium is a Nash equilibrium for an arbitrary starting position of the game and ∗

i

∗

F x (s1 , s2 , . . . , sm∗ ) = 0, ∀x ∈ X, i = 1, 2, . . . , m. Therefore, according to Corollary 1, for the auxiliary game there exist the functions εi : X → R,

i = 1, 2, . . . , m

and the values ω 1 , ω 2 , . . . , ωm (ω i = 0, i = 1, 2, . . . , m), that satisfy the conditions of Theorem 4, i.e. P a i 1) µix,a + px,y εy − εix − ωix ≥ 0, ∀x ∈ Xi , ∀a ∈ A(x), i = 1, 2, . . . , m, y∈X P a i where µix,a = px,y cx,y ; y∈X

2)

min {µix,a +

a∈A(x)

P

y∈X

pax,y εiy − εix − ω ix } = 0,

∀x ∈ Xi , i = 1, 2, . . . , m; ∗


and

µjx,a∗ +

X

y∈X

∗

pax,y εjy − εjx − ω j = 0,

y∈X

∀x ∈ Xi , j = 1, 2, . . . , m.

Taking into account that ω ix = 0, and µix,a = µix,a − ωxi (because cix,y = cx,y − ωxi ) we obtain conditions 1 − 3 of the theorem. 4.

Nash Equilibria Conditions for Stochastic Positional Games with Discounted Payoffs

Now we formulate Nash equilibria conditions in the terms of bias equations for stochastic positional games with discounted payoffs. Theorem 6. Let a stochastic positional game (X, A, {Xi }i=1,m , {ci }i=1,m , p, γ, x) with a discount factor 0 < γ < 1 be given. Then there exist the values σxi , i = 1, 2, . . . , m, for x ∈ X that satisfy the following conditions:

210


1) µix,a + γ

P

pax,y σyi − σxi ≥ 0, ∀x ∈ Xi , ∀a ∈ A(x), i = 1, 2, . . . , m, P a i = px,y cx,y .

y∈X

where µix,a

y∈X

P a i 2) min µix,a + γ px,y σy − σxi = 0, a∈A(x)

y∈X

∀x ∈ Xi , i = 1, 2, . . . , m; ∗

3) on each position set Xi , i ∈ {1, 2, . . . , m} there exists a map si : Xi → A such that X a i i i∗ ∗ i px,y σy − σx , ∀x ∈ Xi s (x) = a ∈ Arg min µx,a + γ a∈A(x)

and

µjx,a∗ + γ

X

y∈X ∗

y∈X

∗

pax,y σyj − σxj = 0, ∀x ∈ Xi ,

j = 1, 2, . . . , m.

∗

The set of maps s1 , s2 , . . . , sm ∗ determines a Nash equilibrium situation s∗ = ∗ ∗ (s1 , s2 , . . . , sm∗ ) for the stochastic positional game with discounted payoffs, where ∗ ∗ Fbxi (s1 , s2 , . . . , sm∗ ) = σxi , ∀x ∈ X, 1∗

2∗

Moreover, the situation s = (s , s , . . . , s trary starting position x ∈ X. ∗

m∗

i = 1, 2, . . . , m.

) is a Nash equilibrium for an arbi-

Proof. According to Theorem 3 for the discounted stochastic positional game (X, A, {Xi }i=1,m , {ci }i=1,m , p, γ, x) there exists a Nash equilibrium s∗ = ∗ ∗ (s1 , s2 , . . . , sm∗ ) which is a Nash equilibrium for an arbitrary starting position x ∈ X of the game. Denote ∗ ∗ σxi = Fbxi (s1 , s2 , . . . , sm ∗ ), ∀x ∈ X, i = 1, 2, . . . , m. ∗

∗

∗

∗

Let us fix the strategies s1 , s2 , . . . , si−1 , si+1 , . . . , sm∗ of the players 1, 2, . . . , i− 1, i + 1, . . . , m and consider the problem of determining the expected total discounted cost with respect to player i. Obviously, the optimal stationary strat∗ egy for this problem is si . Then according to the properties of the bias equations for this Markov decision problem with discounted costs there exist the values σxi , i = 1, 2, . . . , m, for x ∈ X that satisfy the conditions: P a i 1) µix,a + γ px,y σy − σxi ≥ 0, ∀x ∈ Xi , ∀a ∈ A(x), i = 1, 2, . . . , m; y∈X

P a i 2) min µix,a + γ px,y σy − σxi = 0, a∈A(x)

y∈X

∗

∗

∀x ∈ Xi i = 1, 2, . . . , m. ∗

∗

∗

Moreover, for fixed strategies s1 , s2 , . . . , si−1 , si , si+1 , . . . , sm ∗ of the corre∗ sponding players 1, 2, . . . , i − 1, i + 1, . . . , m we can select the strategy si of the player i where X ∗ si (x) ∈ Arg min µix,a + γ pax,y σyi − σxi a∈A(x)

and

y∈X

∗ ∗ Fbxi (s1 , s2 , . . . , sm ∗ ) = σxi , ∀x ∈ X, i = 1, 2, . . . , m.

This means that the conditions 1)–3) of the theorem hold.

Nash Equilibria Conditions for Stochastic Positional Games 5.

211

Saddle Point Conditions for Antagonistic Stochastic Positional Games

The antagonistic stochastic positional game with the average payoff corresponds to the case of the game from Section 2 in the case m = 2 when c = c1 = −c2 . So, we have a game (X, A, X1 , X2 , c, p, x) where the stationary strategies s1 and s2 of the players are defined as two maps s1 : x → a ∈ A1 (x) for x ∈ X1 ; s2 : x → a ∈ A1 (x) for x ∈ X2 . and the payoff function Fx (s1 , s2 ) of the players is determined by the values of average costs ωxs in the Markov processes with the corresponding probability matrices ∗ ∗ P s induced by the situations s = (s1 , s2 ) ∈ S. For this game saddle points s1 , s2 always exists (Lozovanu and Pickl, 2014) , i.e. for a given starting position x ∈ X holds ∗ ∗ Fx (s1 , s2 ) = min max2 Fx (s1 , s2 ) = max min1 Fx (s1 , s2 ). 1 1 2 2 2 1 s ∈S s ∈S

s ∈S s ∈S

Theorem 7. Let (X, A, X1 , X2 , c, p, x) be an arbitrary antagonistic stochastic positional game with an average payoff function Fx (s1 , s2 ). Then the system of equations  P a   εx + ωx = max µx,a + px,y εy , ∀x ∈ X1 ;    a∈A(x) y∈X    P a    min µx,a + px,y εy ,  εx + ωx = a∈A(x) y∈X

∀x ∈ X2 ;

has solution under the set of solutions of the system of equations  P a   ω = max p ω ∀x ∈ X1 ;  x x,y x ,   a∈A(x) y∈X    P a    px,y ωx ,  ωx = min a∈A(x)

y∈X

∀x ∈ X2 ,

i.e. the last system of equations has such a solution ωx∗ , x ∈ X for which there exists a solution ε∗x , x ∈ X of the system of equations  P a  ∗  ε + ω = max µ + p ε ∀x ∈ X1 ;  x x,a x x,y y ,   a∈A(x) y∈X    P a   ∗  px,y εy ,  εx + ωx = min µx,a + a∈A(x)

y∈X

∀x ∈ X2 .

The optimal stationary strategies of the players

s1 ∗ : x → a1 ∈ A(x) f or x ∈ X1 ; s2 ∗ : x → a2 ∈ A(x) f or x ∈ X2

in the antagonistic stochastic positional game can be found by fixing arbitrary maps s1 ∗ (x) ∈ A(x) for x ∈ X1 and s2 ∗ (x) ∈ A(x) for x ∈ X2 such that

212


s1 ∗ (x) ∈

P a ∗ T P a ∗ Arg max px,y ωx Arg max µx,a + px,y εy , a∈A(x)

y∈X

a∈A(x)

y∈X

∀x ∈ X1

and P a ∗ T P a ∗ px,y ωx s2 (x) ∈ Arg min Arg min µx,a + px,y εy . ∗

a∈A(x)

y∈X

a∈A(x)

y∈X

∀x ∈ X2

∗

∗

∗

∗

For the strategies s1 , s2 the corresponding values of the payoff function Fx (s1 , s2 ) coincides with the values ωx∗ for x ∈ X and ∗

∗

Fx (s1 , s2 ) = min max2 Fx (s1 , s2 ) = max min1 Fx (s1 , s2 ) ∀x ∈ X. 1 1 2 2 2 1 s ∈S s ∈S

.

s ∈S s ∈S

Based on the constructive proof of this theorem (see Lozovanu and Pickl, 2014) an algorithm for determining the saddle points in antagonistic stochastic positional games has been elaborated. The saddle point conditions for antagonistic stochastic positional games with a discounted payoff can be derived from Theorem 6. 6.

Conclusion

Stochastic positional games with average and discounted payoffs represent a special class of Shapley stochastic games that extends deterministic positional games. For the considered class of games Nash equilibria conditions have been formulated and proven. Based on these results new algorithms for determining the optimal stationary strategies of the players can be elaborated. References Condon, A. (1992). The complexity of stochastic games. Informations and Computation. 96(2), 203–224. Ehrenfeucht, A., Mycielski, J. (1979). Positional strategies for mean payoff games. International Journal of Game Theory, 8, 109–113. 8–113. Filar, J.A., Vrieze, K. (1997). Competitive Markov Decision Processes. Springer, 1997. Gillette, D. (1957) Stochastic games with zero stop probabilities. Contribution to the Theory of Games, vol. III, Princeton, 179-187. Gurvich, V.A., Karzanov, A.V., Khachian, L.G. (1988). Cyclic games and an algorithm to find minimax cycle means in directed graphs. USSR, Computational Mathematics and Mathematical Physics, 28, 85–91. Howard, R.A. (1960). Dynamic Programming and Markov Processes. Wiley. Lal, A.K., Sinha S. (1992) Zero-sum two person semi-Markov games, J. Appl. Prob., 29, 56-72. Lozovanu, D. (2011) The game-theoretical approach to Markov decision problems and determining Nash equilibria for stochastic positional games. Int. J. Mathematical Modelling and Numerical Optimization, 2(2), 162–164. Lozovanu, D., Pickl, S. (2006) Nash equilibria conditions for Cyclic Games with p players. Eletronic Notes in Discrete Mathematics, 25, 117–124.


213

Lozovanu, D., Pickl, S. (2009) Optimization and Multiobjective Control of Time-Discrete Systems. Springer. Lozovanu, D., Pickl, S., Kropat, E. (2011) Markov decision processes and determining Nash equilibria for stochastic positional games, Proceedings of 18th World Congress IFAC-2011, 13398–13493. Lozovanu, D., Pickl, S. (2014) Antagonistic Positional Games in Markov Decision Processes and Algorithms for Determining the Saddle Points. Discrete Applied Mathematics, 2014 (Accepted for publication). Mertens, J.F., Neyman, A. (1981) Stochastic games. International Journal of Game Theory, 10, 53-66. Moulin, H. (1976) Prolongement des jeux a deux joueurs de somme nulle. Bull. Soc. Math. France., Mem 45. Nash, J.F. (1951) Non cooperative games. Annals of Mathematics, 2, 286–295. Neyman, A., Sorin, S. (2003). Stochastic games and applications. NATO ASI series, Kluver Academic press. Puterman, M. (2005). Markov Decision Processes: Stochastic Dynamic Programming. John Wiley, New Jersey. Shapley L. (1953). Stochastic games. Proc. Natl. Acad. Sci. U.S.A. 39, 1095-1100.

Pricing in Queueing Systems M/M/m with Delays⋆ Anna V. Melnik St.Petersburg State University, Faculty of Applied Mathematics and Control Processes, Universitetskii pr. 35, St.Petersburg, 198504, Russia E-mail: [email protected]

Abstract A non-cooperative m-person game which is related to the queueing system M/M/m is considered. There are n competing transport companies which serve the stream of customers with exponential distribution with parameters µi , i = 1, 2, ..., m respectively. The stream forms the Poisson process with intensity λ. The problem of pricing and determining the optimal intensity for each player in the competition is solved. Keywords: Duopoly, equilibrium prices, queueing system.

1.

Introduction

A non-cooperative n-person game which is related to the queueing system M/M/m is considered. There are n competing transport companies, which serve the stream of customers with exponential distribution with parameters µi , i = 1, 2, ..., m respectively. The stream forms the Poisson process with intensity λ. Suppose that m P λ< µi . Let companies declare the price for the service. Customers choose the i=1

service with minimal costs. This approach was used in the Hotelling’s duopoly (Hotelling, 1929; D’Aspremont, Gabszewicz, Thisse, 1979; Mazalova, 2012) to determine the equilibrium price in the market. But the costs of each customer are calculated as the price for the service and expected time in queue. Thus, the incoming stream is divided into m Poisson flows with intensities λi , i = 1, 2, ..., m, m P where λi = λ. So the problem is following, what price for the service and i=1

the intensity for the service is better to announce for the companies. Such articles as (Altman, Shimkin, 1998; Levhari, Luski, 1978; Hassin, Haviv, 2003), and (Mazalova, 2013; Koryagin 2008; Luski, 1976) are devoted to the similar gametheoretic problems of queuing processes. 2.

The competition of two players

Consider the following game. There are two competitive transport companies which serve the stream of customers with exponential distribution with parameters µ1 and µ2 respectively. The transport companies declare the price of the service c1 and c2 respectively. So the customers choose the service with minimal costs, and the incoming stream is divided into two Poisson flows with intensities λ1 and λ2 , where λ1 + λ2 = λ. In this case the costs of each customer will be ci + ⋆

λi , µi (µi − λi )

i = 1, 2,

This work was supported by the St. Petersburg State University under grants No. 9.38.245.2014

215

Pricing in Queueing Systems M/M/m with Delays

where λi /µi (µi − λi ) is the expected time of staying in a queue (Taha, 2011). So, the balance equations for the customers for choosing the service are c1 +

λ1 λ2 = c2 + . µ1 (µ1 − λ1 ) µ2 (µ2 − λ2 )

So, the payoff functions for each player are H1 (c1 , c2 ) = λ1 c1 ,

H2 (c1 , c2 ) = λ2 c2 ,

We are interested in the equilibrium in this game. Nash equilibrium. For the fixed c2 the Lagrange function for finding the best reply of the first player is defined by λ1 λ2 L1 = λ1 c1 + k c1 + − c2 − + γ(λ1 + λ2 − λ). (1) µ1 (µ1 − λ1 ) µ2 (µ2 − λ2 ) For finding the local maxima by differentiating (1) we get ∂L1 = λ1 + k = 0 ∂c1 ∂L1 k kλ1 = c1 + + +γ =0 ∂λ1 µ1 (µ1 − λ1 ) µ1 (µ1 − λ1 )2 ∂L1 k kλ2 =− − +γ =0 ∂λ2 µ2 (µ2 − λ2 ) µ2 (µ2 − λ2 )2

from which c1 = λ1

1 1 λ1 λ2 + + + µ1 (µ1 − λ1 ) µ2 (µ2 − λ2 ) µ1 (µ1 − λ1 )2 µ2 (µ2 − λ2 )2

,

Symmetric model. Start from the symmetric case, when the services are the same, i. e. µ1 = µ2 = µ. It is obvious from the symmetry of the problem, that in equilibrium c∗1 = c∗2 = c∗ and λ1 = λ2 = λ2 . So ! λ 2 λ ∗ c = + . (2) 2 µ(µ − λ2 ) µ(µ − λ2 )2 So, if one of the players uses the strategy (2), the maximum of payoff of another player is reached at the same strategy. That means that this set of strategies is equilibrium. Asymmetric model. Assume now, that transport services are not equal, i. e. µ1 6= µ2 , suppose that µ1 > µ2 . Find the equilibrium in the pricing problem in this case. The system of equations that determine the equilibrium prices of transport companies is λ1 λ2 c∗1 + = c∗2 + µ1 (µ1 − λ1 ) µ2 (µ2 − λ2 ) 1 1 λ1 λ2 c∗1 = λ1 + + + , µ1 (µ1 − λ1 ) µ2 (µ2 − λ2 ) µ1 (µ1 − λ1 )2 µ2 (µ2 − λ2 )2

216

Anna V. Melnik 1 1 λ1 λ2 c∗2 = λ2 + + + , µ1 (µ1 − λ1 ) µ2 (µ2 − λ2 ) µ1 (µ1 − λ1 )2 µ2 (µ2 − λ2 )2 λ1 + λ2 = λ.

In Table 1 the values of the equilibrium prices with different µ1 , µ2 at λ = 10 and are given. Table 1: The value of (c∗1 , c∗2 ), (p∗1 , p∗2 ) and (λ1 , λ2 ) at λ = 10 µ2 6

µ1 7 (c1 ;c2 ) (λ1 ;λ2 ) 8 (c1 ;c2 ) (λ1 ;λ2 ) 9 (c1 ;c2 ) (λ1 ;λ2 ) 10 (c1 ;c2 ) (λ1 ;λ2 )

3.

7

(5,41;5,1) (5,15;4,85) (4,04;3,64) (5,25;4,75) (3,4;2,98) (5,33;4,67) (3,06;2,62) (5,39;4,61)

(2,5;2,5) (5;5) (1,75;1,65) (5,14;4,86) (1,4;1,26) (5,27;4,73) (1,21;1,04) (5,36;4,64)

8

9

10

(1,11;1,11) (5;5) (0,87;0,82) (0,625;0,625) (5,14;4,86) (5;5) (0,73;0,66) (0,52;0,59) (0,4;0,4) (5.26;4,74) (5,13;4,87) (5;5)

The competition of m players.

Let us increase the number of players. There are m competitive transport companies which serve the stream of customers with exponential distribution with parameters µi , i = 1, 2, ..., m respectively. The transport companies declare the price of the service ci , i = 1, 2, ..., m and the customers choose the service with minimal costs. The incoming stream is divided into n Poisson flows with intensities λi , i = 1, 2, ..., m, m P where λi = λ. Thus, the balance equations for the customers for choosing the i=1

service are

c1 +

λ1 λi = ci + , µ1 (µ1 − λ1 ) µi (µi − λi )

i = 1, ..., m.

The payoff functions for each player are

Hi (c1 , ..., ci ) = λi ci ,

i = 1, ..., m.

Find the equilibrium in this game.For the fixed ci , i = 2, ..., m the Lagrange function for finding the best reply of the first player is defined by L1 = c1 λ1 +

m X i=2

ki c1 +

λ1 λi − ci − µ1 (µ1 − λ1 ) µi (µi − λi )

Differentiating (3),we find m X ∂L1 = λ1 + ki = 0, ∂c1 i=2

m X + γ( λi − λ). i=1

(3)

217

Pricing in Queueing Systems M/M/m with Delays m P

m P

ki λ1 ∂L1 i=2 = c1 + + i=2 + γ = 0, ∂λ1 µ1 (µ1 − λ1 ) µ1 (µ1 − λ1 )2

from which

ki

∂L1 ki ki λi =− − + γ = 0, ∂λi µi (µi − λi ) µi (µi − λi )2 c∗i

c∗i +

= λi

i = 2, ..., m.

1 1 Pm + 2 (µi − λi )2 j=0,j6=i (µj − λj )

λi λi+1 = c∗i+1 + , µi (µi − λi ) µi+1 (µi+1 − λi+1 ) m X

!

,

i = 0, ..., m − 1

(4)

λi = λ.

i=1

4.

The competition of 2 players on graph.

Fig. 1: Competition of 2 players on graph G1

Consider competition on the graph G1 , which is equivalent to a part of the Helsinki Metro. Let’s define the game as Γ =< I, II, G1 , Z1 , Z2 , H1 , H2 >, where I, II are 2 competitive transport companies which serve the stream of customers with exponential distribution with parameters µi , i = 1, 2 on graph G1 =< V, E >. V = {1, 2, 3, 4} is the set of vertices, E = {e12 , e23 , e24 } - the set of edges. Zi = {R1i , R2i }

218

Anna V. Melnik

is the set of routes of player i. Each rout is a sequence of vertices. So there are two routs R1i = {1, 2, 3} and R2i = {1, 2, 4}, i = 1, 2. The stream of passengers forms the Poisson process with intensity Λ, where 0 Λ= 0 0 0

λ12 0 0 0

λ13 λ23 0 0

λ14 λ24 0 0

The transport companies declare the price of the service cikj , i = 1, 2, k = 1, 2, j = 2, 3, 4, j 6= k and the customers choose the service with minimal costs. The incoming stream Λ is divided into two Poisson flows with intensities λkj = λ1kj +λ2kj , k = 1, 2, j = 2, 3, 4, j 6= k. We are interested in equilibrium in this game. The balance equations are c112 + a11 = c212 + a21 , c123 + a12 = c223 + a22 , c124 + a13 = c224 + a23 , c113 + a11 + a12 = c213 + a21 + a22 , c114 + a11 + a13 = c214 + a21 + a23 , λkj = λ1kj + λ2kj , where ai1 =

k = 1, 2,

j = 2, 3, 4,

j 6= k,

λi12 + λi13 + λi14 , − λi12 − λi13 − λi14 )

µi (µi

ai2 = ai3 =

λi13 + λi23 µi µi 2 ( 2

− λi13 − λi23 )

λi14 + λi24 µi µi 2 ( 2

The payoff functions for each player are Hi ((c1R1 , c1R2 , c2R1 , c2R2 ) =

− λi14 − λi24 )

2 4 X X

, .

λikj cikj ,

i = 1, 2.

k=1 j=2,j6=k

The Lagrange function for finding the best reply of the first player is defined by L1 =

2 4 X X

k=1 j=2,j6=k

λikj cikj + k1 c112 + a11 − c212 − a21 + k2 c123 + a12 − c223 − a22 +

219

Pricing in Queueing Systems M/M/m with Delays +k3 c124 + a13 − c224 − a23 + k4 c113 + a11 + a12 − c213 − a21 − a22 + +k5 c114 + a11 + a13 − c214 − a21 − a23 .

Differentiating this equation we find

∂L1 = λ112 + k1 = 0, ∂c112

∂L1 = λ123 + k2 = 0, ∂c123

∂L1 = λ124 + k3 = 0, ∂c124

∂L1 = λ113 + k4 = 0, ∂c113

∂L1 = λ114 + k5 = 0. ∂c114 Since λ1kj = λkj − λ2kj , k = 1, 2, j = 2, 3, 4, j 6= k, we get ∂L1 = c112 + (k1 + k4 + k5 ) ∂λ112 ∂L1 = c123 + (k2 + k4 ) ∂λ123 ∂L1 = c124 + (k3 + k5 ) ∂λ124

∂a11 ∂a21 + 1 ∂λ12 ∂λ212

∂a12 ∂a22 + 1 ∂λ23 ∂λ223 ∂a13 ∂a23 + ∂λ124 ∂λ224

∂L1 = c113 + (k1 + k4 + k5 ) ∂λ113

∂a11 ∂a21 + 1 ∂λ13 ∂λ213

∂L1 = c114 + (k1 + k4 + k5 ) ∂λ114

∂a11 ∂a21 + ∂λ114 ∂λ214

,

,

,

+ (k2 + k4 )

∂a12 ∂a22 + 1 ∂λ13 ∂λ213

,

+ (k3 + k5 )

∂a12 ∂a22 + ∂λ114 ∂λ214

,

Symmetric model. Consider symmetric case, when the services are the same, i. e. µ1 = µ2 = µ. It is obvious from the symmetry of the problem, that in equilibrium λkj 2∗ ∗ 1 2 c1∗ kj = ckj = ckj and λkj = λkj = 2 , k = 1, 2, j = 2, 3, 4, j 6= k. So c∗12 =

λ12 + λ13 + λ14 (λ12 + λ13 + λ14 )2 + 2 λ12 +λ13 +λ14 µ µ− 2µ µ − λ12 +λ213 +λ14 2

c∗23 =

c∗24 =

c∗13 =

µ 2

µ 2

λ23 + λ13 (λ23 + λ13 )2 + 2 µ λ23 +λ13 13 µ µ2 − λ23 +λ 2 − 2 2 λ24 + λ14 (λ24 + λ14 )2 + 2 µ λ24 +λ14 14 µ µ2 − λ24 +λ 2 − 2 2

(λ12 + λ13 + λ14 )2 λ12 + λ13 + λ14 + 2 + λ12 +λ13 +λ14 µ µ− 2µ µ − λ12 +λ213 +λ14 2

220

Anna V. Melnik

+µ 2

c∗14 =

λ23 + λ13 (λ23 + λ13 )2 + 2 µ λ23 +λ13 13 µ µ2 − λ23 +λ 2 − 2 2

λ12 + λ13 + λ14 (λ12 + λ13 + λ14 )2 + 2 + λ12 +λ13 +λ14 µ µ− 2µ µ − λ12 +λ213 +λ14 2

λ24 + λ14 (λ24 + λ14 )2 + 2 µ λ24 +λ14 14 µ µ2 − λ24 +λ 2 − 2 2 In Table 2 the values of the equilibrium prices with different µ, at λ12 = 1, λ23 = 1, λ24 = 2, λ13 = 3, λ14 = 1 are given. µ 2

Table 2: The value of equilibrium prices at λ12 = 1, λ23 = 1, λ24 = 2, λ13 = 3, λ14 = 1 µ

5.

prices

10

11

12

13

14

15

c∗12 c∗23 c∗24 c∗13 c∗14

0,089 0,44 0,24 0,53 0,33

0,069 0,327 0,188 0,396 0,258

0,055 0,25 0,15 0,305 0,204

0,045 0,198 0,12 0,243 0,165

0,038 0,16 0,089 0,199 0,137

0,032 0,13 0,083 0,16 0,115

Conclusion

It is seen from the table, that the higher the intensity of service is, the lower price this transport company declare. But the prices c23 and c24 , that correspond to the edges, where the pass is divided on two roads, are greater, that c12 , because after this division the intensity of service is divided too. References Hotelling, H. (1929). Stability in Competition. Economic Journal, 39, 41–57. D’Aspremont, C., Gabszewicz, J., Thisse, J.-F. (1979). On Hotelling’s “Stability in Competition”. Econometrica, 47, 1145–1150. Mazalova, A. V. (2012). Hotelling’s duopoly on the plane with Manhattan distance. Vestnik St. Petersburg University, 10(2), 33–43. (in Russian). Altman, E., Shimkin, N. (1998). Individual equilibrium and learning in processor sharing systems. Operations Research, 46, 776–784. Levhari, D., Luski, I. (1978). Duopoly pricing and waiting lines. European Economic Review, 11, 17–35. Hassin, R., Haviv, M. (2003). To Queue or Not to Queue / Equilibrium Behavior in Queueing Systems, Springer. Luski, I. (1976). On partial equilibrium in a queueing system with two services. The Review of Economic Studies, 43, 519–525. Koryagin, M. E. (1986). Competition of public transport flows. Autom. Remote Control, 69:8, 1380–1389. Taha, H. A. (2011). Operations Research: An Introduction, ; 9th. Edition, Prentice Hall. Mazalova, A. V. (2013). Duopoly in queueing system. In: Vestnik St. Petersburg University, 10(4), 32–41. (in Russian).

How to arrange a Singles’ Party: Coalition Formation in Matching Game⋆ Joseph E. Mullat Tallinn Technical University, Faculty of Economics, Estonia E-mail: [email protected] http://datalaundering.com/author.htm Residence: Byvej 269, 2650 Hvidovre, Denmark

Abstract The study addresses important issues relating to computational aspects of coalition formation. However, finding payoffs−imputations belonging to the core−is, while almost as well known, an overly complex, NPhard problem, even for modern supercomputers. The issue becomes uncertain because, among other issues, it is unknown whether the core is nonempty. In the proposed cooperative game, under the name of singles, the presence of non-empty collections of outcomes (payoffs) similar to the core (say quasi-core) is fully guaranteed. Quasi-core is defined as a collection of coalitions minimal by inclusion among non-dominant coalitions induced through payoffs similar to super-modular characteristic functions (Shapley, 1971). As claimed, the quasi-core is identified via a version of P-NP problem that utilizes the branch and bound heuristic and the results are visualized by Excel spreadsheet. Keywords: stability; game theory; coalition formation.

1.

Introduction

It is almost a truism that many university and college students abandon schooling soon after starting their studies. While some students opt for incompatible education programs, the composition of students following particular programs may not be optimal; in other words, students and programs are mutually incompatible. Indeed, so-called mutual mismatches of priorities were among the reasons (Võhandu, 2010) behind the unacceptably high percentage of students in Estonian universities and colleges dropping out of schools, wasting their time and the entitlement to government support. However, matching students and education programs more optimally could mitigate this problem. Similar problems have been thoroughly studied (Roth, 1990; Gale, 1962; Berge, 1958...) leading, perhaps, L. Võhandu to propose a way, in this wide area of research, to solve the problem of students and programs mutual incompatibility by introducing “matching total” as the sum of duplets−priorities selected within two directions—horizontal priorities of students towards programs, and vertical priorities of programs towards students. The best solution found among all possible horizontal and vertical duplet assignments, according to LV, is where the sum reaches its minimum. Finding the best solution, however, is a difficult task. Instead, LV proposed a greedy type workaround. In the author’s words, the best solution to the problem ⋆

A thesis of this paper was presented at the Seventh International Conference on Game Theory and Management (GTM2013), June 26-28, 2013, St. Petersburg, Russia.

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of matching students and programs will be close enough (consult with Carmen et al., 2001) to a sum of duplets accumulated while moving along ordering of duplets in non-decreasing direction. It seems that LV’s proposal to the solution is a typical approach in the spirit of classical utilitarism, when the sum of utilities has to be maximized or minimized (Bentham, The Principals of Morals and Legislation, 1789; Sidgwick, The Methods of Ethics, London 1907). As noted by Rawls in "Theory of Justice", the main weakness of utilitarian approach is that, when the total max or min has been reached, those members of society at the very low utility levels will still be receiving very low compensations for incapacity, such as transfer payments to the poor. Arguing for the principal of "maxima of the lowest", referred to as the "Second Principal of Justice", Rawls suggested an alternative to the utilitarian approach. The motive driving this study is similar. We address by example an alternative to conventional core solution in cooperative games, along the lines of monotonic game (Mullat, 1979), whereby the lowest incentive/compensation should be maximized. The reader studying matching problems can also find useful information about these issues, where a number of ways of constructing an optimal matching strategy have been discussed (Veskioja, 2005). Learning by example is of high value because the conventional core solution in cooperative games cannot be clearly explained unless the readers are sufficiently familiar with utopian reality−a reality that sometimes does not exist. Thus, a rigorous set up of a simple game will be presented here, aiming to explain the otherwise rather complicated intersection of interests. More specifically, we hope to shed light on what we call a Singles-Game. It should be emphasized that, even though the game primitives represent an independent mathematical object in a completely different context, we have still “borrowed” the idea of LV duplets to estimate the benefits of matching. For this reason, we changed the nomenclature of duplets to mutual risks in order to justify the scale of payoffs−the incentives and compensations. The rest of the paper is organized as follows. We start with the preliminaries, where the game primitives are explained. In Section 3, we introduce the core concept of conventional stability in relation to the Singles-game. In Section 4, the reader will come across an unconventional theory of kernel coalitions, and nuclei coalitions, minimal by inclusion in accordance with the formal scheme. In Section 5, we continue explaining our techniques and procedures used to locate stable outcomes of the game. The study ends with conclusions and suggestions for future work, which are presented in Section 6. Appendix contains a visualization, which brings to the surface the theoretical foundation of coalition formation. Finally, interested readers would benefit from exploring the Excel spreadsheet, which helps visualize a "realistic" intersection of interests of 20 single women and 20 single men. The addendum provides a sketched outline for the evidence of some propositions. 2.

Preliminaries

Five single women and five single men are ready to participate in the Singles Party. It is assumed that all participants exhibit risk-averse behavior towards dating. To cover dating bureau expenses, such as refreshments, rewards, etc., the entrance fee is set at 50 e 1 . Thus, the cashier will be at the disposal of an amount of 500 e. All the guests have been kindly asked to take part in a survey, helping determine the attributes they look for in their prospective partner. Those who choose to provide 1

Note that red colour points at negative number.

How to arrange a Singles’ Party: Coalition Formation in Matching Game

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this information have been promised to collect a Box of Delights 2 and are hereafter referred to as participants, while others are labeled as dummies, by default, and cannot participate in the game. In addition to the delights, promised to those willing to reveal their priorities, we continue setting the rules of payoffs in the form of incentives and mismatch compensations. However, if all participants decide to date, as no reasonable justification exists for incentives and compensations, the game terminates immediately. We use index i for the women, and an index j for the men taking part in the dating party. Assuming that all the guests have agreed to participate in the game, there are {1, ..., i, ...5} women and {1, ..., j, ...5} men, resulting in 2 × 5 × 5 combinations. Indeed, when priorities have been revealed, they can form two 5 × 5 tables, W = kwi,j k, and M = kmi,j k, indicating that each woman i, i = 1, 5 revealed her priorities positioned in the rows of table W towards men as horizontal permutations wi of numbers h1, 2, 3, 4, 5i. Similarly, each man j, j = 1, 5, revealed his priorities positioned in columns of the table M towards women mj .

as vertical permutations As can be seen in Table-1, priorities wi,j (numbers 1, 5 = 1, 2, 3, 4, 5 ) might repeat in both the columns of the table W and in the rows of the table M . To be sure, more than one man may prefer the same woman at priority level wi,j , and many women, accordingly, may prefer the same man at the level mj,i . Thus, duplets or mutual risks ri,j = wi,j + mi,j occupy the cells in table R = kri,j k. Table 1.

Noting the assumption that all participants are risk-averse, some lucky couples with lower level of mutual risks start dating. These lucky couples will receive an incentive, such as a prepaid ticket to an event, free restaurant meal, etc. On the other hand, unlucky participants—i.e., those that did not find a partner—may claim a compensation, as only high-level mutual risk partners remained, given that the eligible participants at the low level of mutual risk have been matched. If no one has found a suitable partner, the question is—should the party continue? Apparently, given that the original data that failed to produce matches might have not been completely truthful, it would be unwise to offer compensation in proportion to mutual risks ri,j . Nonetheless, let us assume that the compensation equals 1/2r .10 e. In that case, couple’s [5, 5] profit may reach 50 e! Instead, the dating i,j bureau decides to organize the game, encouraging the players to follow Rawls second principle of justice. In Table-1, the minimum−the lowest mutual risk among all participants−is r1,4 = 3. Following the principle, the compensation to all unlucky participants will be equal to 1/2r1,4 .10=15 e. This setting is also fiscally reasonable from the cashier’s point of view. The balance of payoffs for all participants, will be 2

In case the Box is undesirable it will be possible to get 10 e in return.

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25 e, as 50 e paid as entrance fee will be reduced by 15 e compensation amount, and additionally by 10 e, i.e., inclusive of the cost of collected delights. Further on, we assume that each member of a dating couple will receive an incentive that is offered to all dating couples and is equal to double the compensation amount. What happens when the couple [1, 4] decides to date? The entire table R should be dynamically transformed to reflect the fact that the participants [1, 4] are matched. Indeed, as the women {2, 3, 4, 5} and men {1, 2, 3, 5} can no longer count on [1, 4] as their potential partners, the priorities will decline, whereby the scale h1, 2, 3, 4, 5i dynamically shrinks to h1, 2, 3, 4i3 . To reflect this, Table-1 transforms into Table-2: Table 2.

The minimum mismatch compensation did not change and is still equal to 15 e. However, couple’s [1, 4] potential balance 50 e+10 e+2.15 e=10 e of payoffs improves (W1 and M4 each receive 30 e as an incentive to date, based on the rule that it is equal to twice the mismatch compensation). For those not yet matched, the balance remains negative (in deficit) and equals 15 e. On the other hand, if, for example, the couple [3, 5] decides to date, the balance of payoffs improves as well. The party is over and the decisions have been made about who will date and who will leave the party without a partner. The results are passed in writing to the dating bureau. What would be the best collective decision of the participants based on the principle of "maxima of the lowest" in accord with the rules of singles-game? 3.

Conventional stability4

In this section, the aim is to present the well-established solution to the singlesgame by utilizing the conventional concept, called the core. First, without any warranty, it is helpful to focus on the core stability. In order to meet this aim, the original dating party arrangement is expanded to a more general case. The game now has n × m participants, of whom n are single women h1, ..., i, ...ni and m are single men h1, ..., j, ...mi. Some of the guests expressed their willingness to participate in the game and have revealed their priorities. Those who refused, in line with the above, are referred to as dummy players. All those who agreed to play the game will be arranged by default into the grand 4

3

4

To highlight theoretical results of mutual risks, incitements or compensations, or whatever the scales we use, the dynamic quality of monotonic scales is the only feature fostering the birth of MS − the "monotone system." Otherwise, the MS terminology, if used in any type of serialization methods applied for data analysis, will remain sterile. Terminology, which we shall use below, is somewhat conventional but mixed with our own.


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coalition P, |P| 6 n + m. Thus, indices i, j and labels α, ..., σ ∈ P are used to annotate the guests participating in the game. Only the guests in P are regarded as participants, whereas couples [i, j] are referred to as α, ..., σ. This differentiation not only helps make notations short, when needed, but can also be used in reference to an eventual match or a couple without any emphasis on gender. In the singles-game, we focus on the participants D ⊆ P who are matched. Having formed a coalition, we suppose that coalition D has the power and is in a position to enforce its priorities. It is assumed that participants in D can persuade all those in X = P\D, i.e., participants that are not yet matched, to leave the party without a partner and thus receive compensation. However, it is realistic to assume that the suppression of interests of participants’ in X is not always possible. It is conceivable that, those in the coalition D′ ⊆ X, whose interests would be affected (suppressed), will still be capable to receive as much as the participants in D. However, we exclude this opportunity, as it is better that no one expects that coalition D′ can be realized concurrently with D and act as its direct competition. Insisting on this restriction, however, we still assume that others−those participants suppressed in X−have not yet found their suitable partners and have agreed to form their own coalition, even though they could receive compensation equal to 50% of the incentives in D. A realistic situation may occur when all participants in P are matched, D = P, or, in contrast, no one decides to date, D = ∅. It is also reasonable that, after revealing their priorities, some individuals might decide not to proceed with the game and will, thus, be labeled as a dummy player δ ∈ / P. Among all coalitions D, we usually distinguish rational coalitions. Couple α, joining the coalition D, extracts from the interaction in the coalition a benefit that satisfies α ∈ D. In the singles-game, we anticipate that the extraction of benefits, i.e., the incentives and mismatch compensations, strictly depend on the membership−couples in D or participants of coalition X. Using the coalition membership D ⊆ P, we can always construct a payoff x to all participants P, i.e., we can quantify the positions of all participants. The inverse is also true. Given a payoff x, it is easy to establish which couple belongs to the coalition D and identify those belonging to the coalition X = P\D. We label this fact as Dx . Recall that couples of the coalition Dx receive an incentive to date, which is equal to the double amount of the mismatch compensation. Thus, the allocation Dx may provide an opportunity for some participants σ ∈ P to start, or initiate, new matches, thus moving to better positions. We will soon see that, while the best positions induced by special coalitions N, called the nuclei, have been reached, this movement will be impossible to realize. 5 The inability of players to move to better positions by "pair comparisons" is an example of stability. In the work "Cores of Convex games", convex games have been studied (Shapley, 1971); these are so-called games with a non-empty core, where similar type of stability exists. The core forms a convex set of end-points (imputations) of a multidimensional octahedron, i.e., a collection of available payoffs to all players. Below, despite the players’ asymmetry with respect to Dx = P\X, we focus on their payoffs driving their collective behavior as participants P to form a coalition Dx , Dx ⊆ P; here, X = Dx is called an anti-coalition to X. In contrast to individual payoffs improving or worsening the positions of participants, when playing a coalition game, the total payment to a coalition X as a whole is referred to the characteristic function v(X) > 0. In classical coop5

Our terminology is unconventional in this connection.

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erative game theory, the payment v(X) to coalition X is known with certainty, whereby the variance v(X) − v(X\ {σ}) provides a marginal utility π(σ, X). Inequality π(α, X\ {σ}) 6 π(α, X) of the scale of risks expresses a monotonic decrease (increase) in marginal utilities of the membership for α ∈ X. S The monotonicity T is equivalent to the supermodularity v(X1 ) + v(X2 ) 6 v(X1 X2 ) + v(X1 X2 ) (Nemhauser et al., 1978). Thus, any characteristic function v(X), payment on which is built according to the scale of risks, is supermodular. The inverse submodularity was used to find solutions of many combinatorial problems (Edmonds, 1970; Petrov and Cherenin, 1948). In general, such a warranty cannot be given. Recall that we eliminated all rows and columns in tables W = kwi,j k, M = kmi,j k in line with X = Dx . Table R = kπ(α, X) = wi,j (X) + mi,j (X)k, α = [i, j] ∈ X reflects the outcome of shrinking priorities wi,j , mi,j when some σ ∈ X have found a match and have formed a couple. Priorities wi,j , mi,j are consequently P decreasing. Given in the form of characteristic function, e.g., the value v(X) = α∈X π(α, X) sets up a coalition game.6 An imputation for the game ν(X) is defined by a |P|P vector fulfilling two conditions: (i) α∈P xα = v(P), (ii) xα > v({α}), for all α ∈ P. Condition (ii) clearly stems from repetitive use of monotonic inequality π(α, X\ {σ}) 6 π(α, X). A significant shortcoming of the cooperative theory given in the form of the characteristic function stems from its inability to specify a particular imputation as a solution. However, in our case, such imputation can be defined in an intuitive way. In fact, the concept of risk scale determines a popularity index of players. More specifically, the lower the risk of engagement π(α, X) of σ ∈ X, the more reliable the couple’s α coexistence is. Therefore, we set up a popularity index pi of a woman P i among men in the coalition X as number pi (X) = j∈X mi,j . P The index number pj of a man j among women, accordingly, is given by pj (X) = i∈X wi,j . We intend to redistribute the total payment v(X) in proportion to the components of the vector p(X) = hpi (X), pj (X)i, or as the vector p(X). Hereby we can prove, owing to monotonicity of the scale of priorities, that the payoffs in imputation p(P) cannot be improved by any coalition X ⊂ P. Therefore, the game solution, among popularity indices, will be the only imputation p(P). In other words, popularity indices core of the cooperative game v(X) consists of only one point p(P). In line with the terminology used above, we draw the readers’ attention to the fact that the singles-game considered next is not a game given in the form of a characteristic function. The above discussion was presented as the foundation for the course of further investigation only. 4.

Concept of a kernel

In the view of "monotone system" (Mullat, 1971-1995) exactly as in Shapley’s convex games, the basic requirement of our model validity emerges from an inequality of monotonicity π(α, X\ {σ}) 6 π(α, X). This means that, by eliminating an element σ from X, the utilities (weights) on the rest will decline or remain the same. In particular, a class of monotone systems is called p-monotone (Kuznetsov et al., 1982, 1985), where the ordering hπ(α, X)i on each subset X of utilities (weights) follows the initial ordering hπ(α, W)i on the set W. The decline of the utilities on p-monotone system does not change the ordering of utilities on any subset X. 6

ν(X) = |X|2 · (|X| + 1) . Check that v(P) = 150 for 5 × 5 -game, or use the Table-1.


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Thus, serialization (greedy) methods on p-monotone system might be effective. Behind a p-monotone system is the fact that an application of Greedy framework can simultaneously accommodate the structure of all subsets X ⊂ W. Perhaps, for different reasons, many will argue that p-monotone systems are rather simplistic and fail to compare to the serialization method. Nonetheless, many economists, including Narens and Luce (1983), almost certainly, will point out that subsets X of p-monotone systemsSperform on interpersonally compatible scales. An inequality F (X1 X2 ) > min hF (X1 ), F (X2 )i holds for real valued set function F (X) = minα∈X π(α, X), referred to as quasi-convexity (Malishevski, 1998). We observed monotone systems, which we think is important to distinguish. The system is non quasi-convex when two coalitions contradict the last inequality. We consider such systems as non-quasi-convex, which applies to the singles-game case. The ordering of priorities in singles-games−i.e., what men look for in women, and vice versa−remain intact within an arbitrary coalition X. However, in these systems, the ordering of mutual risks kri,j k on grand coalition P does not necessarily hold for some X ⊂ P. Contrary to initial ordering on R(P) = kπ(α, P) = ri,j k, the ordering of mutual risks on R(X) = kπ(α, X)k may be inverse of the ordering on R(P) for some couples. In that case, e.g., the ordering of two couples’ mutual risks can turn "upside down" while the risk scale is shrinking compared to the original ordering on the grand coalition P. Thus, in general, the mutual risks scale is not necessarily interpersonally compatible. In other words, interpersonal incompatibility of this risk scale radically differs from the p-monotone systems. This difference became apparent when it was no longer possible to find a solution using Greedy type framework of so-called defining chain algorithm−i.e., the monotone system was non-quasi-convex. Before proceeding with the formal side of these processes, it is informative to understand the nature of the problem. Definition 1. By kernel coalition we call a coalition K ∈ arg maxX⊆P F (X); {K} is the set of all kernels. Recalling the main quality of defining a chain−a sequence of elements of a monotone system—it is possible to arrange the elements α ∈ W, i.e., the couples α ∈ P of players by a sequence hα1 , ..., αk i, k = 1, n. The sequence follows the lowest risk ordering in each step k corresponding to sequence of coalitions hHk i, H1 = P, Hk+1 ← Hk \ {αk }, αk = arg minα∈Hk π(α, Hk ). Given any arbitrarily coalition X ⊆ P, we say that the defining sequence obeys the left concurrence quality if there exists a superset Ht such that Ht ⊇ X, t = 1, k, where the first element αt ∈ Ht to the left in the sequence hα1 , ..., αk i belongs to the set X, αt ∈ X as well. S On the condition that the element αt is not a member of the superset H = {K ∈ arg maxX⊆P F (X)} including all kernels K, αk ∈ / H, we observe that π(αt , X) < π(αt , Ht ). Hereby, we can conclude that F (X) 6 π(αt , Ht ) is strictly less than the global maximum of the set function F (X) = minα∈X π(α, X). The left concurrence quality guarantees that the sequence can potentially be used for finding the largest kernel H. Due to non-quasi-concavity, the left concurrence quality is no longer valid. Eliminating a couple αk = [i, j], see above, we delete the row i and the column j in the mutual risks table R. Thus, the operation Hk+1 ← Hk \ {αk } is not an exclusion of a couple αk ∈ Hk , given that the couple αk = [i, j] is about to start dating, but rather an exclusion of adjacent couples α in [i, ∗]-row and [∗, j]column. We annotate the engagement as Hk+1 ← Hk − αk or as an equal notation Dk+1 ← Dk + αk .

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In conclusion, note, once again, that, despite the properties of monotone system remaining intact, the chain algorithm, assembling the defining sequence of elements α ∈ P, cannot guarantee the extraction of the supposedly largest kernel H, particularly in the form given by Kempner et al. (2008). Thus, we need to employ special tools for finding the solution. To move further in this direction, we are ready to formulate some propositions for finding kernels K by branch and bound algorithm types. The next step will require a modified variant of imputation (Owen, 1982). We define an imputation as the outcome connected to the singles-game in the form of a |P|vector of payoffs to all participants. More specifically, the outcome is a |P|-vector, where each partner in a couple σ ∈ X receives the lowest mismatch compensation F (X), whereas each partner in the couple σ ∈ / X belonging to the anti-coalition X = Dx receives the incentive to date, which is equal to twice that amount, i.e., 2 · F (X), cf. Tables 3,4. The concept of outcome (imputation) in this form is not common because the amount to be claimed by all participants is not fixed and equals |P| + F (X) · |X| + 2 · X . Thus, it is likely that participants will fail to reach an understanding, and will claim payoffs obtaining less than available total amount (n + m)·50 e. The situation, in contrast, when participants will claim more than total amount, is also conceivable. Any coalition X induces a |P|-vector x = hxσ i as an outcome x:7 xσ =

X 2 + F (X) if σ ∈ X, → xσ = |P| + F (X) · |X| + 2 · X . 2 · (1 + F (X)) if σ ∈ / X. σ∈P

In this case, xσ is a quasi-imputation. This definition of outcome is used later, adapting the concept of the quasi-imputation for the purpose of the singles-game. We say that an arbitrary coalition X induces an outcome x. Computed and prescribed by coalition X, the components of x consist of two distinct values F (X) and 2 · F (X). Participants σ ∈ X could not form a couple, while participants σ ∈ Dx were able to match. Recall that the notation for X is also used as a mixed notation for dating couples Dx . Before we move further, we will try to justify our mixed notation X. Although a coalition X = Dx uniquely defines both those Dx among participants P who went on dating, and those X = P\Dx who did not, the coalition X does not specifically indicate matched couples. In contrast, using the notation Dx , we indicate that all participants in Dx are matched, whereas a couple σ ∈ Dx also indicates an individual decision how to match. More specifically, this annotation represents all men and all women in Dx standing in line facing one member of the opposite sex, with whom they are matched. However, any matching or engagement among couples belonging to Dx , or whatever matches are formed in Dx , does not change the payoffs xσ valid for the outcome x. In other words, each particular matching Dx induces the same outcome x. Decisions in Dx with respect to how to match provide an example of individual rationality, while the coalition Dx formation, as a whole, is an example of collective rationality. Therefore, in accordance with payoffs x, the notation Dx subsumes two different types of rationality−the individual and the collective rationality. In that case, the outcome x accompanying Dx represents 7

Further, we follow the rule that capital letters represent coalitions X, Y, ..., K, H, ... while lowercase letters x, y, ..., k, h, ... represent outcomes induced by these coalitions.


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the result of a partial matching of participants P. Propositions below somehow bind the individual rationality with the collective rationality. One of the central issues in the coalition game theory is the question of the possible formation of coalitions or their accessibility, i.e., the question of coalition feasibility. While it is traditionally assumed that any coalition X ⊆ P is accessible or available for formation, such an approach is generally unsatisfactory. We will try to associate this issue with a similar concept in the theory of monotone systems. The issue of accessibility of subsets X ⊂ W in the literature of monotone systems has been considered not only in the context of the totality 2W of its subsets X ∈ 2W but also with respect to special collections of subsets F ⊂ 2W . A singleton chain αt adding elements step-by-step, starting with the empty set ∅, can, in principle, access any set X ∈ F, or access the set X by removing the elements starting with the grand set W−so called upwards or downwards accessibility. Definition 2. Given coalition X ⊆ P, where P is the grand coalition, we call the collection of pairs C(X) = {arg minα∈X π(α, X)} naming C(X) as best potential couples, capable of matching with the lowest mutual risk, within the coalition X. Consider a coalition Dx , generated by the formation by a chain of steps Dk+1 ← Dk + hαk i. Let X1 = P, Xk = P\Dk , where Dk are participants trying to match during the step k; C(Xk ) are couples in Xk with the lowest mutual risk among couples not yet matched in steps k = 1, n, Xn+1 = ∅. Coalitions in the chain Xk+1 = Xk − αk are arranged after the rows and columns, indicated by couple αk , have been removed from W , M and R. Mutual risks R have been recalculated accordingly. Definition 3. Given the sequence hα1 , ..., αk i of matched couples, where X1 = P, Xk+1 = Xk − αk , we say that coalition Dx = X = P\X of matched (as well as X of not yet matched) participants is feasible, when the chain T hX1 , ..., Xk+1 = Xi complies with the rational succession C(Xk+1 ) ⊇ C(Xk ) Xk+1 . We call the outcome x, induced by sequence hα1 , ..., αk i, a feasible payoff, or a feasible outcome. Proposition 1. The rational succession rationality necessarily emerges from the condition that, under the coalition Dx formation a couple αk does not decrease the payoffs of couples hα1 , ...αk−1 i formed in previous steps. The accessibility or feasibility of coalition Dx formation offers convincing interpretation. In fact, the feasibility of coalition Dx means that the coalition can be formed by bringing into it a positive increment of utilities to all participants P, or by improving the position of existing participants having already formed a coalition when new couples enter the coalition in subsequent steps. We claim that, in such a situation, coalitions are formed by rational choice. The rational choice C(X) satisfies so-called heritage or succession rationality described by Chernoff (1954), Sen (1970), and Arrow (1959). Below, we outline the heritage rationality in the form suitable for visualization. The proposition states that, in matches, the individual decisions are also rational in a collective sense only when all participants in Dx individually find a suitable partner. We can use different techniques to meet the individual and collective rationality by matching all participants only in Dx , which is akin to the stable marriage procedure (Gale & Shapley, 1962). In contrast, the algorithm below provides an

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optimal outcome/payoff accompanied by partial matching only−i.e., only matching some of participants in P as participants of Dx ; once again, this is in line with the Greedy type matching technique. Proposition 2. The set {K} of kernels in the singles-game arranges feasible coalitions. Any outcome κ induced by a kernel K ∈ {K} is feasible. At last, we are ready to focus on our main concept. Definition 4. Given a pair of outcomes x and y, induced by coalitions X and Y , an outcome y dominates the outcome x, x ≺ y: (i) ∃S ⊆ P| ∀σ ∈ S → xσ < yσ , (ii) the outcome y is feasible. Condition (i) states that participants/couples σ ∈ S ⊂ P receiving payoffs xσ can break the initial matching in Dx and establish new matches while uniting into Dy . Alternatively, some members of X, i.e., not yet matched participants in S, can find suitable partners amid participants in Dy , or, even their compensations in Y may be higher than their incentives in x. Thus, by receiving yσ instead of xσ the participants belonging to S are guaranteed to improve their positions. The interpretation of the condition (ii) is obvious. Thus, the relation x ≺ y indicates that participants in S can cause a split (bifurcation) of Dx , or are likely to undermine the outcome x. Definition 5. A kernel N ∈ {K} minimal by inclusion is called a nucleus−it does not include any other proper kernel K ⊂ N: K 6⊂ N is true for all K 6= N. Proposition 3. The set {n} of outcomes, induced by nuclei {N}, arranges a quasicore of the singles-game. Outcomes in {n} are non-dominant upon each other, i.e., n ≺ n′ , or n ≻ n′ is false. Thus, the quasi-core is internally stable. The proposition above clearly indicates that the concept of internal stability is based on "pair comparisons" (binary relation) of outcomes. The traditional solution of coalition games recognizes a more challenging stability, known as NM solution, which, in addition to the internal stability, demands external stability. External stability ensures that any outcome x of the game outside NM -solution cannot be realized because there is an outcome n ∈ {n}, which is not worse for all, but it is necessarily better for some participants in x. Therefore, most likely, only the outcomes n that belong to NM -solution might be realized. The disadvantage of this scenario stems from the inability to specify how it can occur. In contrast, in the singles-game, we can define how the transformation of one coalition to another takes place, namely, only along feasible sequence of couples. However, it may happen that for some coalitions X outside the quasi-core {N}, feasible sequence may stall unable to reach any nucleus N ∈ {N}, whereby starting at X the quasi-core is feasibly unreachable. This is a significant difference with respect to the traditional NM -solution. 5.

Finding the quasi-core

In general, when using Greedy type algorithms, we gradually improve the solution by a local transformation. In our case, a contradiction exists because nowhere is stated that local improvements can effectively detect the best solution−the best outcome or payoffs to all players. The set of best payoffs, as we already established above,


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arranges a quasi-core of the game. Usually, finding the core in the conventional sense is a NP-hard task, as the number of "operations" increases exponentially, depending on the number of participants. In the singles-game, or in almost all other types of coalition games, we observe an extensive family of subsets constituting traditional core imputations. Even if it is possible to find all the payoff vectors in the core, it is impractical to do so. We thus posit that it is sufficient to find some feasible coalitions belonging to the quasi-core and the payoffs induced by these coalitions. This can be accomplished by applying a procedure of strong improvements of payoffs, and several gliding procedures, which do not worsen the players’ positions under coalition formation. Indeed, based on rationality, known as the rational succession, Definition 3, it is not rational in some situations to use the procedure of strong improvements, as these do not exist. However, using gliding procedures, we can move forward in one of the promising directions to find payoffs not worsening the outcome. Experiments conducted using our polynomial algorithm show that, while using a mixture of improvement procedure and gliding procedures, combined with the succession condition, one can take the advantage of backtracking strategy, and might find feasible payoffs of the singles-game belonging to the quasi-core. We use five procedures in total—one improvement procedure and four variants of gliding procedures. Combining these procedures, the algorithm below is given in a more general form. While we do not aim to explain in detail how to implement these five procedures, in relation to rational succession, it will be useful to explain beforehand some specifics of the procedures because a visual interaction is best way to implement the algorithm. In the algorithm, we can distinguish two different situations that will determine in which direction to proceed. The first direction promises an improvementTin case the couple α ∈ X decides to match. We call the situation when C(X T − α) C(X) = ∅ as a potential improvement situation. Otherwise, when C(X − α) C(X) 6= ∅, it is a potential gliding direction. Let CH(X) be the set of rows C(X), the horizontal routes in the table R, which contain the set C(X). By analogy CV (X) represents the vertical routes, the set of columns, C(X) ⊆ CH(X)×CV (X). To apply our strategy upon X, we distinguish four cases of four non-overlapping blocks in the mutual risk table R:CH(X) × CV (X); CH(X) × CV (X); CH(X) × CV (X); CH(X) × CV (X). Proposition 4. An improvement in payoffs for all participants in the singles-game may occur only when a couple α ∈ X complies with the potential improvement T situation in relation to the coalition X, the case of C(X − α) C(X) = ∅. The couple α ∈ X is otherwise in a potential gliding situation. The following algorithm represents a heuristic approach to finding a nucleus n among nuclei {N} of the singles-game. Input Build the mutual risks table, R = W + M −a simple operation in Excel spreadsheet. Recall the notation P of players as the game participants. Set k ← 1, X ← P in the role of not yet matched participants, i.e., as players available for potential matching. In contrast to the set X, allocate indicating by Dx ← ∅ the initial status of matched participants. Do Step up: S Find a match αk ∈ CH(X) × CV (X), Dx ← Dx + αk , such that F (X) < F (X − αk ), X ← X − αk , Xk = X, k = k + 1, otherwise Track Back. Gliding: D Find a match αk ∈ CH(X) × CV (X), Dx ← Dx + αk , such that F (X) = F (X − αk ), X ← X − αk , Xk = X, k = k + 1, otherwise Track Back.

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Joseph E. Mullat

F Find a match αk ∈ CH(X) × CV (X), Dx ← Dx + αk , such that F (X) = F (X − αk ), X ← X − αk , Xk = X, k = k + 1, otherwise Track Back. G Find a match αk ∈ CH(X) × CV (X), Dx ← Dx + αk , such that F (X) = F (X − αk ), X ← X − αk , Xk = X, k = k + 1, otherwise Track Back. H Find a match αk ∈ CH(X) × CV (X), Dx ← Dx + αk , such that F (X) = F (X − αk ), X ← X − αk , Xk = X, k = k + 1, otherwise Track Back. Loop Until no couples to match can be found in accordance with cases S, D, F, G and H. Output The set Dx has the form Dx = hα1 , ..., αk i. The set N = P\Dx represents a nucleus of the game while the payoff n induced by N belongs to the quasi-core. In closing, it is worth noting that a technically minded reader would likely observe that coalitions Xk are of two types. The first case is X ← X − αk operation when the mismatch compensation increases, i.e., F (Xk ) < F (Xk − αk ). The second case occurs when gliding along the compensation F (Xk ) = F (Xk − αk ). In general, independently of the first or the second type, there are five different directions in which a move ahead can proceed. In fact, this poses a question—in which sequence couples αt should be selected in order to facilitate the generation of the sequence Dx = hα1 , ..., αk i? We solved the problem for singles-games underpinning our solution by backtracking. It is often clear in which direction to move ahead by selecting improvements, i.e., either a strict improvement by s) or gliding procedures though d), f ), g) or h). However, a full explanation of backtracking is out of the scope of our current investigation. Thus, for more details, one may refer to similar techniques, which effectively solve the problem (Dumbadze, 1989). 6.

Conclusions

The uniqueness of singles-game lies in the dynamic nature of priorities. As the construction of the matching sequence proceeds, priorities dynamically shrink, and finally converge at one point. Dynamic transformation, or the monotonic (dynamic) nature of priorities, enabled constructing a game based on so-called monotone system, or MS. One disadvantage behind the use of the MS-system is its drawback in the respective interpretation of the analysis results. More specifically, when the process of extracting the core terminates, the interpretation requires further corrections. However, with regards to the choice of the best variants, i.e., the choice of the best matches in the singles-game, the paper reports a scalar optimization in line with "maxima of the lowest" principle, or rather an optimal choice of partial matching. This view opens the way to consider the best partial matching as the choice of the best variants−alternatives—and to explore the matching process from the perspective of a choice problem. Usually, when trying to analyze the results, a researcher must rely on the common sense. Therefore, applying the well-known and well thought out concepts and categories that have been successfully applied in the past, we can move forward in the right direction. Our advantage was that this relation was found, and was transformed into a shape similar to the core, which is known concept in the theory of stability of collective behavior, e.g., in the theory of coalitional games. Irrespective of the complexity of intersections in the interests of players, deftly twisted rules for compensations in unfortunate circumstances, incitements, etc., singles-game, as it seems, makes a point. However, this is not enough in social sciences, especially in economics, when a formal scheme rarely depicts the reality,


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e.g., the difference in political views and positions of certain groups of interest, etc. Perhaps, the individual components of the game will still be helpful in moving closer to answering the question of what is right or wrong, or what is good and what is bad, which would be a fruitful path to explore in future studies of this type. Appendix Visualization Recall that, in the singles-game, the input to the algorithm presented in the main paper contains two tables: W = wi,j −priorities wi the women specify with the respect to the characteristics the men should possess, in the form of permutations of numbers 1, n in rows, and the table M = mj,i −priorities mj the men specify with the respect to the characteristics the women should possess, in the form of permutations of numbers1, m in columns. These tables, and tabular information in general, are well-suited for use in Excel spreadsheets that feature calculation, graphing tools, pivot tables, and a macro programming language called VBA−Visual Basic for Applications. A spreadsheet was developed in order to present our idea visually, i.e., the search for nuclei of the singles-game, and the stable coalitions with outcomes belonging to the quasi-core induced by these coalitions. The spreadsheet takes for granted the Excel functions and capabilities. Tables W , M and R of 20 × 20 dimensions can be downloaded from http://www.datalaundering.com/download/singles-game.xls. We first provide the user with the list of macros written in VBA. Then, we supply tables W , M and R extracted from the spreadsheet by comments. We also hope that the spreadsheet exercise will be useful in enhancing the understanding of our work. In particular, we focus on the technology of backtracking, given by macros TrackR and TrackB. The list of macro-programming routines is in line with the steps of the algorithm presented in Section 5. • CaseS. Ctrl+s Trying to move by improvement along the block CH(X) × CV (X) of cells [i,j] by" pm and excludes ballet from the woman‘s possible strategies if pm > pw . The formalization of this game is: P ΓBoS = h{man = 1, woman = 2}, ({ballet, soccer}, {batter, soccer}), (u1, u2 ), (p1 , p2 ), Pi Y Si = {(B, B), (B, S), (S, B), (S, S)} = S i∈In

The set of parts of S, 2S , is given by:   Φ, S, {(B, B)}, {(B, S)}, {(S, B)}, {(S, S)}, {(B, B), (B, S)},       {(B, B), (S, B)}, {(B, B), (S, S)}, {(B, B), (B, S), (S, B)}, S 2 = {(B, B), (B, S), (S, S)}, {(B, B), (S, B), (S, S)}, {(B, S), (S, B), (S, S)}       {(B, S)}, {(S, B)}, {(B, S), (S, S)}, {(S, B)}, {(S, S)} The power map P : R2 → 2S is:   {(B, S), (S, S)} if x < y P(x, y) = {(B, S), (B, B)} if x > y  φ if s = y

where x = p1 = man power, and y = p2 = woman power. Assuming that P constrains the set of strategies, as shown below: Y Y Y Si |P(p1 ) = Si |P(p2 ) = Si \P(p) i∈In

i∈In

i∈In

In the game of the BoS, the available strategies are: Y Si |P(xy) = S\{(B, S), (B, B)} = {(S, B)}, {(S, S)}

Example 3 (Prisonerï£¡s Dilemma). Supposing that criminal A is more powerful than B, A will use his power to influence the decisions (set of strategies) of agent B. For this particular situation, (Williamson (2010), p. 26) offered an explanation regarding the relation between the prisoners that implicitly uses the concept of power: "rather than assume that players are accepting of the coercive payoffs that are associated with the prisoners’ dilemma - according to which each criminal is

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Rodolfo Coelho Prates

induced to confess, whereas both would be better off if they could commit not to confess - Transaction Costs Economy assumes that the criminals (or their handlers, such as the mafia) can, upon looking ahead, take ex ante actions to alter the payoffs by introducing private ordering penalties to deter defections. This latter is a governance move, variants of which can be introduced into many other bad games". The ex ante actions are defined exactly by the power relations between the criminals, which could make use of condign, compensatory or conditioned power. Assuming asymmetric forces between both, the criminal with more power imposes the strategy not to confess on the man with less power, he chooses the strategy not to confess, while the criminal with more power chooses the strategy to confess. In this situation, the criminal with more power goes free while the man with less power will be punished. Taking into account that power is present in the relationship, there is no more Nash equilibrium. The examples above show that the powerful agent constrains the set of strategies of the other agent. In this sense, the other agent does not have all the options available to choose, only part of them. In this new condition, the equilibrium is not the same that would be achieved if the options were unconstrained. Due to this, it is possible to think of a measure or index of power which the powerful player has over the other. In a two person game, and supposing that both have mixed strategies, the measure of power of the powerful agent (player i) can be described as: ui |Pij IPi = ui Where ui is the expected utility for the player i and uj |Pij is the expected utility for the player i when he uses his power (Pij ) to constrain the strategy set of player j. The power is the "force" that the powerful player uses to restrict the choices of the other player. In fact, by the definition, IPi lies in the interval [1, ∞]. If IPi = 1, it means that ui |Pij is equal to ui , and in this situation the function wij does not restrict the strategy set of player j. Thus, player i does not have power over player j. On the other hand, if ui |wij is greater than ui , IPi will be greater than one. In this sense Pij acts and restricts the strategy set of player j. When the power is acting, some of the options are not available in the strategy set, meaning that this player does not choose the option that he would if all the options were available. Example 4. The matrix below shows the same game as example one, but here the strategies are mixed, with equal probabilities for both players and for all strategies. Table 1: Example 4

A

a1 a2 a3

b1 2.4 2.3 1.5

B b2 3.2 4.1 2.2

b3 2.1 3.0 1.5

The expected utility for player B is 2.55. Assuming again that player B has power over player A, and that he uses this power, it is clear that the strategy a2 is restricted by the power. In this way, player A cannot choose this strategy (a2 ) because it is not available, and player B is free to choose any strategy in his set.

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Afterwards, the new expected utility for player B is 3.16, and the measure of power is 1.23. This measure shows an increase in expected utility when the powerful player exercises his power. 3.

Conclusions

Power is not a philosophical or remote subject, despite the fact that usually no one can see or measure it. The understanding of this subject has been developed, over many years, mainly by social scientists, including economists. Game theory is an applied branch of mathematics that analyses the rational interaction between agents, but the models constructed have not embodied the asymmetric forces between them. In this way, it is possible to understand in a formal model regarding game theory how power can affect the relationship between players, assuming that one of the agents has more forces than others, and uses his forces to guarantee that his will is easily achieved. The model was constructed taking into account that power should be analyzed in the strategic interaction between individuals. In games where there are one or more Nash equilibria, the power relationship between the players removes one or all of the equilibria due to the fact that one of the players is not doing what is best, and does exactly is imposed on him. Finally, this paper unites two different sciences that developed separately: sociology and game theory. References Balzer, W. (1992). Game theory and power theory: a critical comparison. In: Wartenbert, T. E. rethinking power. Albany, NY: Suny Press. Dahl, R. A. (1957). The concept of power. Behavioral Science. Foucault, M. (1978). The history of sexuality. New York: Pantheon Books. Galbraith, J. K. (1983). The Anatomy of Power. John Kenneth Galbraith, Boston: Houghton Mifflin. Hobbes, T. (1968). (editor: MacPherson, Crawford Brow,) Leviathan, or The Matter, Forme & Power of a Common-wealth Ecclesiastical and Civill, London, (original edition: London, 1651). Nye, J. S. (2006). Soft Power, Hard Power and Leadership. Retrieved: March, 15,2013, from http://www.hks.harvard.edu/netgov/files/talks/docs/11_06_06_seminar_Nye_HP _SP_Leadership.pdf Nye, J. S. (2011). The future of power. Public Affairs/Perseus Book Group. Osborne, M., Rubinstein, A. (1994). A Course in Game Theory. Cambridge, MA: MIT Press. Wartenbaerg, T. E. (1990). The Forms of Power: From Domination to Transformation. Philadelphia: Temple University Press. Weber, M. (1968). Economy and Society. Bedminster Press, New York. Wiese, H. (2009). Applying cooperative game theory to power relations. Quality and Quantity, 43(4), 519–533. Williamson, O. E. (2010). Transaction cost economics: an overview. In: The Elgar Companion to Transaction Cost Economic (Peter G. Klein and Michael E. Sykuta, eds).

Completions for Space of Preferences Victor V. Rozen Saratov State University, Astrakhanskaya St. 83, Saratov, 410012, Russia E-mail: [email protected]

Abstract A preferences structure is called a complete one if it axiom linearity satisfies. We consider a problem of completion for ordering preferences structures. In section 2 an algorithm for finding of all linear orderings of finite ordered set is given. It is shown that the indicated algorithm leads to construction of the lattice of ideals for ordered set. Further we find valuations for a number of linear orderings of ordered sets of special types. A problem of contraction of the set of linear completions for ordering preferences structures which based on a certain additional information concerning of preferences in section 4 is considered. In section 5, some examples for construction and evaluations of the number of all linear completions for ordering preferences structures are given. Keywords: preferences structure, ordering preferences structure, completion of preferences structure, a valuation for the number of linear completions.

1.

Introduction

A space of preferences (or preferences structure) can be defined as a triplet of the form hA, α, βi , (1) where α and β are binary relations on a set A satisfying the following axioms: 1. α ∩ α−1 = ∅ −1

2. β = β 3. ∆A ⊆ β

4. α ∩ β = ∅

(asymmetry); (symmetry); (reflexivity);

(2)

(disjointness).

We mean A as a set of alternatives; α as a strict preference relation; β as an indifference relation. As usually we put ρ = α ∪ β and use the notation: ρ

df

α

β

a . b ⇔ a < b or a ∼ b. Then a space of preferences can be written as a pair hA, ρi, where the strict preference relation and the indifference relation can be presented as α = ρ\ρ−1 , β = ρ ∩ ρ−1 .

(3)

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Completions for Space of Preferences

The main special properties for preferences structures are the following ones: Transitivity : Antisymmetry : Linearity :

ρ

ρ

ρ

a1 . a2 , a2 . a3 ⇒ a1 . a3 ;

(Tr)

a1 . a2 , a2 . a1 ⇒ a1 = a2 ;

(Antsym)

ρ

ρ

a1 . a2

ρ

ρ

or a2 . a1 .

(Lin)

Definition 1. Preferences structure satisfying the conditions (Tr) and (Antsym) is called an ordering preferences structure and satisfying the conditions (Tr), (Antsym) and (Lin) is called a linear (or complete) ordering preferences structure. Definition 2. A preferences structure hA, α1 , β1 i is called a completion of a preferences structure hA, α, βi if inclusions α ⊆ α1 , β ⊆ β1

(4)

hold and at least once of these inclusions is strict. Remark 1. A preferences structure hA, α, βi has not completions if and only if it is a linear one. Thus the most interesting are completions of a preferences structure to a linear preferences structure. In this paper, we study some questions concerning of completions for ordering preferences structure. The main problems of our investigation are: (PI) The problem of description of all completions for ordering preferences structure to a linear one and (PII) The problem of contraction of the set of linear completions based on certain additional information concerning of these completions. 2. 2.1.

Linear orderings of ordered sets An algorithm for finding of all linear orderings

It is well known the following classical result (Birkhoff, 1967). Szpilrajn Theorem. Any partial ordering can be enlarged to a linear ordering. Thus in terms of our paper, any ordering preferences structure has a completion to a linear one. However, Szpilrajn theorem is not a constructive propositional since it does not indicate a method for construction of linear completions. Consider an ordering preferences structure which on a set of alternatives A is given. In algebra terminology, such a structure can be presented as an ordered set (A, ≤) (i.e. ≤ is a binary relation on A satisfying conditions reflexivity, antisymmetry and transitivity). In this notations, the strict preference relation α coincides with strict order < and the indifference relation β is identity relation. We now state an algorithm for finding of all linear orderings of a finite ordered set that is an algorithm for finding of all linear completions of an ordering preferences structures. Remark that formally a linear ordering of k-element subset B ⊆ A can be represented as one-one isotonic function ϕ from B into {1, . . . , k}, where ϕ (a) is a number of element a ∈ B under this linear ordering. The required algorithm is based on the following lemma.

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Lemma 1. Suppose an ordered set A contains n elements and a∗ is a maximal element. Assume that we have a linear ordering ϕ of a subset A\a∗ (by numbers 1, 2, . . . , n − 1). Preserve the function ϕ for elements of A\a∗ and put ϕ (a∗ ) = n then ϕ becomes a linear ordering of all set A. Thus we can obtain all completions of the set A, having completions of subsets which are a result of extraction of maximal elements. Further we use the same method for these subsets until the empty set ∅ appears. To realize this algorithm we need in the following steps. Step 1. Define an auxiliary graph γ by the following rule. Vertexes of graph γ γ are some subsets of A and for two subsets A1 , A2 ⊆ A put A1 ≺ A2 if and only if A2 is a result of extraction of some maximal element belonging to A1 . Then starting of the set A, we construct some sequence of conjugate subsets with respect to graph γ. It is evident that in graph γ the length of any path is equal to n. Step 2. For each one element subset which is a vertex of graph γ write its single linear ordering. Step 3. Let B be a k-element subset (k = 2, . . . , n) which is a vertex of graph γ. Assume we have a linear ordering for each subset of the form B\a, where a is a maximal element of B. Then we preserve these linear orderings for elements belonging B\a and set ϕ (a) = k. Step 4. As the final step of this algorithm we obtain all linear orderings for set A which is a vertex of graph γ. An example for finding of all completions of ordered set in section 5 is given. 2.2.

Ideals of ordered set

Definition 3. Let hA, ≤i be an arbitrary ordered set. A subset B ⊆ A is called an ideal in hA, ≤i if the following condition a ∈ B, a′ ≤ a ⇒ a′ ∈ B holds. For any subset X ⊆ A we define a set of its minorants X ↓ by setting X ↓ = {a ∈ A : (∃x ∈ X) a ≤ x} .

(5)

For any X ⊆ A, subset X ↓ is the smallest (under inclusion) ideal which contains X; if X is an ideal then X ↓ = X. It is said that X ↓ is the ideal generated by subset X. Particularly an ideal generated by one element subset {a} is called a main ideal and denoted by a↓ . A mapping X → X ↓ which every subset X ⊆ A put in correspondence the ideal generated by this subset is a closure operation, hence the set Id (A) of all ideals of ordered set hA, ≤i forms (under inclusion) a complete lattice in the sense (Birkhoff, 1967). Since the intersection and the union of any family of ideals is an ideal also then the lattice of ideals Id (A) is distributive. We now indicate some method for construction of the lattice Id (A). Theorem 1. Let hA, ≤i be a finite ordered set. Then 1. A subset which is a result of extraction from ideal its maximal element is an ideal also; 2. Any ideal can be realized from ideal A with help of procedure of extraction of maximal elements by a finite number steps.


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Proof (of theorem 1). 1. Let B ⊆ A be an ideal of the ordered set hA, ≤i and b∗ ∈ B a maximal element of B. Show that B\b∗ also is an ideal. Indeed suppose a ∈ B\b∗ and a′ ≤ a; since subset B is an ideal and a ∈ B then a′ ∈ B. Assumption a′ = b∗ implies b∗ ≤ a. The equality b∗ = a is false since we obtain b∗ ∈ B\b∗ . Then b∗ < a that is impossible for maximal element b∗ ∈ B. Thus a′ 6= b∗ hence a′ ∈ B\b∗ . We now state the following lemma. Lemma 2. Let B ⊆ A be an ideal in ordered set hA, ≤i and B 6= A. Then there exists such element a1 ∈ A\B that α) subset B ∪ {a1 } is an ideal and β) the element a1 is a maximal one in B ∪ {a1 }. Proof (of lemma 2). Consider any ideal B in ordered set hA, ≤i where B 6= A. Fix some minimal element a1 of non-empty set A\B. Check that B ∪ {a1 } is an ideal. Suppose a ∈ B ∪ {a1 } and a′ < a. If a ∈ B then a′ ∈ B by definition of ideal hence a′ ∈ B∪{a1 }. In the case a = a1 assume a′ ∈ / B. Then a′ ∈ A\B and we have a1 > a′ that is false since element a1 is minimal in A\B. Thus a′ ∈ B ⊆ B ∪ {a1 } and α) is proved. Show β). The assumption b > a1 for some b ∈ B implies by definition of ideal the inclusion a1 ∈ B in contradiction with a1 ∈ A\B and lemma 2 is proved. ⊔ ⊓ We now prove the proposition 2 of theorem 1. Let B ⊆ A be an ideal in ordered set hA, ≤i and B 6= A. By lemma 2 there exists such element a1 ∈ A\B that the subset B ∪ {a1 } is an ideal and a1 is a maximal element in B ∪ {a1 }. If B ∪ {a1 } = A then the ideal B is a result of extraction of maximal element a1 from A and our proposition is proved. If B ∪ {a1 } 6= A then using lemma 2 once more we obtain that there exists such a2 ∈ A\ (B ∪ {a1 }) that the subset B ∪ {a1 , a2 } is an ideal and a2 is a maximal element in B ∪ {a1 , a2 }. Consider two cases: B ∪ {a1 , a2 } = A and B ∪ {a1 , a2 } 6= A etc. Since the set A is finite, we have a sequence of the kind {a1 , a2 , . . . , at } where as ∈ A\ (B ∪ {a1 , . . . , as−1 }) and the following conditions hold (s = 1, . . . , t): α∗ ) B ∪ {a1 , . . . , as } is an ideal; β ∗ ) as is a maximal element in B ∪ {a1 , . . . , as }; γ ∗ ) B ∪ {a1 , . . . , at } = A. Thus the ideal B is a result of extraction of maximal elements {at , at−1 , . . . , a1 } from a chain of ideals starting of A which was to be proved. Finally for B = A the proposition 2 of Theorem 1 is evident since in this case the required number of extractions of maximal elements is equal to zero. ⊔ ⊓ According to theorem 1, we remark that vertexes of an auxiliary graph γ are precisely ideals of ordered set hA, ≤i. Hence we obtain Corollary 1. We can identify auxiliary graph γ of ordered set hA, ≤i with lattice Id (A) of its ideals Id (A). Namely, the set of vertexes of graph γ coincides with the set of ideals and the canonical order relation of lattice Id (A) can be presented as following: B1 ⊇ B2 if and only if there exists a path from B1 to B2 in graph γ. We now remark that for finite ordered set hA, ≤i the procedure of finding its linear orderings can be reduced to finding of maximal chains in the lattice of ideals Id (A). Further using the indicated algorithm for construction of linear orderings, we have

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Corollary 2. For finite ordered set hA, ≤i, there exists one-one correspondence between its linear orderings and maximal chains in the lattice Id (A) of its ideals. Hence the number of linear completions of an ordering preferences structure hA, ≤i coincides with the number of maximal chains in the lattice Id (A). Remark 2. Indicated correspondence can be realized in the following manner. For linear ordering {ai1 < ai2 < . . . < ain } of A, the corresponding maximal chain in the lattice of ideals is {∅ ⊂ {ai1 } ⊂ {ai1 , ai2 } ⊂ . . . ⊂ {ai1 , ai2 , . . . , ain }} . 3.

A valuation for a number of linear orderings

3.1.

A finding of the number of linear ordering with help of auxiliary graph γ Using inductive algorithm for construction of auxiliary graph γ (see 2.1.), we can find the number N (A) of all linear orderings of a finite ordered set without of finding of these orderings. As a first step, we need to construct the auxiliary graph γ. Denote by N (B) the number of all linear orderings for arbitrary set B which is a vertex of graph γ. Since linear orderings of B are extensions of linear orderings of conjugate with B vertexes then we obtain the following recurrent formula: X N (B) = N (B\a) , (6) where a is an arbitrary maximal element of subset B. Since every subset which is a final vertex in graph γ is one element hence it has a single linear ordering. Using formula (6) we can find the number of all linear orderings for any vertex of graph γ. In particular we can find the required number N (A). An example for count of the number of all linear orderings of ordered set in section 5 will be given. 3.2.

A valuation of the number of linear orderings for some special cases Remark that formula (6) for finding of N (A) can be used only in the case the graph γ (i.e. the lattice of ideals of ordered set) is given. However a practical construction of the graph γ for ordered set which contains some tens of elements is very hard. Further we consider certain methods for finding N (A) in some special cases. Let hAk , ωk i (k = 1, . . . , r) be a family of ordered sets and hA, ωi is the discrete sum of this family. Denote by Ndis the number of all linear orderings for hA, ωi. Then we have the following formula (see Rozen, 2013): Ndis =

n! N1 · N2 · . . . · Nr , n1 !n2 ! . . . nr !

where nk = |Ak | (k = 1, . . . , r) , n =

r P

k=1

(7)

nk .

This formula is proved by induction on r. For r = 1 the right part of (7) is equal to N1 = Ndis . Let us show that (7) is truth for r = 2. Indeed, consider two ordered sets A and B, where the first set contains n1 elements and the second set n2 elements. Let (a1 , a2 , . . . , an1 ) and (b1 , b2 , . . . , bn2 ) be their linear orderings, respectively. Then we can obtain a linear ordering for discrete sum A ∪ B in the following manner. Fix a subset {i1 , i2 , . . . , in1 } in the set {1, 2, . . . , n1 + n2 } and


295

let {in1 +1 , . . . , in1 +n2 } be its complement (suppose these sequences are increasing). Then by setting ϕ (as ) = is (s = 1, . . . , n1 ) and ϕ (bt ) = in1 +t (t = 1, . . . , n2 ) we obtain a linear ordering of discrete sum A ∪ B. Hence every pair of linear orderings 2 )! of A and B generates Cnn11+n2 = (nn11+n !n2 ! of linear orderings for their discrete sum A ∪ B. Denote by N1 the number of linear orderings of A and by N2 the number of linear orderings of B, then the number of pairs of linear orderings is equal to N1 ·N2 ; 2 )! thus we obtain (nn11+n !n2 ! N1 · N2 of linear orderings for A ∪ B. For r = 2 formula (7) is shown. Remark now that discrete sum of r ordered sets can be represented as a discrete sum of two ordered sets: A1 ∪ . . . ∪ Ar−1 ∪ Ar = (A1 ∪ . . . ∪ Ar−1 ) ∪ Ar . Using our assumption for r = 2, we obtain the required proposition in general case, that is (7). As a corollary of formula (7) we now obtain a valuation for N (A) in the case A is a tree ordered set. Consider a tree T with a root a0 . Then we can define on the set A of tree vertexes the tree order by the rule: a1 ≤ a2 if and only if there exists a path from a1 to a2 . Remark that a0 is the greatest element under order ≤. For each ak ∈ A the set Tak consisting of vertexes a ≤ ak forms a tree with root ak ; it is called subtree with root ak . Particularly, T = Ta0 . Corollary 3. Let Ta0 be a tree and {Ta0 , Ta1 , . . . , Tar } all its subtrees having not less than two vertexes. Then a number N (Ta0 ) of all linear orderings of tree Ta0 is defined by formula: |Ta0 |! NTa0 = , (8) |Ta0 | · . . . · |Tar | where |Tak | denotes a number of elements of subtree Tak .

Proof of corollary 3 is given by induction on numbers of levels of tree. To prove induction step one can use that if to eliminate the greatest element of tree then we obtain a discrete sum of tree orders for which formula (7) is true, and the number of linear orderings for these tree orders can be founded by assumption (8). 4.

A contraction of the set of linear completions

We now consider a problem of contraction of the set of linear completions for ordering preferences structures which based on some additional information concerning of preferences. Suppose an ordering preferences structure in the form hA, ωi is given where ω is an order relation on the set of alternatives A. We consider here additional information of the following types. Type 1: Information under strict preferences This information with binary relation δ ⊆ A2 can be given where the assertion (a1 , a2 ) ∈ δ means that alternative a2 is strict better than alternative a1 . Such information does not contradict with an ordering ω if and only if the relation ω ∪ δ is acyclic; in this case ω1 = tr (ω ∪ δ) is an ordering of A which contains previous ordering ω and the relation δ also. Further finding linear completions of ordering ω1 we obtain some part of all linear completions for ordering ω. Completions of ordering ω1 are completions for ordering ω which conform with additional information in the form of binary relation δ. Type 2: Information under indifference relation In this case, additional information in the form of an equivalence relation ε ⊆ A2 is given.

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Victor V. Rozen

Definition 4. Let ϕ be an isotonic function from ordered set A in some chain C. Put εϕ = {(a1 , a2 ) : ϕ (a1 ) = ϕ (a2 )}. An equivalence ε ⊆ A2 is said to be ranged equivalence if ε = εϕ for some isotonic function ϕ. We assume here that ε is a ranged equivalence. Then we factorize relation ω under equivalence ε and obtain factor-ordering T r (ω/ε) on the factor-set A/ε. Let C1 , . . . , Cr be all classes of equivalence ε and ωk is a restriction of ω on subset Ck (k = 1, . . . , r); put Nk the number of linear completions for ωk and N (ω) the number of linear completions for hA, ωi. Then we have the following evaluation for the number N (ω, ε) of linear completions for the factor-structure: N (ω, ε) ≤

N (ω) . N1 · N2 · . . . · Nr

(9)

The inequality (9) shows that additional information of type 2 implies a strong contraction for the number of linear completions. Remark 3. Conditions concerning of equivalence ε ⊆ A2 under which there exists the unique linear completions for factor-structure A/ε (i.e. N (ω, ε) = 1) is given in (Rozen, 2011). 5.

Examples

Example 1. Finding of all linear completions for ordering preferences structures Consider an ordering preferences structure consisting of 6 alternatives A = {a, b, c, d, e, f } presented by a diagram (fig. 1).

Fig. 1

To construct all its linear orderings, we need in the following steps. Step 1. Using a procedure of extraction of maximal elements (see 2.1.), we obtain an auxiliary graph γ (fig. 2).

297


Fig. 2

Step 2. We now construct Table 1 whose rows are vertexes of graph γ (i.e. ideals) and for each ideal all its linear completions are given. Starting of one element ideals, we receive at last linear orderings of ideal A = {a, b, c, d, e, f } in lower block of Table 1.

298

Victor V. Rozen Table 1.

a {a} {d} {a, d} {a, b} {a, d, e} {a, b, d}

{a, b, c} {a, b, d, e}

{a, b, c, d}

b

c

d

e

1 1 2 1 1 2 1 2 1 1 1 2 1 2 1 1 2 1 1

f

a

b

c

d

e

{a, b, c, d, e}

1 2 1 2 1 1 2 1 1

4 4 3 3 2 3 3 2 2

5 5 5 5 5 4 4 4 3

2 1 2 1 3 2 1 3 4

3 3 4 4 4 5 5 5 5

A = {a, b, c, d, e, f }

1 2 1 2 1 1 2 1 1

4 4 3 3 2 3 3 2 2

5 5 5 5 5 4 4 4 3

2 1 2 1 3 2 1 3 4

3 3 4 4 4 5 5 5 5

1 2 1 2

3 3 2 2 4 4 3 3 2 3 3 2 2

2 1 2 1 3

3 3

2 1 2 1 3 2 1 3 4

3 3 4 4 4

3

4 4 4 3

f

6 6 6 6 6 6 6 6 6

Step 3. All linear orderings which are completions of the ordering preferences structure in fig. 1 on the following diagram are given (fig. 3)

Fig. 3

Example 2. A count of the number of all linear completions for ordering preferences structure Consider an ordering preferences structure in the fig. 1. To count all its linear completions we need in construction of auxiliary graph γ only. Since each subset

299


which is a final vertex of graph γ consists of one element, it has single linear completion, hence we write 1 near every final vertex of graph γ (fig. 4). Further we write a number N (B) near others vertexes B of graph γ in accordance with formula (6). The number N (B) indicates a number of all linear completions for subset B. In particularly, N (A) is a number of all completions for the set of all alternatives A = {a, b, c, d, e, f }.

Fig. 4

References Birkhoff, G. (1967). Lattice theory. Amer. Math. Soc., Coll. Publ., Vol. 25. Rozen, V. (2011). Decision making under many quality criteria.. Contribution to game theory and management, vol. 5. /Collected papers presented on the Fifth International Conference Game Theory and management-SPb.: Graduate School of Management, SPbU, 2012, p.257-267. Rozen, V. (2013). Decision making under quality criteria (in Russian). Palmarium Academic Publishing. Saarbrucken, Deutschland.

Bridging the Gap between the Nash and Kalai-Smorodinsky Bargaining Solutions Shiran Rachmilevitch Department of Economics, University of Haifa, Mount Carmel, Haifa, 31905, Israel E-mail: [email protected] Web: http://econ.haifa.ac.il/∼shiranrach/

Abstract Bargaining solutions that satisfy weak Pareto optimality, symmetry, and independence of equivalent utility representations are called standard. The Nash (1950) solution is the unique independent standard solution and the Kalai-Smorodinsky (1975) solution is the unique monotonic standard solution. Every standard solution satisfies midpoint domination on triangles, or MDT for short. I introduce a formal axiom that captures the idea of a solution being “at least as independent as the Kalai-Smorodinsky solution.” On the class of solutions that satisfy MDT and independence of non-individually-rational alternatives, this requirement implies that each player receives at least the minimum of the payoffs he would have received under the Nash and Kalai-Smorodinsky solutions. I refer to the latter property as Kalai-Smorodinsky-Nash robustness. I derive new axiomatizations of both solutions on its basis. Additional results concerning this robustness property, as well as alternative definitions of “at least as independent as the Kalai-Smorodinsky solution” are also studied. Keywords: Bargaining; Kalai-Smorodinsky solution; Nash solution.

1.

Introduction

Nash’s (1950) bargaining problem is a fundamental problem in economics. Its formal description consists of two components: a feasible set of utility allocations, each of which can be achieved via cooperation, and one special utility allocation—the disagreement point—that prevails if the players do not cooperate. A solution is a function that picks a feasible utility allocation for every problem. The axiomatic approach to bargaining narrows down the set of “acceptable” solutions by imposing meaningful and desirable restrictions (axioms), to which the solution is required to adhere. Weak and common restrictions are the following: weak Pareto optimality—the selected agreement should not be strictly dominated by another feasible agreement; symmetry—if the problem is symmetric with respect to the 45◦ -line then the players should enjoy identical payoffs; independence of equivalent utility representations— the selected agreement should be invariant under positive affine transformations of the problem. I will call a solution that satisfies these three restrictions standard. Two additional restrictions that will be considered in the sequel are the following. Midpoint domination requires the solution to provide the players payoffs that are at least as large as the average of their best and worst payoffs. Every standard solution satisfies midpoint domination on triangular problems, which are the simplest kind of bargaining problems: each such problem is a convex hull of the disagreement point, the best point for player 1, and the best point for player 2. Independence of

Bridging the Gap between the Nash and Kalai-Smorodinsky Bargaining Solutions 301 non-individually-rational alternatives requires the solution to depend only on those options that provide each player at least his disagreement utility; in other words, outcomes that clearly cannot be reached by voluntary behavior should not matter for bargaining. Two other known principles are independence and monotonicity. Informally, the former says that if some options are deleted from a given problem but the chosen agreement of this problem remains feasible (it is not deleted) then this agreement should also be chosen in the problem that corresponds to the post-deletion situation; the latter says that if a problem “expands” in such a way that the set of feasible utilities for player i remains the same, but given every utility-payoff for i the maximum that player j 6= i can now achieve is greater, then player j should not get hurt from this expansion. Within the class of standard solutions, these principles are incompatible: Nash (1950) showed that there exists a unique standard independent solution, while Kalai and Smorodinsky (1975) showed that a different solution is the unique standard monotonic one. The reconciliation of independence and monotonicity in bargaining, therefore, is a serious challenge. One possible response to this challenge is to give up the restriction to standard solutions.1 Here, however, I consider only standard solutions. When attention is restricted to standard solutions, the aforementioned challenge can, informally, be expressed in the form of the following question: How can we “bridge the gap” between the Nash and Kalai-Smorodinsky solutions? Motivated by this question, I introduce an axiom that formalizes the idea of a solution being “at least as independent as the Kalai-Smorodinsky solution.” 2 I denote this axiom by IIA KS.3 I also consider the following requirement, which refers directly to both solutions: in every problem each player should receive at least the minimum of the payoffs he would have received under the Nash and KalaiSmorodinsky solutions. I call this property Kalai-Smorodinsky-Nash robustness, or KSNR. On the class of solutions that satisfy midpoint domination on triangles and independence of non-individually-rational alternatives, IIA KS implies KSNR. KSNR, in turn, captures much of the essence of the Nash and Kalai-Smorodinsky solutions: the Nash solution is characterized by KSNR and independence and the Kalai-Smorodinsky solution—by KSNR and monotonicity. Both characterizations hold on the class of all solutions, not only the standard ones, not only the ones that satisfy midpoint domination on triangles.4 The rest of the paper is organized as follows. Section 2 describes the model. The axiom IIA KS is defined in Section 3. KSNR is introduced and discussed in Section 4. Section 5 elaborates on midpoint domination and its non-trivial connections to the Nash and Kalai-Smorodinsky solutions. Section 6 considers an alternative definition of “at least as independent as the Kalai-Smorodinsky solution.” Section 7 concludes with a brief discussion. 1

2 3 4

For example, the egalitarian solution (due to Kalai (1977)) satisfies all the above mentioned requirements except independence of equivalent utility representations. A formal definition of this solution will be given in Section 4 below. The precise meaning of this term will be given in Section 3. The rationale behind this notation will be clarified in Sections 2 and 3. The class of solutions that satisfy midpoint domination on triangles contains the class of standard solutions.

302 2.

Shiran Rachmilevitch Preliminaries

A bargaining problem is defined as a pair (S, d), where S ⊂ R2 is the feasible set, representing all possible (v-N.M) utility agreements between the two players, and d ∈ S, the disagreement point, is a point that specifies their utilities in case they do not reach a unanimous agreement on some point of S. The following assumptions are made on (S, d): – S is compact and convex; – d < x for some x ∈ S.5 Denote by B the collection of all such pairs (S, d). A solution is any function µ : B → R2 that satisfies µ(S, d) ∈ S for all (S, d) ∈ B. Given a feasible set S, the weak Pareto frontier of S is W P (S) ≡ {x ∈ S : y > x ⇒ y ∈ / S} and the strict Pareto frontier of S is P (S) ≡ {x ∈ S : y x ⇒ y ∈ / S}. The best that player i can hope for in the problem (S, d), given that player j obtains at least dj utility units, is ai (S, d) ≡ max{xi : x ∈ Sd }, where Sd ≡ {x ∈ S : x ≥ d}. The point a(S, d) = (a1 (S, d), a2 (S, d)) is the ideal point of the problem (S, d). The Kalai-Smorodinsky solution, KS, due to Kalai and Smorodinsky (1975), is defined by KS(S, d) ≡ P (S) ∩ [d; a(S, d)].6 The Nash solution, N , due to Nash (1950), is defined to be the unique maximizer of (x1 − d1 ) × (x2 − d2 ) over Sd . Nash (1950) showed that N is the unique solution that satisfies the following four axioms, in the statements of which (S, d) and (T, e) are arbitrary problems. Weak Pareto Optimality (WPO): µ(S, d) ∈ W P (S). Let FA denote the set of positive affine transformations from R to itself.7 Independence of Equivalent Utility Representations (IEUR): f = (f1 , f2 ) ∈ FA × FA ⇒ f ◦ µ(S, d) = µ(f ◦ S, f ◦ d).8 Let π(a, b) ≡ (b, a). Symmetry (SY): [π ◦ S = S]&[π ◦ d = d] ⇒ µ1 (S, d) = µ2 (S, d). Independence of Irrelevant Alternatives (IIA): [S ⊂ T ]&[d = e]&[µ(T, e) ∈ S] ⇒ µ(S, d) = µ(T, e). Whereas the first three axioms are widely accepted, criticism has been raised regarding IIA. The idea behind a typical such criticism is that the bargaining solution could, or even should, depend on the shape of the feasible set. In particular, Kalai and Smorodinsky (1975) noted that when the feasible set expands in such a way that for every feasible payoff for player 1 the maximal feasible payoff for player 2 increases, it may be the case that player 2 loses from this expansion under the Nash 5

6 7 8

Vector inequalities: xRy if and only if xi Ryi for both i ∈ {1, 2}, R ∈ {>, ≥}; x y if and only if x ≥ y & x 6= y. Given two vectors x and y, the segment connecting them is denoted [x; y]. i.e., the set of functions f of the form f (x) = αx + β, where α > 0. If fi : R → R for each i = 1, 2, x ∈ R2 , and A ⊂ R2 , then: (f1 , f2 ) ◦ x ≡ (f1 (x1 ), f2 (x2 )) and (f1 , f2 ) ◦ A ≡ {(f1 , f2 ) ◦ a : a ∈ A}.

Bridging the Gap between the Nash and Kalai-Smorodinsky Bargaining Solutions 303 solution. Given x ∈ Sd , let giS (xj ) be the maximal possible payoff for i in S given that j’s payoff is xj , where {i, j} = {1, 2}. What Kalai and Smorodinsky noted, is that N violates the following axiom, in the statement of which (S, d) and (T, d) are arbitrary problems with a common disagreement point. Individual Monotonicity (IM): [aj (S, d) = aj (T, d)]&[giS (xj ) ≤ giT (xj ) ∀x ∈ Sd ∩ Td ] ⇒ µi (S, d) ≤ µi (T, d). Furthermore, they showed that when IIA is deleted from the list of Nash’s axioms and replaced by IM, a characterization of KS obtains.9 Following Trockel (2009), a solution that satisfies the common axioms—namely WPO, SY, and IEUR—will be referred to in the sequel as standard. Most solutions from the bargaining literature (standard or not) also satisfy the following axiom, in the statement of which (S, d) is an arbitrary problem. Independence of Non-Individually-Rational Alternatives (INIR): µ(S, d) = µ(Sd , d).10 3.

Relative independence

Consider the following partial order on the plane, . Given x, y ∈ R2 , write x y if x1 ≤ y1 and x2 ≥ y2 . That is, x y means that x is (weakly) to the north-west of y. Let µ be a solution and let (S, d) ∈ B be a problem such that µ(S, d) = KS(S, d). Say that µ is at least as independent as KS given (S, d) if the following is true for every (Q, e) ∈ B such that e = d, Q ⊂ S, and KS(S, d) ∈ Q: 1. KS(Q, e) KS(S, d) ⇒ KS(Q, e) µ(Q, e) KS(S, d), and 2. KS(S, d) KS(Q, e) ⇒ KS(S, d) µ(Q, e) KS(Q, e). That is, µ is at least as independent as KS given (S, d) if in every relevant “subproblem” the solution point according to µ is between the solution point of KS and the solution point “of IIA.” A solution, µ, is at least as independent as KS if it is at least as independent as KS given (S, d), for every (S, d) such that µ(S, d) = KS(S, d). Denote this property (or axiom) by IIA KS. There is no shortage of standard solutions satisfying this property. Obviously, KS is such a solution and every standard IIA-satisfying solution is such a solution. However, there are many others. For describing such a solution, the following notation will be useful (it will also turn out handy in the next Section). For each (S, d) ∈ B and each i, let: mi (S, d) ≡ min{Ni (S, d), KSi (S, d)}. 9

10

In many places throughout the paper I refer to “individual monotonicity” and “independence of irrelevant alternatives” simply as “monotonicity” and “independence.” (see, e.g., the Introduction). The longer names for these axioms, and their respective abbreviations IM and IIA, are presented here in order to distinguish them from other monotonicity and independence axioms from the literature. To the best of my knowledge, the earliest paper that utilizes this axiom is Peters (1986).

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Shiran Rachmilevitch

Now consider the following solution, µ∗ :

∗

µ (S, d) ≡

m(S, d) if N (S, d) = KS(S, d) KS({x ∈ S : x ≥ m(S, d)}, m(S, d)) otherwise

It is easy to see that µ∗ is a standard solution which is at least as independent as KS, it is different from KS, and it violates IIA. The property IIA KS is, of course, not the only possible formalization of the idea “at least as independent as KS.” Moreover, it is not immune to criticism. I will discuss one of its drawback and propose an alternative formal definition for “at least as independent as KS” later in the paper, in Section 6. Finally, it is worth noting that one may very well question the validity of taking the monotonic solution and employing it as the “measuring stick” for the extant to which independence can be violated. Why not prefer the analogous criterion, where N is taken to be the measuring stick for the degree to which monotonicity can be compromised? In principle, this alternative approach expresses a sensible consideration, but in practice it is problematic. To see this, consider S ≡ conv hull{0, (0, 1), (1, 1), (2, 0)} and T ≡ {x ∈ R2+ : x ≤ (2, 1)}.11 Note that when we move from S to T the feasible set “stretches” in the direction of coordinate 1 and hence, by IM, player 1 should not get hurt from this expansion. Accordingly, an “at least monotonic as N ” relation would naturally impose that player 1’s benefit from the change (S, 0) 7→ (T, 0) would be at least as large as the one he would have obtained under N . However, even KS fails this test. 4.

Kalai-Smorodinsky-Nash robustness

Consider the following axiom, in the statement of which (S, d) is an arbitrary problem. Midpoint Domination (MD): µ(S, d) ≥ 12 d + 12 a(S, d). This axiom is due to Sobel (1981), who also proved that N satisfies it. The idea behind it is that a “good” solution should always assign payoff that Pareto-dominate “randomized dictatorship” payoffs. Anbarci (1998) considered a weakening of this axiom, where the the requirement µ(S, d) ≥ 21 d + 12 a(S, d) is applied only to problems (S, d) for which S = Sd and is a triangle. I will refer to this weaker axiom as midpoint domination on triangles, or MDT.12 It is easy to see that every standard solution satisfies MDT. Next, consider the following axiom, in the statement of which (S, d) is an arbitrary problem. Kalai-Smorodinsky-Nash Robustness (KSNR): µ(S, d) ≥ m(S, d). KSNR implies MD since both N and KS satisfy MD.13 Therefore, the following implications hold: 11 12 13

0 ≡ (0, 0). Anbarci calls it midpoint outcome on a linear frontier. It is straightforward that KS satisfies MD; as mentioned above, the fact that N satisfies it was proved by Sobel (1981).

Bridging the Gap between the Nash and Kalai-Smorodinsky Bargaining Solutions 305

KSNR⇒ MD⇒ MDT.

(1)

Now recall that the big-picture goal in this paper is to offer a reconciliation of monotonicity and independence within the class of standard solutions; this class, in turn, is a subclass of the MDT-satisfying solutions, so MDT essentially expresses no loss of generality for our purpose. With the additional (weak) restriction of INIR, the following theorem says that the aforementioned reconciliation, as expressed by IIA KS, implies KSNR. Theorem 1. Let µ be a solution that satisfies independence of non-individuallyrational alternatives and midpoint domination on triangles. Suppose further that it is at least as independent as the Kalai-Smorodinsky solution. Then µ satisfies Kalai-Smorodinsky-Nash robustness. Proof. Let µ satisfy INIR, MDT and IIA KS. Let (S, d) be an arbitrary problem. By INIR we can assume S = Sd . Let x ≡ µ(S, d), k ≡ KS(S, d), and n ≡ N (S, d). We need to prove that xi ≥ min{ni , ki }. Let T = conv hull{d, (2n1 − d1 , d2 ), (d1 , 2n2 − d2 )}. By MDT, µ(T, d) = KS(T, d) = N (T, d) = n. If k = n, then by IIA KS it follows that x = k = n. Now consider k 6= n; wlog, k1 < n1 . In this case IIA KS dictates that k x n, which implies that xi ≥ min{ni , ki }. Let A = {INIR, MDT, IIA KS, KSNR} and let A denote a generic axiom in A. Theorem 1 says that A =KSNR is implied by A\{A}. From (1) we see that the same is trivially true for A =MDT. On the other hand, for A =IIA KS or A =INIR, an analogous conclusion cannot be drawn. Consider first A =IIA KS. Let Q ≡ conv hull{0, (0, 1), (1, 1), (2, 0)}, Q′ ≡ {x ∈ Q : x1 ≤ KS1 (Q, 0)}, and consider the following solution, µ∗∗ . For (S, d) such that S = Sd = S0 , µ∗∗ (S, d) = N (Q′ , 0)(= (1, 1)) if S = Q′ and µ∗∗ (S, d) = KS(S, d) otherwise; for other problems the solution point is obtained by a translation of d to the origin and deletion of non-individually-rational alternatives. It is obvious that µ∗∗ satisfies MDT, INIR, and KSNR. However, in the move from (Q, 0) to (Q′ , 0), µ∗∗ violates IIA KS. Regarding A =INIR, consider the following solution, µ∗∗∗ : µ∗∗∗ (S, d) ≡

N (S, d) if {x ∈ S : x < d} 6= ∅ KS(S, d) otherwise

It is immediate to see that µ∗∗∗ satisfies KSNR and MDT, and it is also not hard to check that it also satisfies IIA KS. Finally, A =KSNR is not implied by any strict subset of A\{A}. The egalitarian solution (due to Kalai (1977)), E, which is defined by E(S, d) ≡ W P (Sd ) ∩ {d + (x, x) : x ≥ 0}, satisfies IIA (and thereforeIIA KS) as well as INIR, but does not satisfy MDT or KSNR. The Perles-Maschler solution (Perles and Maschler (1981)), P M , is an example of a solution that satisfies MDT and INIR, but not IIA KS or KSNR. For (S, d) with d = 0 and P (S) = W P (S), it is defined to be Ru R (a ,0) √ √ the point u ∈ P (S) such that (0,a2 ) −dxdy = u 1 −dxdy, where a = a(S, d); for other problems it is extended by IEUR and continuity is an obvious fashion. This is a standard solution, and therefore it satisfies MDT; in fact, it actually

306


satisfies MD.14 It is easy to see that it violates IIA KS; to see that it violates KSNR, look at S ∗ = conv{0, (0, 1), ( 34 , 0), ( 34 , 14 )}: it is easily verified that P M1 (S ∗ , 0) = 38 < 37 = KS1 (S ∗ , 0) < 36 = N1 (S ∗ , 0). Finally, the following solution, µ∗∗∗∗ , satisfies IIA KS and MDT, but not INIR or KSNR.15

µ

∗∗∗∗

(S, d) ≡

KS(S, d) if d ∈ / intS 1 K(S, d) otherwise 2

KSNR captures much of the essence of the Nash and Kalai-Smorodinsky solutions. This is expressed in the following theorems. Theorem 2. The Nash solution is the unique solution that satisfies Kalai-SmorodinskyNash robustness and independence of irrelevant alternatives. Theorem 3. The Kalai-Smorodinsky solution is the unique solution that satisfies Kalai-Smorodinsky-Nash robustness and individual monotonicity. The proofs of Theorems 2 and 3 follow from the combination of two results from the existing literature. Before we turn to these results, one more axiom needs to be introduced. This axiom, which is due to Anbarci (1998), has a similar flavor to that of MD, but the two are not logically comparable. Balanced Focal Point (BFP): If S = d + conv hull{0, (a, b), (λa, 0), (0, λb)} for some λ ∈ [1, 2], then µ(S, d) = d + (a, b).16 The justification for this axiom is that the equal areas to the north-west and southeast of the focal point d+(a, b) can be viewed as representing equivalent concessions. Similarly to MD, BFP is implied by KSNR and implies MDT: KSNR⇒ BFP⇒ MDT.

(2)

The first implication is due to the fact that every standard solution satisfies BFP, and both N and KS are standard. The second implication follows from setting λ = 2 in BFP’s definition. Anbarci (1998) showed that KS is characterized by IM and BFP. His work was inspired by that of Moulin (1983), who in what is probably the simplest and most elegant axiomatization of N , proved that it is the unique solution that satisfies IIA and MD. Combining the results of Moulin (1983), Anbarci (1998), and the implication KSNR⇒ [MD, BFP], one obtains a proof for the theorems. It is worth noting that whereas the implication KSNR⇒ [MD, BFP] is true, not only the converse is not true, but, moreover, even the combination of MD and “standardness”(and, therefore, the combination of MD and BFP) does not imply KSNR. The Perles-Maschler solution, P M , is an example. 14 15 16

See, e.g., Salonen (1985). In the definition of this solution, int stands for “interior.” Anbarci assumes the normalization d = 0; the version above is the natural adaptation of his axiom to a model with an arbitrary d.

Bridging the Gap between the Nash and Kalai-Smorodinsky Bargaining Solutions 307 5.

Midpoint domination

Both N and KS are related to MD. Regarding N , we already encountered the results of Moulin (1983) and Sobel (1981). Additionally, a related result has been obtained by de Clippel (2007), who characterized N by MD and one more axiom— disagreement convexity.17 As for KS, Anbarci (1998) characterized it by BFP and IM, and we already noted the relation between BFP and MD—each can be viewed as an alternative strengthening of MDT which is also a weakening of KSNR. More recently, I characterized KS by MD and three additional axioms (see Rachmilevitch (2013)). Finally, Chun (1990) characterized the Kalai-Rosenthal (1978) solution— which is closely related to KS—on the basis of several axioms, one of which is a variant of MD. Below is an example for another non-trivial link between MD and N/KS. In it, KSNR takes center stage. The following family of standard solutions is due to Sobel (2001). For each number a ≤ 1 corresponds a solution, which is defined on normalized problems— (S, d) for which d = 0 and a(S, d) = (1, 1). Given a ≤ 1, this solution is: 1 1 1 W (S, a) ≡ arg maxx∈S [ xa1 + xa2 ] a .18 2 2 On any other (not normalized) problem, the solution point is obtained by appropriate utility rescaling. In light of the resemblance to the well-known concept from Consumer Theory, I will call these solutions normalized CES solutions. Both N and KS are normalized CES solutions: N corresponds to lima→0 W (., a) and KS corresponds to lima→−∞ W (., a).19 Thus, Sobel’s family offers a smooth parametrization of a class of solutions, of which N and KS are special members. It turns out that on this class KSNR and MD are equivalent. Theorem 4. A normalized CES solution satisfies Kalai-Smorodinsky-Nash robustness if and only if it satisfies midpoint domination. 6.

An alternative “at least as independent as KS” relation

Recall the basic definition from Section 3: µ is at least as independent as KS given (S, d) if for every relevant “sub-problem” (Q, e) the solution point according to µ is between the solution point of KS and the solution point “of IIA.” Underlying this definition is a notion of “betweenness,” a notion which is not immune to criticism. Specifically, it suffers the following drawback. Consider the case where KS(Q, e) is to the left (and north) of KS(S, d), µ(Q, e) is to the right (and south) of KS(S, d), but the distance between µ(Q, e) and KS(S, d) is only a tiny ǫ > 0; namely, the solution µ hardly changed its recommendation in the move from (S, d) to (Q, e), 17

18

19

See his paper for the definition of the axiom. de Clippel’s result is related to an earlier characterization which is due to Chun (1990). The maximizer is unique for a < 1; a = 1 corresponds to the relative utilitarian solution, which (in general) is multi-valued. Maximizing W (S, a) describes a well-defined method for solving arbitrary bargaining problems, not necessarily normalized. In this more general case, lima→−∞ W (., a) corresponds to the egalitarian solution (Kalai (1977)). See Bertsimas et al (2012) for a recent detailed paper on the matter.

308


but this change is opposite in its direction to that of KS. According to the aforementioned definition, such a solution µ is not at least independent as KS given (S, d). Therefore, it is not at least as independent as KS even if it coincides with KS (or with an IIA-satisfying solution) on all other problems. Thus, one may argue that an appropriate definition of “more independent than” should not rely on direction (and hence should not rely on betweenness) and should take distance into consideration. This definition, therefore, should be based on an appropriate metric of “IIA violations.” Here is one such definition. Say that µ is metrically at least as independent as KS given (S, d), if µ(S, d) = KS(S, d) ≡ x and for every relevant sub-problem (Q, e) it is true that: maxi∈{1,2} |µi (Q, e) − xi | ≤ maxi∈{1,2} |KSi (Q, e) − xi |.

Metrically at least as independent as KS means that the corresponding property holds for every (S, d) such that µ(S, d) = KS(S, d). In Theorem 1, “at least as independent as KS” can be replaced by “metrically at least as independent as KS” provided that attention is restricted solutions that (i) are efficient, and (ii) satisfy the following axiom, in the statement of which (S, d) is an arbitrary problem. Weak Contraction Monotonicity (WCM): For every i and every number r, µi (V, d) ≤ µi (S, d), where V ≡ {x ∈ S : xi ≤ r}.20 Note that WCM is implied (separately) by IIA and by IM. Theorem 5. Let µ be solution that satisfies weak Pareto optimality, midpoint domination on triangles, independence of non-individually-rational alternatives, and weak contraction monotonicity. Suppose that µ is metrically at least as independent as the Kalai-Smorodinsky solution. Then µ satisfies Kalai-Smorodinsky-Nash robustness. Finally, we also have the following result. Theorem 6. A normalized CES solution that satisfies weak contraction monotonicity also satisfies Kalai-Smorodinsky-Nash robustness.21 7.

Discussion

Motivated by the goal to reconcile independence and monotonicity in bargaining within the class of standard solutions, I have introduced the requirement that the bargaining solution be “at least as independent as KS.” That is, I took the monotonic standard solution as the measuring stick for how far one can depart from independence. Weaker versions of IIA have previously been considered in the literature. The best known axiom in this regard is Roth’s (1977) independence of irrelevant alternatives other than the disagreement point and the ideal point, which applies IIA only to pairs of problems that, in addition to a common disagreement point, share the 20

21

Implicit here is the assumption that (V, d) ∈ B. Obviously, V = ∅ for a sufficiently small r. It is an open question whether the converse is true; namely, whether WCM and KSNR are equivalent on the class of normalized CES solutions.

Bridging the Gap between the Nash and Kalai-Smorodinsky Bargaining Solutions 309 same ideal point. Thomson (1981) generalized Roth’s axiom to a family of axioms, parametrized by a reference function. Such an axiom applies IIA only to pairs of problems that share the same reference-function-value. Though these type of weakening of IIA are more common in the literature, they seems not be a good fit for the task of reconciling independence and monotonicity: Roth’s axiom seems too weak and Thomson’s seems too abstract. The property “at least as independent as KS,” or IIA KS, though having a somewhat special structure, seems more well-suited for the goal of the current paper. This property, in turn, when considered in combination with MDT and INIR (a combination which is almost without loss of generality within the class of standard solutions), implies KSNR. KSNR, in turn, is of interest in its own right. On its basis I have derived axiomatizations of the two solutions: N is characterized by KSNR and IIA and KS is characterized by KSNR and IM. These new characterizations are not a consequence of the known theorems of Nash (1950) and Kalai-Smorodinsky (1975), since they hold on the entire class of bargaining solutions, not only on the class of the standard ones. Note also that the fact that a solution satisfies KSNR does not imply that it is standard. For example, the following is a non-standard solution that satisfies KSNR: E KSN R (S, d) ≡ m(S, d) + (e, e), where e is the maximal number such that the aforementioned expression is in S. Appendix Proof of Theorem 4 : It is known that MD is equivalent to a ≤ 0 (see Sobel (1981)). I will prove that KSNR is also equivalent to a ≤ 0. It is enough to prove this equivalence for a < 0 (because a = 0 corresponds to the Nash solution). For simplicity (and without loss), I will consider only (normalized) strictly comprehensive problems—those for which S = {(x, f (x)) : x ∈ [0, 1]}, where f is a differentiable strictly concave decreasing function; the case of an arbitrary problem follows from standard limit arguments. The parameter a corresponds to the solution 1 1 1 W (S, a) ≡ arg maxx∈S0 [ xa1 + xa2 ] a . 2 2 In the case of a normalized strictly comprehensive problem, the object of interest is 1 1 1 arg max0≤x≤1 [ xa + f (x)a ] a ≡ W (x, a). 2 2

d Therefore dx W (x, a) = e[ 2a xa−1 + a2 f (x)a−1 f ′ (x)], where e is a shorthand for a strictly positive expression. At the optimum, the derivative of this expression is zero (i.e., an FOC holds).

[

f (x(a)) 1−a ] = −f ′ (x(a)), x(a)

where x(a) is player 1’s solution payoff given the parameter a. The derivative of the RHS with respect to a is −f ′′ (x(a))x′ (a), hence the sign of x′ (a) is the same as the sign of the derivative of the LHS. To compute the latter, recall the formula Ψ ′ (a) = Ψ (a) × {h′ (a)log[g(a)] +

h(a)g ′ (a) }, g(a)

310


where Ψ (a) ≡ [g(a)]h(a) . Taking g(a) ≡ f (x(a)) x(a) and h(a) ≡ 1 − a, we see that the signs of the derivative of the LHS is the same as the sign of

−log[

(1 − a) f (x(a)) f (x(a)) ]+ x′ (a)(f ′ (x(a))x(a)−f (x(a))) ≡ −log[ ]+Zx′ (a), x(a) f (x(a))x(a) x(a)

where Z is a shorthand for a negative expression. Therefore, the sign of x′ (a) is the ′ same as that of −log[ f (x(a)) x(a) ] + Zx (a). Let k ≡ KS1 (S, 0) and n ≡ N1 (S, 0). f (x(a)) Case 1: x(a) < k. In this case, f (x(a)) x(a) > 1, hence −log[ x(a) ] < 0. This means that x′ (a) < 0. To see this, assume by contradiction that x′ (a) ≥ 0. This means ′ ′ that sign[−log[ f (x(a)) x(a) ] + Zx (a)] = −1 = sign[x (a)], a contradiction. Therefore n < x(a) < k, so KSNR holds. Case 2: x(a) > k. In this case, f (x(a)) < 1, hence −log[ f (x(a)) x(a) x(a) ] > 0. This ′ means that x (a) > 0. To see this, assume by contradiction that x′ (a) ≤ 0. This ′ ′ means that sign[−log[ f (x(a)) x(a) ]+ Zx (a)] = 1 = sign[x (a)], a contradiction. Therefore k < x(a) < n, so KSNR holds. ⊓ ⊔ Proof of Theorem 5 : Let µ be a solution that satisfies the axioms listed in the theorem and let (S, d) ∈ B. Assume by contradiction that KSNR is violated for (S, d). Wlog, we can assume that Sd is not a rectangle and that d = 0. Let T ≡ conv hull{(2n1 , 0)(0, 2n2 ), 0}, where n ≡ N (S, 0). By MDT, µ(T, 0) = N (T, 0) = KS(T, 0) = n. Let k ≡ KS(S, 0). Wlog, suppose that x1 < min{n1 , k1 }. Let simply denote the “metrically at least as independent as KS” relation by . Note that because of , n1 = k1 is impossible. Therefore n1 6= k1 . Case 1: k1 < n1 . Here we have |x1 − n1 | > |k1 − n1 | and hence, because of , |k2 − n2 | > |k1 − n1 |. Case 1.1: |x1 − n1 | ≥ |x2 − n2 |. In this case |k2 − n2 | ≤ |x2 − n2 | ≤ |x1 − n1 | ≤ |k2 − n2 |. The first inequality follows from the fact that x (which, by WPO, is on the boundary) is to the north-west of k, the second inequality is because we are in Case 1.1, and the third is by . Case 1.2: |x1 − n1 | < |x2 − n2 |. By , |x2 − n2 | ≤ |k2 − n2 |. indent Whatever the case—1.1 or 1.2—it follows that |x2 − n2 | = |k2 − n2 |, which is impossible, since k is in the relative interior of P (S). Case 2: k1 > n1 . Let r ∈ (n1 , k1 ) be such that KS1 (V, 0) < n1 , where V ≡ {x ∈ S : x1 ≤ r}. Invoking WCM and applying the arguments from Case 1 to V completes the proof. ⊓ ⊔ Proof of Theorem 6 : I will prove that if a normalized CES solution violates KSNR— that is, if its defining parameter is a > 0—then it also violates weak contraction monotonicity. Fix then 0 < a < 122 and let µ denote the corresponding solution. 22

The case a = 1 can be easily treated separately (i.e., it is easy to show that the relative utilitarian solution violates WCM; for brevity, I omit an example); focusing on a < 1 allows for “smooth MRS considerations” as will momentarily be clear.

Bridging the Gap between the Nash and Kalai-Smorodinsky Bargaining Solutions 311 Claim: There is a unique k = k(a) ∈ (0, 1) such that: (3)

1 − k = k 1−a .

Proof of the Claim: The RHS is decreasing in k, the LHS is increasing, and opposite strict inequalities obtain at k = 0 and k = 1. QED From here on, fix k = k(a). Consider S = conv hull{0, (1, 0), (0, 1), (1, k)}. Due to (3), a and k are such that the slope of the line connecting (0, 1) and (1, k) (i.e., the slope of the strict Pareto frontier of S) is |k − 1|, which equals the “MRS” of the objective that µ maximizes, when this MRS is evaluated at the corner (1, k). Therefore, µ(S, 0) = (1, k). Now let t ∈ (0, 1). Note that (t, (k−1)t+1) ∈ P (S). Let us “chop” S at the height (k − 1)t + 1; namely, consider V = V (t) ≡ {(a, b) ∈ S : b ≤ (k − 1)t + 1}. To show a violation of WCM, I will find a t ∈ (0, 1) for which µ2 (V (t), 0) > k = µ2 (S, 0). To this end, let us consider the normalized feasible set derived from V = V (t), b ) : (a, b) ∈ V (t)}. By call it Q = Q(t). Given t ∈ (0, 1), Q = Q(t) = {(a, (k−1)t+1 SINV, µ2 (Q(t), 0) =

µ2 (V (t),0) (k−1)t+1 ,

or µ2 (V (t), 0) = [(k − 1)t + 1]µ2 (Q(t), 0). Therefore,

k I will show that [(k − 1)t + 1]µ2 (Q(t), 0) > k, or µ2 (Q(t), 0) > (k−1)t+1 . Note that k the south-east corner of P (Q) is (1, (k−1)t+1 ). Therefore, it is enough to show that the MRS at this corner is smaller than the slope of P (Q). Namely, that

[

(k − 1)t + 1 a−1 1−k ] < , k (k − 1)t + 1

for a suitably chosen t. Note that for t ∼ 1 the LHS is approximately one, so we are 1 1 done if 1 < 1−k k , or k < 2 . Hence, it is enough to prove that k(a) < 2 for a > 0. To this end, let us re-write (3) as G ≡ k 1−a + k − 1 = 0.

Since k(0) = 12 , it is enough to prove that k ′ < 0. By the Implicit Function Theorem, ∂G ∂G −a the sign of k ′ is opposite to the sign of [ ∂G +1 > 0 ∂k ]/[ ∂a ]. Note that ∂k = (1−a)k and that the sign of

∂G ∂a

is the same is that of

∂logk1−a ∂a

= −logk > 0. ⊓ ⊔

Acknowledgments: I am grateful to Nejat Anbarci, Youngsub Chun, Uzi Segal, Joel Sobel, William Thomson, and an anonymous referee for helpful comments. References Anbarci, N. (1998) Simple characterizations of the Nash and Kalai/Smorodinsky solutions. Theory and Decision, 45, 255–261. Bertsimas, D., Farias, F.V., and Trichakis, N. (2012). On the efficiency-fairness trade-off. Management Science, in press. Chun, Y. (1990). Minimal cooperation in bargaining. Economics Letters, 34, 311–316. de Clippel, G., (2007). An axiomatization of the Nash bargaining solution. Social Choice and Welfare, 29, 201–210. Kalai, E. (1975). Proportional solutions to bargaining situations: Interpersonal utility comparisons. Econometrica, 45, 1623–1630.

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Kalai, E., and Rosenthal, R. W. (1978). Arbitration of two-party disputes under ignorance. International Journal of Game Theory, 7, 65–72. Kalai, E. and Smorodinsky, M. (1975). Other solutions to Nash’s bargaining problem. Econometrica, 43, 513–518. Moulin, H. (1983). Le choix social utilitariste. Ecole Polytechnique Discussion Paper. Nash, J. F. (1950). The bargaining problem. Econometrica, 18, 155–162. Perles, M. A. and Maschler, M. (1981). The super-additive solution for the Nash bargaining game. International Journal of Game Theory, 10, 163–193. Peters, H. (1986). Characterization of bargaining solutions by properties of their status quo sets, Research Memorandum, Department of Economics, University of Limburg. Rachmilevitch, S. (2013). Randomized dictatorship and the Kalai-Smorodinsky bargaining solution. Theory and Decision, forthcoming. Roth, A. E. (1977). Independence of irrelevant alternatives, and solutions to Nash’s bargaining problem. Journal of Economic Theory, 16, 247–251. Salonen, H. (1985). A solution for two-person bargaining problems. Social Choice and Welfare, 2, (1985), 139-146. Sobel, J. (1981). Distortion of utilities and the bargaining problem. Econometrica, 49, 597–620. Sobel, J. (2001). Manipulation of preferences and relative utilitarianism. Games and Economic Behavior, 37, 196–215. Thomson, W. (1981). A class of solutions to bargaining problems. Journal of Economic Theory, 25, 431–441. Trockel, W. (2009). An axiomatization of the sequential Raiffa solution, Working paper.

Unravelling Conditions for Successful Change Management Through Evolutionary Games of Deterrence Michel Rudnianski1 and Cerasela Tanasescu2 LIRSA, CNAM, Paris E-mail: [email protected] 2 ESSEC Business School Av. B. Hirsch, 95000 Cergy Pontoise, France E-mail: [email protected] 1

Abstract The paper proposes analyze the conditions for successful change management requiring information transmission and transformation of the information received into change implementation. To that end, starting from an elementary standard matrix game considering only information transmission, the paper will extend the case by considering that stakeholders have to simultaneously take decisions concerning the two above dimensions. A dynamic approach supported by the Replicator Dynamics model will then be proposed, aiming at analyzing asymptotic behaviors. The difficulties often met when trying to solve differential systems will be pointed out. Therefore a new method will be developed, leaning on a bridging in the evolutionary context between standard games and a particular type of qualitative games, called Games of Deterrence, and which object is to analyze strategies playability. Through the equivalence between the two types of games, the methodology will enable to remove some question marks in the analysis of asymptotic behaviors, thus contributing to a better understanding of conditions fostering change pervasion, and in particular of the role played by incentives. Keywords: change, deterrence, evolution, incentives, playability, Replicator Dynamics, stability.

1.

Introduction

The ever increasing pace of ICT development and globalization generates a dramatic shortening of the products’ life cycle, which in turn decreases the possibility of sustainable competitive advantages for the firms. Richard D’Aveni (D’Aveni, 1994) has analyzed this phenomenon distinguishing different arenas of hyper-competition. One of the major elements highlighted is the core capacity of managing breakthroughs, in particular through the mastering of timing and knowhow associated with products and services. In this respect, there is no doubt that change management is a core competency for a firm which aim is sustainable development. Now implementation of change management may present a variety of difficulties, among which reluctance to share information and to change (Fichman and Kemener, 1997; Rogers, 1983). The works developed in Experimental Psychology, and especially Kahneman and Tversky’s Prospect Theory (Kahneman and Tversky, 1979), have highlighted decision biases like anchoring, procrastination, sensitivity to loss, stubbornness, mirroring, or status quo. All of them may lead the individual under consideration to take inappropriate decisions, which most of the time have as hidden objective to comfort his/her position and hence not accept to change. Whence the necessity for the

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Michel Rudnianski, Cerasela Tanasescu

firm’s management to develop an accurate cost-benefit analysis of change versus status quo for each decision maker. As a result of this analysis the firm’s management may decide to allocate incentives to the personnel concerned. A game theoretic model of the issues at stake has already been developed, considering a firm structured in departments, each one having a relative autonomy in terms of information sharing and change adoption (Rudnianski and Tanasescu, 2012). This model has considered several assumptions about the consequences for a department of receiving information relevant from the company’s global perspective. The starting point was to consider that a department i can decide to send or not to send to a neighboring department j an information pertaining to a possible change in the conduct of affairs. Similarly, department j, when receiving the information, may decide or not to act accordingly and especially to implement change that might be triggered by the information received. At the most elementary level, the problematic can be analyzed through a series of standard 2x2 games in which the players’ strategic sets could refer, either to information sending or to change adoption. Various cases have been considered, depending on the respective values for each player of costs, incentives and benefits received from the company’s general management. At a second level, the model used the Replicator Dynamics to define conditions under which cooperation between connected departments can prevail. At a third level, the initial model was extended to matrix games in which each party should simultaneously consider whether to send information or not, and whether to adopt changes stemming from the information received or not. A general evolutionary analysis of these games was not performed due to the difficulties to find an analytical solution of the dynamic system. Now this obstacle can be removed, thanks to the results recently found by Ellison and Rudnianski, about the existence of equivalence relations between standard quantitative games and a particular type of qualitative games called Games of Deterrence (Ellison and Rudnianski, 2009; Ellison and Rudnianski, 2012). More precisely these equivalence relations enable to translate standard evolutionary games into evolutionary Games of Deterrence which display identical asymptotic properties. In turn, it has been shown (Ellison and Rudnianski, 2012) that the asymptotic properties of these Evolutionary Games of Deterrence may be derived from the playability properties of the players’ strategies in the corresponding matrix Games of Deterrence. There is then no need to solve the original dynamic system. On these bases, the present paper will in a first part recall the results available in the analysis of conditions required for successful change management through the standard game theoretic approach. In a second part, after having recalled the core properties of matrix Games of Deterrence, the paper will develop the equivalences between evolutionary standard games and evolutionary Games of Deterrence. A third part will then use these equivalences to analyze the conditions of success in non-elementary issues of change pervasion. In particular, success of change pervasion will be associated with the playability properties of the Games of Deterrence under consideration. 2.

Conditions for successful change management through the standard game theoretic approach

The present global context can be characterized as highly dynamic with high failure rates. The continuous increase in the rythm of technological innovations translates into a dramatic shortening of products life cycles and a higher frequency of orga-

Unravelling Conditions for Successful Change Management

315

nizational change. One of the consequences being that the windows of opportunity to make profit from innovation and change open more frequently, but for a shorter time. In order to overcome these difficulties one idea could be to develop a set of incentives, such that the personnel of the organization accepts and contributes to the implementation of the change. In this section we start with an elementary model which will help to better understand the context, and then we shall develop the general model of information exchange between two departments. 2.1.

An introductive elementary example

Let us consider two departments i and j of the firm such that each one can decide to send (S) or not to send (S) information to the other. Let us furthermore assume that for each of the two departments: – receiving information generates a profit of 3 – sending information generates a cost of 2. The question is : should a department send information (strategy S) or not (strategy S)? To find the answer, one may resort to a game theoretic approach characterized by the matrix hereunder: S - sending S - not sending

S - sending S - not sending (1,1) (-2,3) (3, -2) (0,0) Fig. 1

It can be easily seen that the game displays a unique Nash equilibrium (S, S). In other words the example shows that despite the fact that the two departments would benefit from information exchange, this exchange cannot occur. In fact, this paradoxical conclusion just reflects the fact that the game is a Prisonner’s Dilemma. Whence the perpetual question: what conditions could make cooperation prevail and result into information exchange between the two parties? A possible answer is to develop a set of incentives that will push each department to send information to the other one. In our example, if the company rewards the sending of information by an incentive of 3, the game is then represented by the matrix hereunder: S S S (4,4) (1,3) S (3, 1) (0,0) Fig. 2

The unique Nash Equilibrium here is (S, S). Thus by the use of incentives the company is sure that information exchange will pervade. 2.2. General case: results from a standard approach Let us generalize the case here above by considering a company’s department with k employees, and which wants to implement a change possibly stemming from a technological innovation. Let us furthermore assume that :

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Michel Rudnianski, Cerasela Tanasescu

– the department’s incentives policy is decided by its manager – the case is symmetric: benefits, incentives and costs resulting from information sharing and change adoption are the same for all employees. – success is an increasing function of information sharing and the resulting change adoption – the value for the department of change adoption by k employees is an increasing function of k. Each employee i generating change or possessing relevant information about such change can decide to send or not to send this information to other employees. Likewise each employee i receiving information from employee j can decide to adopt or not to adopt the change made possible through reception of this information. For each employee i there are four possible sets of actions, which are given by the table hereunder: S S A SA SA A SA S A Fig. 3

For employee i, the results of the different possible interactions with employee j are given by the following table: Employee i Benefit Cost Incentive Sending information to j bs cs βs n n Not sending information to j b c βn a a Adopting change b c βa Not adopting change 0 0 0 Fig. 4

This means that for employee i, the payoff resulting from: – sending an information to another employee, is: s = β s + bs − cs – not sending information, is: n = bn − cn – adopting change made possible by the information received from another employee, is:a = (β n + bn − cn ). In case employee j doesn’t send information, the result for employee i of adopting is the same than the result of not adopting. – not adopting change made possible by the information received from another employee, is 0. Of course this is just an assumption. One could consider situations in which not adopting is associated with a non-zero payoff, for instance a negative one, meaning that by adopting this attitude employee i is a loser. We shall follow two approaches: – the static one based on non-repeated standard matrix games – the dynamic one based on the Replicator Dynamics. 2.3.

Static Approach

The starting point is the matrix of table 2.3.

Unravelling Conditions for Successful Change Management

317

SA SA SA SA SA (a+s,a+s) (a+s,s) (s, a+n) (s,n) SA (s,s+a) (s,s) (s, a+n) (s,n) SA (a+n,s) (a+n,s) (n,n) (n,n) SA (n,s) (n,s) (n,n) (n,n) Fig. 5 Conditions n < s < n+a < s+a

A1 A2 A3

n < n+a < s < s+a a>0

s < s+a < n < n+a

A4 B1

s 0

(5)

sn = 0

(6)

µt ∈ {0, 1}

(7)

t = i, i + 1, i + 2, ..., n

(8)

si , with:

For the problem definition see for example Tempelmeier (Tempelmeier, 2006, p. 138) or Neumann (Neumann, 2004, p. 594 – 595). The objective function K ∗ is composed of the ordering costs and the storage costs. The procurement costs as a product of the quantity to be procured and the price do not have to be considered in more detail in the procurement quantity calculation and do not need to be a component of the objective function K ∗ since they are a constant over the whole period under review. Consequently they have no influence on minimising the overall costs K (see for example Neumann, 2004, p. 594). In case of static demand, bt , is equal to the mean demand b. The first side condition (equation 2) is also called storage-balance-equation. It says that the inventory st at the end of period t is a result of the inventory st−1

Applying Game Theory in Procurement

331

at the end of period t − 1, the quantity to be procured in period t (xt ) and the demand for period t (bt ) (see for example Neumann, 2004, p. 593–594). The binary variable µt is 1, if an order is put in period t and 0 otherwise. This is achieved by equation (3) in connection with the minimising provision of the objective function (cf. Tempelmeier, 2006, p. 139). Thereby let M be an arbitrary big figure, which has to be at least so big that the quantity to be procured in each period t (xt ) is not restricted (cf. Tempelmeier, 2006, p. 139). The third side condition (equation 4) states that there are no negative quantities to be procured. By equation (5) is stated that inventory cannot be negative. The model presented in equation (1) to (8) describes the procurement quantity calculation and is called single-level uncapacitated lot sizing problem (SLULSP) (cf. Tempelmeier, 2006, p. 138). For the purpose of analysing the influence of different logistics parameters and random incidents the above described assumptions have to be extended as in reality for example stock-outs occure. These and other criterions which have to be considered but are not incorporated in the SLULSP have been taken into account by use of an additional model. 5.

Research study: Applicability of methods for procurement quantity calculation under different conditions

In the literature exist a lot of different heuristic methods and one optimal method for solving the presented SLULSP. As the SLULSP describes the problem of procurement quantity calculation, these heuristics are methods for procurement quantity calculation. The most established methods are the method from Wagner and Within (WW), the Least Unit Cost Method (LUC), the heuristic from Silver and Meal (SM), the heuristic from Groff (Groff), the Least Total Cost Method (LTC), the Incremental Order Quantity Method (IOQ), the method of Periodic Order Quantity (POQ) and the McLaren Order Moment (MOM). Existing surveys regarding the applicability of the heuristics do not analyse all possible types of demand (static demand, seasonal fluctuating demand, sporadic demand and trend in the demand) and especially uncertainty in logistics parameters such as • the demand calculation was not right • the supplier delivered less items than ordered • the delivery arrives delayed have not been in focus in existing surveys, but exactly these uncertainties become more relevant in practice nowadays. Game theoretic solution concepts can help especially under dynamic conditions like these as it was pointed out at the beginning of this paper. For setting up a decision model thus in a first step the influence of these logistics parameters was investigated. As the assumptions of the SLULSP as described in the previous section are not sufficient for these investigations, the model had been expanded to analyse the influence of different logistics parameters and random incidents. As mentioned above for example stock-outs need to be integrated into the model by use of a further model which displays the inventory. By use of these two models for the approach presented in this paper, the above given methods of procurement quantity calculation have been implemented to investigate the influence of the logistics parameters for the different methods on the

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Günther Schuh, Simone Runge

Fig. 1: Impact of the type of demand per method of procurement quantity calculation

result of the objective function of the SLULSP. First of all it could be seen, that the types of demand have considerably impact on the results (see Figure 1).

The logistics parameters under consideration are listed in the following. Additionally the range for the variation is given for each parameter (see table 2). Table 2: Logistics parameters under review in the survey and their range Logistics parameters minimum Maximum price (monetary units) 1 50 storage costs (percentage) 12 35 fixed costs for procurement (monetary units) 437,5 5292 stock-out costs (percentage) 50 500 replacement time (periods) 1 21 planning interval (periods) 25 90 Length of the planning horizon (periods) 140 365 delay in delivery (probability; periods) 0; 0 0.5; 10 shortshipment (probability; percentage) 0; 0 0,5; 0,5 deviation of demand (expected value; standard 0; 0 0; 10 deviation) The results of the study confirm the relevance of combining these well-known methods of procurement quantity calculation by use of game theoretic solution concepts. The sequence of applicability of the methods could be seen with regard to the result in the objective function expressed in monetary units. For example considering different length of the planning horizon from 140 periods up to 365 periods when the demand is stationary, seasonal or has a trend the objective value of the IOQ-Method becomes higher (see Figure 2). In contrast the objective value becomes lower when the methods LUC, SM, LTC, Groff, POQ or MOM are applied under a seasonal demand. Considering a trend in demand it could be seen, that the method POQ causes the second highest costs when the length of the planning horizon is 365 periods. With a shorter planning horizon of 140 periods the methods WW, Groff and SM all cause higher costs than the method POQ. So in this case


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when all the other parameters remain stable and only the length of the planning horizon is varied the applicability of the methods POQ, Groff and SM is inverted.

Fig. 2: Influence of the length of the planning horizon on the objective value in different demand situations

In real-life situations in most cases more than only one parameter changes. For this purpose the correlations between different logistics parameters have been investigated as well. In Figure 3 for example the correlation of the planning interval and the deviation of demand could be seen. A deviation of demand is given if the demand of the product was incorrect beforehand. When the method WW is applied and there is a trend in demand the correlation of these two parameters is marginal (see Figure 3, left side) but if the method IOQ is applied in the same situation the correlation of these two parameters is much bigger (see Figure 3, right side). In the second case when there is no deviation in demand (e.g. the demand was right beforehand) the objective value is much higher with a bigger planning interval. In contrast in case the demand was not calculated right beforehand the spread of the objective value is way smaller.

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Fig. 3: Correlation Diagrams: Influence of length of the planning interval combined with devation of demand with a given trend in demand

As it could be seen by these two examples, the objective values generated by the methods differ depending on the given environmental conditions. A sophisticated analysis of the influence of the logistics parameters is thus an essential finding for the decision model which will be developed and later on solved by game theoretic solution concepts. By means of the conducted investigations it became clear that none of the wellknown methods for procurement quantity calculation provides the best solution for all combinations of the logistics parameters under review. Thus these parameters, their combination as well as uncertainties have got an effect on the appropriate method for procurement quantity calculation. Furthermore it could be seen that the influence of the logistics parameters under consideration differs from method to method. The influence on the objective value of the parameters differs from method to method. This could be seen for example in Figure 2. Thus through the investigations it was approved that game theoretic solution concepts should support in choosing the appropriate method under given conditions, as it will be done by the approach presented in this paper. 6.

A structure for classifying game theoretic models for choosing appropriate solution concepts

Nowadays in game theory there exist a lot of different alternative models that are suited for representing diverse real-life-situations. It is beyond dispute that the type of game (i.e. the game theoretic model) has essential influence on the appropriate game theoretic solution concept (see for example Schiml, 2008, p. 26). Thus when applying game theory to real-life decision-problems the first very important step is to decide which solution concept is the most suitable for the decision-problem at hand. To reach this goal, a scheme which can help classifying game theoretic solution concepts is needed. This scheme has to be based on the different attributes of game theoretic models as these models have essential influence on the solution concepts. It has been found out, that no such scheme which is comprehensive exists in game theoretic literature. A lot of authors list different attributes of game theoretic models with some corresponding characteristics. In the majority of cases these explanations take place only in a textual way (see for example Herbst, 2007, p. 84 – 86; Kaluza,


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1972, p. 21 – 49). Other authors give some game theoretic models in form of a list (see for example Lasaulce and Tembine, 2011, p. 15; Kuhn, 2007, p. 50). For each of the considered types of games Vogt lists two different corresponding characteristics (see Vogt, 2013, p. 300). Only two authors describe the different game theoretic models and their corresponding schemes in form of a structured scheme: Marchand gives such a scheme but for each type he depicts only two different corresponding characteristics (see Marchand, 2012, p. 36). Pickel et al. sum up different game theoretic models with their characteristics in a diagram, but this diagram is not structured consistently (see Pickel et al., 2009, p. 67). Independent from the form of representation none of the above mentioned compositions give an exhaustive description of all existing game theoretic models and their corresponding characteristics. Thus to obtain a comprehensive and well-structured description of the attributes and corresponding characteristics of game theoretic models a morphology has been developed in the context of the research work presented in this paper. The morphology is depicted in Figure 4 and is described in the following.

Fig. 4: Structure for classifying game theoretic models for choosing appropriate solution concepts

Ten attributes have been identified which differentiate game theoretic models. By the number of decision makers it is meant how many people participate in the decision. If there is only one decision maker the model could rather be assigned to decision theory than game theory. But as some authors assign these models to game theory, this characteristic is listed in the morphology presented here. In game

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theoretic models not in all cases only two decision makers are incorporated in the model. There are a lot of situations with more than two decision makers as well. Another attribute when analysing decision situations is the distribution of profits and losses, which could be expected as a result of the decision. The corresponding characteristics of this attribute were derived from the types of games zero-sum-games, nonzero-sum-games, constant-sum-games and strictly competitive games as all of these models could be differentiated by the attribute of how the profits and losses of the decision makers are distributed. The existence of agreements is the third attribute which has been identified as an essential attribute when classifying game theoretic models and choosing the most suitable game theoretic solution concept. Looking at cooperative game theoretic models, binding agreements between the decision makers are declared. On the other hand in non-cooperative game theory these agreements could be non-binding or the decision makers make no agreements at all. The number of repetitions in a game corresponds to the attribute frequency of decisions in the morphology given in Figure 4. This attribute, which refers to repeated games, has considerable impact on the applicability of the solution concept as well. If the game will not be repeated the decision maker will always chose the best choice for himself. But if a game is repeated all decision makers will be more willing to find best solutions for all of them as they are afraid of retaliation otherwise. In game theory, games with singular decision, a known finite number > 1 of games to be played, a unknown finite number of games to be played > 1 as well as games which have a countable finite number of repetitions could be differentiated. Another important attribute to distinguish game theoretic models is the sequence of decisions. Meant by this attribute is if the decisions of the players in a game are performed simultaneously or successive. This refers to simultaneous and sequential move games. Moreover game theoretic models could be classified by the information available to the players. In game theory there exist games with perfect or imperfect information. This is meant by the recognition of prior actions. Does every player know exactly which situation is on hand (i.e. past and current decisions of the other decision makers are completely known to all players) this is a game with perfect information. If at least one decision maker does not exactly know past decisions of the other players, this is called a game with imperfect information. Further characteristics of this attribute could be that past decisions of the other decision makers are completely known to all players or that no information of past decisions are know. Another deficit in information in game theoretic models could be caused by the availability of the background of other decision makers. As a first corresponding characteristic it is possible that the payoff and possible strategies of all decision makers are known to all players. These are games with complete information where every rational decision maker is able to calculate the best strategy for himself exactly. In contrast to these there are games with incomplete information. The corresponding graduation could be seen in Figure 4. The strategic scope it the set of available strategies for a player. The size of the strategic scope can be distinguished by the criterion if the possible strategies could be depicted reasonable as a decision-matrix or game-tree. Theoretically every game-tree could be depicted unless it is infinite. Surveys show that up to nine alternatives could be processed by humans. Thus for a strategic scope of ten or


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more alternatives it does not seem to be reasonable to depict them as a decisionmatrix or game-tree. For these cases as well as if the strategic scope is countably or even uncountably infinite reaction-functions from game theory could help solving such decision problems. In evolutionary game theory no longer individual decision makers are in focus. Instead these models focus on populations and the members of a population are able to decide only in the way their genetic code allows. Thus regarding the attribute individuality of the decision makers it could be distinguished if the individuality is existent or not. The last attribute in the morphology for classifying game theoretic models is the consideration of random incidents. In contrast to all other attributes this attribute could not be derived from the types of games directly. This attribute could be originary derived from decision-theory but is relevant in game theoretic models as well. If random incidents have to be considered in game theory, a decision maker has to determine the probability for the random incidents if possible. Therefore the corresponding characteristics of this atribute in the morphology are: random incidents are not relevant, random incidents are relevant and the probability for these could be determined and random incidents are relevant but the probability for random incidents could not be determined. The developed morphology can support in carving out which game theoretic solution concept will help to solve the described problem statement of combining existing heuristics for the procurement quantity calculation for getting more flexibility in supply chains. Therefore the corresponding characteristic per attribute has to be identified for each game theoretic solution concept which is generally suited for solving the problem on hand. The same has to be done for the real-life-situation under review. Then structural similarities could be seen. This gives a structured basis for deciding which solution concept should be applied for solving the given problem later on. 7.

Conclusion and further research-steps

In this paper at first a short introduction into the idea of applying game theoretic solution concepts for appropriate use of well-known methods of procurement quantity calculation had been given. In the next part in the state of the art in could be seen that there exists a lack in using game theory for improving and adapting decisions over time. Additionally a lack in applying game theory especially in purchasing and inventory had been identified. Solving the problem of procurement quantity calculation means to solve a SLULSP. Therefore the model was introduced in the next part of this paper. Some results of the conducted study regarding the applicability of the methods of procurement quantity calculation have as well been presented in this paper. This study made clear that changing parameters impact the objective value of the different methods for procurement quantity calculation significantly. Thus the relevance of the work presented here is approved. To enable choosing the most suitable method under given conditions the survey will be pursued in more detail to deduce concrete recommendations for switching the methods from these findings. As a last step in this paper a structure for classifying game theoretic models to enable choosing the appropriate solution concept was presented. This morphology can help to answer the question of which game theoretic solution concepts can be

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applied in an application area – for example in procurement quantity calculation as regarded in this paper. For this purpose the solution concepts have to be opposed to the application area by the use of this morphology. By this it is ensured that exactly the solution concepts which are relevant in the application area of procurement are considered in the following research steps. References Abad, P. L. (1994). Supplier pricing and lot sizing when demand is price sensitive. In: European Journal of Operational Research, 78, p. 334–354. Başar, T., Olsder, G. J. (1999). Dynamic noncooperative game theory. 2. ed., Siam, New York. Behnen, D., Brecher, C., Hammann, G., Herfs, W., Koeppe, R., Pauly, D., Possel-Dölken, F., Verl, A. (2011). Potenziale kognitiver Systeme in der Produktionstechnik. In: Wettbewerbsfaktor Produktionstechnik: Aachener Perspektiven. Aachener Werkzeugmaschinenkolloquium. Edit: C. Brecher; F. Klocke; R. Schmitt; G. Schuh (Hg.). Shaker, Aachen, p. 101–123. Brecher, C., Kozielski, S., Karmann, O. (2011). Integrative Produktionstechnik für Hochlohnländer. In: Wettbewerbsfaktor Produktionstechnik: Aachener Perspektiven. Aachener Werkzeugmaschinenkolloquium. Edit.: C. Brecher; F. Klocke; R. Schmitt, G. Schuh. Shaker, Aachen, p. 1–16. Cachon, G. P., Netessine, S. (2004). Game Theory in Supply Chain Analysis. In: Handbook of quantitative supply chain analysis. Modeling in the e-business era. Edit.: D. SimchiLevi, D.; S. D. Wu; Z.-J. Shen. Springer, New York, p. 13–65. Chen, M.-S, Chang, H.-J, Huang, C.-W, Liao, C.-N (2006). Channel coordination and transaction cost: A game-theoretic analysis. In: Industrial Marketing Management 35, p. 178–190. Chiang, W.-C, Fitzsimmons, J., Huang, Z., Li, S. X. (1994). A Game-Theoretic Approach to Quantity Discount Problems. In: Decision Sciences, 25(1), p. 153–168. Corbett, C. J., Groote, X. de (2000). A Supplierťs Optimal Quantity Discount Policy Under Asymmetric Information. In: Management Science, 46(3), p. 444–450. Daxböck, C., Zinn, U., Hunstock, T. (2011). Supply Chain Performance Management. Über die intelligente Verknüpfung von Strategien, Prozessen und Controlling zur Steuerung der Wertschöpfungskette. Edit.: Horváth & Partner, Stuttgart. Drozak, J. (2013). Spieltheorie im Einkauf. Warten auf den Durchbruch. In: Beschaffung aktuell 04, S. 32. Esmaeili, E., Aryanezhad, M.-B, Zeephongsekul, P. (2009). A game thoery approach in seller-buyer supply chain. In: European Journal of Operational Research, 195, p. 442– 448. Fischer, K. (1997). Standortplanung unter Berücksichtigung verschiedener Marktbedingungen. Physica, Heidelberg. Fudenberg, D., Tirole, J. (1991). Game theory. MIT Press, Cambridge [et al.]. Harsanyi, J. C. (1967). Games with incomplete information played by "bayesian" players, I-III. Part I. The basic model. In: Management Science , 14(3), p. 159–182. Hennet, J.-C, Arda, Y. (2008). Supply chain coordination: A game-theory approach. In: Engineering Applications of Artificial Intelligence, 21(3), p. 399–405. Herbst, U. (2007). Präferenzmessung in industriellen Verhandlungen. 1. ed., Gabler, Wiesbaden. Holler, M. J., Illing, G. (2006). Einführung in die Spieltheorie. Springer, Berlin [et al.]. Jordan, P. R., Kiekintveld, C., Wellmann, M. P. (2007). Empirical game-theoretic analysis of the TAC Supply Chain game. In: AAMS 07. Proceedings of the 6th international joint conference on Autonomous Agents and Multiagent Systems. IFAAMAS, [S. l.], p. 1196–1203.


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An Axiomatization of the Myerson Value Özer Selçuk1 and Takamasa Suzuki2 1

2

CentER, Department of Econometrics & Operations Research, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands E-mail: [email protected] CentER, Department of Econometrics & Operations Research, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands E-mail: [email protected]

Abstract TU-games with communication structure are cooperative games with transferable utility where the cooperation between players is limited by a communication structure represented by a graph on the set of players. On this class of games, the Myerson value is one of the most well-known solutions and it is the Shapley value of the so-called restricted game. In this study we give another form of fairness axiom on the class of TU-games with communication structure so that the Myerson value is uniquely characterized by this fainess axiom with (component) efficiency, a kind of null player property and additivity. The combination is similar to the original characterization of the Shapley value. Keywords: Cooperative TU-games, communication structure, Myerson value, Shapley value

Cooperative game theory describes situations of cooperation between players. A cooperative game with transferable utility, TU-game for short, expresses such situations by a finite set of players and a characteristic function that assigns a worth to any subset of players, a coalition. Players within a coalition can freely divide the worth of the cooperation among themselves. The main focuses of TUgames are investigating under which conditions the players cooperate to form the grand coalition of all players and how to divide the worth of this grand coalition into a payoff for each player. A single-valued solution on a class of games assigns as an allocation a payoff vector to each game which belongs to the class. Shapley (1953) introduces one of the most well-known single-valued solution. The solution, the Shapley value, is the average of all marginal vectors of a TU-game, where a marginal vector corresponds to a payoff vector for a permutation on the player set. Each permutation can be seen as an ordering of the players joining to from the grand coalition, and in the marginal vector associated with a permutation each player gets as payoff the difference in worth of the set of players preceding him in the permutation with and without him. While being introduced, the Shapley value is characterized as the unique solution on the class of TU-games that satisfies efficiency, additivity, null player property and symmetry in Shapley (1953). TU-games assume that any coalition can be formed to cooperate and gain its worth of their cooperation, but in many economic situations there exist restrictions which prevent some coalitions from cooperating. A TU-game with this kind of situation is firstly introduced by Myerson (1977) as a TU-game with communication structure. It arises when the restriction is represented by an undirected graph in

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which the vertices represent the players and a link between two players shows that these players can communicate and are able to cooperate by themselves. One of the most well-known single-valued solutions on the class of TU-games with communication structure is the Myerson value (Myerson (1977)), defined as the Shapley value of the so-called Myerson restricted game. By Myerson (1977), the Myerson value is characterized by (component) efficiency and fairness, fair in the sense that if a link is deleted between two players, the Myerson value imposes the same loss on payoffs for each of these two players. Other characterizations of the Myerson value are given in Borm et al. (1992), Brink (2009) for the class of TU-games with cycle-free communication structure. In this study we give an alternative axiomatization of the Myerson value for TU-games with communication structure. Our approach is to give another form of fainess axiom so that the Myerson value is characterized by (component) efficiency, a kind of null player property, additivity and a kind of fairness. The combination is similar to the original characterization of the Shapley value by Shapley (1953). This paper is organized as follows. Section 2 introduces TU-games with communication structure and the Myerson value. In Section 3 an axiomatic characterization for the solution is given. 1.

TU-games with communication structure and the Myerson value

A cooperative game with transferable utility, or a TU-game, is a pair (N, v) where N = {1, . . . , n} is a finite set of n players and v : 2N → R is a characteristic function with v(∅) = 0. For a subset S ∈ 2N , being the coalition consisting of all players in S, the real number v(S) represents the worth of the coalition that can be maximially achived, and can be freely distributed among the players in S. Let GN denote the class of TU-games with fixed player set N . We often identify a TU-game (N, v) by its characteristic function v. A special class of TU-games is the class of unanimity games. For T ∈ 2N , the unanimity game (N, uT ) ∈ GN has characteristic function uT : 2N → R defined as

uT (S) =

1 if T ⊆ S, 0 otherwise.

It is well-known that any TU-game can be uniquely expressed as a linear combination of unanimity games. Let (N, 0) ∈ GN denote the zero game, i.e., 0(S) = 0 for all S ∈ 2N . A payoff vector x = (x1 , ..., xn ) ∈ Rn is an n-dimentional vector and it assigns payoff xi to player i ∈ N . A single-valued solution on GN is a mapping ξ : GN → Rn which assigns to every TU-game (N, v) a payoff vector ξ(N, v) ∈ Rn . The most well-known single-valued solution on the class of TU-games is the Shapley value, see Shapley (1953). It is the average of the marginal vectors induced from the collection of all permutations of players. Let Π(N ) be the collection of all permutations on N . Given a permutation σ ∈ Π(N ), the set of predecessors of any element i ∈ N in σ is defined as Pσ (i) = {h ∈ N | σ −1 (h) < σ −1 (i)}.

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An Axiomatization of the Myerson Value

Given a TU-game (N, v) ∈ GN , for a permutation σ in Π(N ) the marginal vector mσ (N, v) assigns payoff mσi (N, v) = v(Pσ (i) ∪ {i}) − v(Pσ (i)) to agent i = σ(k), k = 1, . . . , n. The Shapley value of (N, v), Sh(N, v), is the average of all n! marginal vectors, i.e., 1 X Sh(N, v) = mσ (N, v). n! σ∈Π(N )

A graph on N is a pair (N, L) where N = {1, . . . , n} is a set of vertices and L ⊆ LcN , where LcN = {{i, j} | i, j ∈ N, i 6= j} is the complete set of undirected links without loops on N and an unordered pair {i, j} ∈ L is called an edge in (N, L). A subset S ∈ 2N is connected in (N, L) if for any i ∈ S and j ∈ S, j 6= i, there is a sequence of vertices (i1 , i2 , . . . , ik ) in S such that i1 = i, ik = j and {ih , ih+1 } ∈ L for h = 1, . . . , k − 1. The collection of all connected coalitions in (N, L) is denoted C L (N ). By definition, the empty set ∅ and every singleton {i}, i ∈ N , are connected in (N, L). For S ∈ 2N , the subset of edges L(S) ⊆ L is defined as L(S) = {{i, j} ∈ L| i, j ∈ S}, being the subset of L of edges that can be established within S. The graph (S, L(S)) is a subgraph of (N, L). A component of a subgraph (S, L(S)) of (N, L) is a maximally connected coalition in (S, L(S)) and bL (S). For a graph (N, L), the collection of components of (S, L(S)) is denoted C if {i, j} ∈ L, then i is called a neighbor of j and vice versa. Given (N, L) and i ∈ N , the collection of neighbors of i is denoted by DiL , that is, DiL = {j ∈ N \ {i} | {i, j} ∈ L}. The collection of neighbors of S ∈ 2N is defined similarly as DSL = {j ∈ N \ S | ∃i ∈ S : {i, j} ∈ L}. The combination of a TU-game and an (undirected) graph on the player set is a TU-game with communication structure, introduced by Myerson (1977) and denoted by a triple (N, v, L) where (N, v) is a TU-game and (N, L) is a graph on N . A link between two players has as interpretation that the two players are able to communicate and it is assumed that only a connected set of players in the graph is able to cooperate to obtain its worth to freely transfer as payoff among the players in cs the coalition. Let GN denote the class of TU-games with communication structure cs cs and fixed player set N . A single-valued solution on GN is a mapping ξ : GN → Rn cs which assigns to every TU-game with communication structure (N, v, L) ∈ GN a n payoff vector ξ(N, v, L) ∈ R . The most well-known single-valued solution on the class of TU-games with communication structure is the Myerson value, see Myerson (1977). It is the Shapley value of the so-called Myerson restricted game. Following Myerson (1977), the restricted characteristic function v L : 2N → R of (N, v, L) is defined as X v(K), S ∈ 2N . v L (S) = b L (S) K∈C

The pair (N, v L ) is a TU-game and is called the Myerson restricted game of (N, v, L), cs and the Myerson value of a game (N, v, L) ∈ GN is defined as µ(N, v, L) =

1 n!

X

σ∈Π(N )

mσ (N, v L ).

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Özer Selçuk, Takamasa Suzuki An axiomatic characterization of the Myerson value

Most of the single-valued solutions proposed in the literature are characterized by axioms which state desirable properties a solution possesses. The most well-known characterization of the Shapley value for TU-games is given by Shapley (1953) as the unique solution on the class of TU-games that satisfies efficiency, additivity, the null player property and symmetry. Other characterizations of the Shapley value are proposed in for example Young (1985) and Brink (2002). While introducing the class of TU-games with communication structure, Myerson (1977) characterizes the Myerson value by component efficiency and fairness axioms. cs Definition 1. A solution ξ : GN → Rn satisfies component efficiency if for any P cs bL (N ). (N, v, L) ∈ GN it holds that i∈Q ξi (N, v, L) = v(Q) for all Q ∈ C

A solution on the class of TU-games with communication structure satisfies component efficiency if the solution allocates to each component as the sum of payoff among its members the worth of the component. cs cs Definition 2. A solution ξ : GN → Rn satisfies fairness if for any (N, v, L) ∈ GN and {i, j} ∈ L it holds that

ξi (N, v, L) − ξi (N, v, L \ {i, j}) = ξj (N, v, L) − ξj (N, v, L \ {i, j}). A solution on the class of TU-games with communication structure satisfies fairness if the deletion of an edge from the game results in the same payoff change for the two players who own the edge. cs Theorem 1. (Myerson, 1977) The Myerson value is the unique solution on GN that satisfies component efficiency and fairness.

For the class of TU-games with cycle-free communication structure, which is a subclass of TU-games with communication structure, other characterizations of the Myerson value are given by Borm et al. (1992) and Brink (2009). The axioms we propose in this study are modified versions of the four axioms used in Shapley (1953), i.e., an efficiency axiom (component efficiency), an additivity axiom, a null player property and a fairness axiom. For any two TU-games v and w in GN , the game v + w is well defined by (v + w)(S) = v(S) + w(S) for all S ∈ 2N . cs Definition 3. A solution ξ : GN → Rn satisfies additivity if for any (N, v, L), cs (N, w, L) ∈ GN it holds that ξ(N, v + w, L) = ξ(N, v, L) + ξ(N, w, L).

Additivity of a solution means that if there are two TU-games with the same communication structure, the resulting payoff vectors coincide when applying the solution to each of the two games and adding the two vectors and when applying the solution to the game which is the sum of the two games. A player i ∈ N is a restricted null player in a TU-game with communication cs structure (N, v, L) ∈ GN if this player never Pcontributes whenever he joins to form a connected coalition, that is, v(S ∪ {i}) − K∈CbL (S) v(K) = 0 for all S ∈ 2N such that i ∈ / S and S ∪ {i} ∈ C L (N ). The restricted null player property says that this player must get zero payoff.

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cs Definition 4. A solution ξ : GN → Rn satisfies the restricted null player property cs if for any (N, v, L) ∈ GN and restricted null player i ∈ N in (N, v, L) it holds that ξi (N, v, L) = 0.

Note that a restricted null player of a TU-game with communication structure is a null player of its Myerson restricted game. The last axiom replaces symmetry. cs Definition 5. A solution ξ : GN → Rn satisfies coalitional fairness if for any ′ cs two TU-games (N, v, L), (N, v , L) ∈ GN and Q ∈ 2N it holds that ξi (N, v, L) − ′ ′ ξi (N, v , L) = ξj (N, v, L) − ξj (N, v , L) for all i, j ∈ Q whenever v(S) = v ′ (S) for all S ∈ 2N , S 6= Q.

Coalitional fairness of a solution implies that given a TU-game with communication structure, if the worth of a single coalition changes, then the payoff change should be equal among all players in that coalition. From additivity and the restricted null player property we have the following lemma.

cs Lemma 1. Let a solution ξ : GN → Rn satisfy additivity and the restricted null player property. Then for any two TU-games with the same communication structure cs (N, v, L), (N, v ′ , L) ∈ GN it holds that ξ(N, v, L) = ξ(N, v ′ , L) whenever v(S) = ′ L v (S) for all S ∈ C (N ).

Proof. Consider the game (N, w, L) where w = v − v ′ . Then every player is a restricted null player in this game because w(S) = 0 for all S ∈ C L (N ). Therefore every player must receive zero payoff, that is, ξ(N, w, L) = 0. From additivity and v = w + v ′ it follows that ξ(N, v, L) = ξ(N, w, L) + ξ(N, v ′ , L) = 0 + ξ(N, v ′ , L) = ξ(N, v ′ , L). ⊔ ⊓

This lemma says that the worth of an unconnected coalition does not affect the outcome of a solution that satisfies additivity and the restricted null player property, which leads to the following corollary. cs Corollary 1. If a solution ξ : GN → Rn satisfies additivity and the restricted null cs player property, then ξ(N, v, L) = ξ(N, v L , L) for any (N, v, L) ∈ GN .

To prove that on the class of TU-games with communication structure the axioms above uniquely define the Myerson value, we consider Myerson restricted unanimity games. Given a unanimity game with communication structure (N, uT , L) ∈ cs GN with T ∈ 2N , the Myerson restricted unanimity game (N, uL T ) ∈ GN is given by b L (S), T ⊆ K, 1 if ∃ K ∈ C uL T (S) = 0 otherwise. L

Given a graph (N, L) and S ∈ 2N , let C (S) denote the collection of connected coalitions which minimally contain S, that is, L

C (S) = {K ∈ C L (N ) | S ⊆ K, K \ {i} ∈ / C L (N ) ∀ i ∈ K \ S}.

Lemma 2. For a unanimity TU-game with communication structure (N, uT , L) ∈ cs GN with T ∈ 2N , it holds that  X L  (−1)|J|+1 u∪j∈J Qj if C (T ) = {Q1 , . . . , Qk },  uL J⊆{1,...,k} T =  L 0 if C (T ) = ∅.

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Proof. First consider the case when C (T ) = ∅. This implies that there exists no b L (N ) which contains T , and from the definition of uL it follows that uL (S) = K∈C T T P L 0 for all S ∈ 2N . Next, let v = J⊆{1,...,k} (−1)|J|+1 u∪j∈J Qj when C (T ) 6= ∅. If L

T ∈ C L (N ), then C (T ) = {T } and therefore it holds that v = uT = uL T . Suppose N T ∈ / C L (N ). It is to show that v(S) = uL T (S) holds for every S ∈ 2 . First take b L (S) satisfying T ⊆ K. This implies that S ∈ 2N such that there is no K ∈ C L Q 6⊂ S for any Q ∈ C (T ), and thus we have u∪j∈J Qj (S) = 0 for all J ⊆ {1, . . . , k}, N which results in v(S) = 0 = uL such that there exists T (S). Next, take any S ∈ 2 L b K ∈ C (S) satisfying T ⊆ K. This K is unique and let M ⊆ {1, . . . , k} be such that Qj ⊆ K for all j ∈ M and Qj 6⊂ K for all j ∈ / M . Among all J ⊆ {1, . . . , k}, it holds that u∪j∈J Qj (S) = 1 only when J ⊆ M , and otherwise u∪j∈J Qj (S) = 0. Let P Pk=m |M | = m. Then v(S) = J⊆M (−1)|J|+1 u∪j∈J Qj (S) = k=1 (−1)k+1 m k = 1 = Pk=m uL the binominal theorem that k=0 (−1)k m T (S), since it is known from k = 0 and Pk=m P k=m m k+1 m k m therefore k=1 (−1) ⊔ ⊓ k=1 (−1) k = − k = 0 = 1.

Note that for any J ⊆ {1, . . . , k}, it holds that ∪j∈J Qj is connected, since for each j ∈ J, the set Qj itself is connected and it also contains T . This lemma shows that any restricted unanimity TU-game with communication structure can be uniquely expressed as a linear combination of unanimity TU-games with the same communication structure for connected coalitions. On the class of unanimity TU-games with communication structure, we have the following expression, which is well known and we present without proof. cs Lemma 3. For any TU-game with communication structure (N, cuT , L) ∈ GN with L T ∈ C (N ), T 6= ∅, and c ∈ R, it holds that c/|T | if j ∈ T, µj (N, cuT , L) = 0 if j 6∈ T.

This lemma says that the Myerson value of a unanimity TU-game with communication structure with a connected coalition assigns the allocation which gives zero payoffs to the players who do not belong to the connected coalition and the worth of the connected coalition is shared equally among those who belong to it. Next, we give a characterization of the Myerson value in the following theorem. cs Theorem 2. The Myerson value is the unique solution on GN that satisfies component efficiency, additivity, the restricted null player property, and coalitional fainess.

Proof. First, we show that the Myerson value satisfies all properties. Component efficiency follows from the fact that all marginal vectors are component efficient by construction. Since all marginal vectors of a TU-game with communication structure are linear in the worths of the connected coalitions and the Myerson value is the average of these vectors, the Myerson value satisfies additivity. If a player is a restricted null player, this player has marginal contribution equal to zero at any permutation and therefore the average is also zero. Finally, suppose there are two cs TU-games with the same communication structure (N, v, L), (N, v ′ , L) ∈ GN and Q ∈ C L (N ) such that v(S) = v ′ (S) for all S ∈ C L (N ), S 6= Q, and take any i ∈ Q. It holds that mσi (N, v, L) = mσi (N, v ′ , L) for any σ ∈ Π(N ) unless Pσ (i) = Q \ {i}.


347

There are (|Q| − 1)!(n − |Q|)! permutations σ such that Pσ (i) = Q \ {i} and for each such σ the marginal contribution of i changes by mσi (N, v, L) − mσi (N, v ′ , L) = (v L (Q)−v L (Q\{i}))−(v ′L (Q)−v L (Q\{i})) = v L (Q)−v ′L (Q), which is independent of i. Therefore every player in Q receives the same change the same number of times and so the change in the Myerson value is the same among all players in Q. cs Second, let ξ : GN → Rn be a solution which satisfies all four axioms. Since ξ satisfies additivity and the restricted null player property, with Corollary 1 and Lemma 2, it suffices to show that for any graph (N, L) it holds that ξ(N, cuT , L) = µ(N, cuT , L) for any T ∈ C L (N ) and c ∈ R. Let (N, L) be any graph on N . cs First consider the zero game (N, 0, L) ∈ GN . In this game all players are restricted null players and therefore it follows from the restricted null player property that ξi (N, 0, L) = 0 = µi (N, 0, L) for all i ∈ N . Next consider the game (N, cuN , L) ∈ cs GN with N ∈ C L (N ). Between the games (N, cuN , L) and (N, 0, L) it holds that cN (N ) = c and cuN (K) = 0(K) = 0 for all K ∈ 2N , K 6= N . From efficiency, coalitional fairness, and Lemma 3, we have ξi (N, cuN , L) =

c = µi (N, cuN , L) ∀ i ∈ N. n

cs Now consider a game (N, cuT , L) ∈ GN with T ∈ C L (N ), |T | = n − 1. It follows from the restricted null player property that player i ∈ / T receives zero payoff, since this player yields zero marginal contribution when joining to any set of players to form a connected coalition. For the games (N, cuT , L) and (N, cuN , L), it holds that cuT (K) = cuN (K) for all K ∈ 2N , K 6= T . Coalitional fairness then implies that

ξi (N, cuT , L) − ξi (N, cuN , L) = ξj (N, cuT , L) − ξj (N, cuN , L) ∀ i, j ∈ T, which, with efficiency and Lemma 3, results in ξi (N, cuT , L) =

c = µi (N, cuT , L) ∀ i ∈ T. |T |

Next, suppose ξ(N, cuT , L) = µ(N, cuT , L) holds for all T ∈ C L (N ), |T | > m > 1. cs Consider (N, cuT , L) ∈ GN with T ∈ C L (N ), |T | = m. For i ∈ / T , it follows from the restricted null player property that ξ (N, cu , L) = 0. By comparing (N, cuT , L) i T P and (N, v, L) with v = ℓ∈DL cuT ∪{ℓ} − (k − 1)cuN where k = |DTL | is the number T of neighbors of T in (N, L), it holds that cuT (S) = v(S) for all S ∈ 2N , S 6= T , and cuT (T ) = c while v(T ) = 0. Then coalitional fairness implies ξi (N, cuT , L) − ξi (N, v, L) = ξj (N, cuT , L) − ξj (N, v, L) ∀ i, j ∈ T. From additivity and the supposition that ξ(N, cuS , L) = µ(N, cuS , L) for all connected S with |S| > m, it follows that X ξi (N, v, L) = ξi (N, cuT ∪{ℓ} , L) − (k − 1)ξi (N, cuN , L) = X

L ℓ∈DT

X

L ℓ∈DT

L ℓ∈DT

µi (N, cuT ∪{ℓ} , L) − (k − 1)µi (N, cuN , L) =

µj (N, cuT ∪{ℓ} , L) − (k − 1)µj (N, cuN , L) = ξj (N, v, L)

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for all i, j ∈ T , and therefore ξi (N, cuT , L) = ξj (N, cuT , L) ∀ i, j ∈ T. By efficiency it holds that ξi (N, cuT , L) = c/|T | for all i ∈ T , which implies ξ(N, cuT , L) = µ(N, cuT , L). When |T | = 1, efficiency and the restricted null player cs property imply that ξ allocates the Myerson value to (N, cuT , L) ∈ GN . Therefore for a multiple of any unanimity TU-game with communication structure for a connected coalition, the four axioms uniquely give the allocation of the Myerson value. Since ξ satisfies additivity and the restricted null player property, it follows cs from Corollary 1 that ξ(N, v, L) = ξ(N, v L , L) for any (N, v, L) ∈ GN . By Lemma L 2 it holds that v can be expressed as a unique linear combination of unanimity cs games for connected coalitions. That is, given any (N, v, L) ∈ PGN there exist unique L L numbers cT ∈ R for T ∈ C (N ), T 6= ∅, such that v = T cT uT . The proof is cs completed since for any (N, v, L) ∈ GN it holds from additivity that X ξ(N, v, L) = ξ(N, v L , L) = ξ(N, cT uT , L) = X

T ∈C L (N ),T 6=∅

ξ(N, cT uT , L) =

X

T ∈C L (N ),T 6=∅

µ(N, cT uT , L) = µ(N, v, L).

T ∈C L (N ),T 6=∅

⊔ ⊓

ToP show the independence of the four axioms, consider the linear solution ξ(N, v, P L) = T ∈C L (N ) f (N, cT uT , L) where v = T ∈C L (N ) cT uT and f (N, cT uT , L) allocates c to the player in T who has the smallest index and 0 to any other player. It only fails coalitional fairness. Next, consider the solution ξ(N, v, L) that allocates payoff vector ξ(N, v, L) as follows. When N = {1, 2}, L = {1, 2}, v L (S) 6= 0 for all S ∈ C L (N ), and further v(S) 6= v(T ) for all distinct T, S ∈ 2N , then it gives ξj (N, v, L) = v(N )/2, and in any other case it gives ξ(N, v, L) = µ(N, v, L). This solution satisfies all axioms except additivity. The equal sharing solution, where each agent receives v(N )/n, satisfies every axiom except the restricted null player property. Finally, the solution where each agent receives zero payoff only fails efficiency. References Borm, P., G. Owen, and S. Tijs (1992). On the position value for communication situations. SIAM Journal of Discrete Mathematics, 5, 305–320. Brink, R. van den (2002). An axiomatization of the Shapley value using a fairness property. International Journal of Game Theory, 30, 309–319. Brink, R. van den (2009). Comparable axiomatizations of the Myerson value, the restricted Banzhaf value, hierarchical outcomes and the average tree solution for cycle-free graph restricted games. Tinbergen Institute Discussion Paper 2009–108/1, Tinbergen Institute, Amsterdam. Myerson, R. B. (1977). Graphs and cooperation in games, Mathematics of Operations Research, 2, 225–229. Shapley, L. (1953). A value for n-person games. In: Kuhn, H.W. and A.W. Tucker (eds.), Contributions to the Theory of Games, Vol. II, Princeton University Press, Princeton, pp. 307–317. Young, H. P. (1985). Monotonic solutions of cooperative games. International Journal of Game Theory, 14, 65–72.

Multi-period Cooperative Vehicle Routing Games Alexander Shchegryaev and Victor V. Zakharov St.Petersburg State University, Faculty of Applied Mathematics and Control Processes, Universitetsky Prospect 35, St.Petersburg, Peterhof, 198504, Russia E-mail: [email protected], [email protected]

Abstract In the paper we treat the problem of minimizing and sharing joint transportation cost in multi-agent vehicle routing problem (VRP) on large-scale networks. A new approach for calculation subadditive characteristic function in multi-period TU-cooperative vehicle routing game (CVRG) has been developed. The main result of this paper is the method of constructing the characteristic function of cooperative routing game of freight carriers, which guarantees its subadditive property. A new algorithm is proposed for solving this problem, which is called direct coalition induction algorithm (DCIA). Cost sharing method proposed in the paper allows to obtain sharing distribution procedure which provides strong dynamic stability of cooperative agreement based on the concept of Sub-Core and time consistency of any cost allocation from Sub-Core in multi-period CVRG. Keywords: VRP, vehicle routing problem, vehicle routing games, heuristics, multi-period cooperative games, dynamic stability, time consistency.

1.

Introduction

When we study collaboration in cargo transportation and routing we have to address the following questions partly discussed in (Agarwal et al., 2009): – How does one evaluate the maximum potential benefit from collaboration of carriers forming coalitions? However, to obtain such a benefit value is not easy because the underlying computational problem is NP hard. – How should a membership mechanism be formed to be stable during sufficiently long period of time, and what are the desired properties that such a mechanism should possess? For logistics applications, this involves issues related to the design of the service network and utilization of assets, such as the allocation of ship capacity among collaborating carriers, assignment and scheduling vehicles on routes. – How should the benefits achieved by collaborating be allocated among the members in a fair way? In the cargo transportation routing setting we investigate what does a fair allocation mean and how such an allocation may be achieved in the context of day-to-day operations to be time consistent during transportation process? – How to overcome these disadvantages? Dynamic cooperative game theory can provide us with models of coordination carrier’s actions in order to reduce transportation costs. Cooperation issues in vehicle routing models are still an insufficiently studied problem. Possible applications of the cooperative game theory for such problems are demonstrated in the papers

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Alexander Shchegryaev, Victor V. Zakharov

(Ergun et al., 2007; Krajewska et al., 2008). The most important object under investigation of the cooperative game theory is the characteristic function of the game which reflects assessment of guaranteed values of total costs of participants united in a coalition. If one constructs a mathematical model of cooperation in practical tasks, it is important to select the method of such function building. Computational difficulties of finding the values of the characteristic function in a cooperative vehicle routing game (CVRG) are caused by the large size of the problem, which makes it unacceptable to use exact methods for solving wide class of routing problems with a comparatively small number of customers to be served (Baldacci et al., 2012; Kallehauge, 2008). At the same time, using of heuristic algorithms in the general case does not allow to guarantee fulfillment of the subadditive property of the characteristic functions, which has crucial importance for achievement of cooperative agreements and total cost reduction. Considering dynamic cooperation models, it is expedient to use imputation distribution procedures (IDP) which were first proposed by L.A.Petrosyan, as well as cooperation stability principles formulated by L.A.Petrosyan and N.A.Zenkevich (2009). In our paper we propose mathematical setting of the freight carriers cooperation problem, a new approach to building the characteristic function of the multi-period CVRG and algorithm for constructing cost sharing scheme providing strong dynamic stability of the Sub-Core to meet condition of time consistency (dynamic stability) of cooperative agreements. 2.

General Problem Statement

In this paper it is presumed that in the transportation service market there are several agents (companies) engaged in cargo transportation on a network. Each agent has a great number of customers located in nodes of network and its own resources, such as a depot and a non-empty fleet of vehicles. These companies consider various options of cooperation to reduce transportation costs. Each coalition meets the demand of customers for transportation services of all companies involved in cooperation using consolidated resources. Thus, within cooperative service, customers can be redistributed between participants in each coalition. In its turn, customers exchange between agents within a coalition would extend the set of feasible routes of consolidated fleet and provide additional possibility to improve current solution in comparison to non-cooperative case. On the other hand, when agents cooperate, the total number of customers that has to be dispatched at once to vehicles substantially increases along with the computational complexity of finding routes minimizing the total transportation costs of the coalition. Therefore, in operative decision-making environment there is a lack of time for quick assignment of customers to optimal routes, since this problem belongs to the class of NP-hard problems. To find a good solution for vehicle routing problem with several depots the adaptation of well-known metaheuristic algorithm proposed by Ropke and Pisinger (2006) may be used for each coalition. Once the routes with minimum transportation costs for each possible coalition are found, the characteristic function value of the cooperative routing game can be calculated. To ensure that the agents have the motivation to form a coalition, the characteristic function has to satisfy subadditivity condition. In general, heuristic algorithms that find minimum of transportation costs of a coalition do not guarantee this property. Therefore, a special metaheuris-

Multi-period Cooperative Vehicle Routing Games

351

tic algorithms providing subadditive property of the characteristic function has to be proposed for VRP. The solution of VRP is a set of vehicle routes, such that all customers are visited exactly once, each route starts and ends in a depot, the length of each route is limited to predetermined value. Additionally, in the vehicle routing problems with time windows each customer has specified service time and must be visited within the specified time interval. Generally, the objective of such problems is to minimize the total length of routes. In real-life cases the number of used vehicles has more significant impact on the total transportation costs, because the cost of using additional vehicle appears to be much higher than benefit from shorter routes. 3.

Mathematical Model of Static CVRG

Let N be a set of companies engaged in transportation service in the same transport network. Each company i ∈ N provides transportation service to the given set of customers Ai . Each customer is served by only one company. Companies are considering possibilities of cooperation to reduce total transportation costs. Let S ⊆ N be a proper coalition of companies (players or agents in the static CVRG with transferable utilities) to be formed. The total cost of the coalition S consists of two parts: costs of used vehicles and direct transportation costs. In this paper two assumptions are made concerning costs: – direct transportation cost is linear function of the total length of routes; – fleet of vehicles of coalition S includes homogenous vehicles of all companies from this coalition, and each vehicle has fixed utilization price. It is also assumed that each coalition has unlimited number of identical vehicles and pays only for those that are used in transportation service. Thus, the cost function may be represented as follows: cost(S, pS ) = aS · N T (S, pS ) + bS · T T C(S, pS ), where pS ∈ PS — feasible routing plan for the vehicles of the coalition S, PS — the finite set of feasible routing plans of the collation S; aS — the cost of one vehicle utilization for the coalition S; N T (S, pS ) — number of vehicles used by the coalition S at the particular routing plan pS ; bS — cost of one unit of the length for the coalition S; T T C(S, pS ) — the total length of routes of the coalition S at the particular routing plan pS . For the sake of simplicity, it is assumed that each company has only one depot. It is also assumed that companies may redistribute transportation costs among the collaborators using some cost sharing procedure. In order to design subadditive characteristic function of CVRG consider for coalition S ⊆ N the costs minimization problem over the set of feasible vehicle routes minpS ∈PS cost(S, pS ) (1)

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Suppose the exact minimum value of the problem (1) is equal to copt (S). In the case of using heuristic algorithm for solving this problem the obtained value of the minimum ch (S) will be not less than copt (S), that is copt (S) ≤ ch (S)

(2)

For two disjoint coalitions S ⊆ N and T ⊆ N , for any pair of feasible routing plans pS ∈ PS , pT ∈ PT , the routing plan (pS , pT ) consisting of the union of routing plans pS and pT is feasible in the routing problem for the joint coalition of carriers S ∪ T , that is (pS , pT ) ∈ PS∪T , moreover PS ∪ PT ⊆ PS∪T and AS ∪ AT = AS∪T , then it is clear that the following inequality holds copt (S ∪ T ) ≤ copt (S) + copt (T ) Taking into account the inequality (2) we have copt (S ∪ T ) ≤ ch (S) + ch (T )

(3)

Last inequality can be rewritten for the arbitrary coalition L ⊆ S and the corresponding values ch (L) and ch (S/L) copt (S) ≤ ch (S/L) + ch (L)

(4)

We define the value of characteristic function c(S) in cooperative CVRG in the following way c(S) = min{minL⊂S {c(S/L) + c(L)}, ch (S)} (5) One can notice that if we start calculation of the characteristic function c(S) with one-element coalitions and then gradually increase the size of coalitions by 1 until we obtain the value for the grand coalition N , the characteristic function designed by using (5) would fulfill the condition of the subadditive, i.e. c(S ∪ T ) ≤ c(S) + c(T ), S ⊆ N, T ⊆ N, S ∩ T = ∅

(6)

We call this algorithm for constructing characteristic function of TU-cooperative CVRG in the form (5) the direct coalition induction algorithm (DCIA). Thus, the following theorem holds. Theorem 1. The characteristic function c(S) defined by (5) of static TU-cooperative VRG and calculated using direct coalition induction algorithm satisfies subadditivity condition (6). 4.

Example of Cooperative Routing

To illustrate the algorithm implementation we consider one artificial problem of cooperation with four transport companies D1, D2, D3, D4 having demand for cargo transportation from 54, 49, 44, 53 customers. The example has been generated using one benchmark (R2_2_1) proposed by Gehring and Homberger to compare heuristic algorithms that solve vehicle routing problems with time windows. Thus, in case of full cooperation, the transportation companies together have to service 200 customers. Clients of each company are distributed evenly throughout the nodes of network where the servicing is provided. And they have wide time

353


service windows (the time interval during which servicing is possible) which allows to use less vehicles, but at the same time increases the computational complexity of the problem. Use of a little number of vehicles is also facilitated by big carrying capacity thereof as compared to the customers’ demand. The total costs will be calculated assuming that the value of use of one vehicle is 5000 condition monetary units, and the value of one unit of the route length is 5 conditional monetary units. The algorithm proposed by Ropke and Pisinger (2006) was used for solving corresponding routing problems. As a basic problem, this algorithm considers the more general problem, the particular case of which is the problem in question. In order to find efficient routes several basic heuristics were united in one algorithm with the help of the simulated annealing. One part of these heuristics removes several customers from the solution, and the other inserts them into the solution again. The adaptive mechanism tracks the performance of basic heuristics and chooses at each step of iterations two certain heuristics using obtained statistics of their previous effectiveness. Such mechanism is based on the special genetic algorithm. To diversify search process and enhance algorithm robustness the noise value is added to the value of objective function. Table 1 shows the solution of respective costs minimization problems for each coalition and values of the game characteristic function calculated using the direct coalition induction algorithm. Table 1: Solutions of costs minimization problems and values of characteristic function Coalition

Vehilces Length Characteristi function value

(D1)

3

2043,4

25217,05

(D2)

3

2013,3

25066,46

(D3)

2

1852,8

19263,80

(D4)

2

2245,9

21229,39

(D1, D2)

3

3879,6

34398,07

(D1, D3)

4

2750,9

33754,39

(D1, D4)

3

3573,3

32866,71

(D2, D3)

3

2949,2

29745,90

(D2, D4)

3

3130,7

30653,36

(D3, D4)

4

2502,3

32511,41

(D1, D2, D3)

4

4127,5

40637,45

(D1, D2, D4)

4

4166,2

40830,99

(D1, D3, D4)

5

3569,8

42848,97

(D2, D3, D4)

4

3570,8

37853,79

(D1, D2, D3, D4)

5

4575,6

47878,11

As one can see in Table 1, the sum of minimum costs of companies, if there is no cooperation, is equal to 90776.70. The minimum total costs in the case of cooperation are equal to 47878.11. Thus, the savings from cooperation in this example are about 47 per cent.

354


To share the total costs among players Shapley value is used. As Table 2 shows, considerable reduction of costs of the companies can be achieved in comparison to their minimum costs prior to cooperation. It should be noted that reduction of costs of each of the companies under cooperation after redistribution of the total costs using the Shapley value was 43 to 54 percent. Table 2: Solution for maximum coalition and cost sharing using Shapley value Coalition Shapley Minimum costs Cost reduction value without cooperation coefficient (D1) (D2) (D3) (D4)

5.

14382,5 11630,3 10571,1 11294,1

25217,0 25066,5 19263,8 21229,4

0,43 0,54 0,45 0,47

Dynamic Model of CVRG

Suppose CVRG has duration from 0 to T . Let the interval [0, T ] be divided by periods t0 , t1 , . . . , tm . That is [0, T ] = (t0 , t1 , . . . , tm ). Cost functions of players for the period [0, T ] and set of feasible routing plans (strategies) are determined in the same way like in section 3. It is assumed that for CVRG starting from origin t0 the characteristic function c(S, 0) is calculated by the direct coalition induction algorithm. For further calculations the following notation will be used: phN (0) — is the optimal routing plan of grand coalition N in original game, which calculated by direct coalition induction algorithm and minimizes the total costs of the coalition for the periods t0 , t1 , . . . , tm ; phS (0) — is the optimal routing plan of coalition S in origin game, which is calculated by direct coalition induction algorithm and minimizes the total costs of the coalition for the periods t0 , t1 , . . . , tm , S ⊂ N ; phS (tk ) — profile of the optimal routing plan of the coalition S in period tk , S ⊆ N , k = 1, 2, . . . , m; phN (0) = (phN (t0 ), . . . , phN (tm )) — vector of profiles of optimal routing plan; phN,i (tk ) — optimal routing plan for vehicles of company i ∈ N in period tk as part of optimal plan phN (tk ); c(S, k, phN (t0 ), . . . , phN (tk−1 )) — value of minimal total costs of coalition S ⊆ N after implementation the optimal routing plan calculated by direct coalition induction algorithm. One of the important issues of successful implementation of the routing plan phN (0) = (phN (t0 ), . . . , phN (tm )) during all periods of the game is optimality of each restriction of the original optimal plan phN (0) on the set of periods tk , tk+1 , . . . , tm for k = 1, 2, . . . , m. We denote this restriction of the plan phN (0) by phN (k)) = (phN (tk ), . . . , phN (tm )). When restriction of original optimal plan phN (0) appears to be not optimal for at least one period tk , k = 1, 2, . . . , m, we call this plan time inconsistent. Notice that unlike Bellman optimality principle it might be happened in routing optimization because of using heuristics instead of exact methods. To make

355


an attempt to overcome time inconsistency of the plan formed by heuristic algorithm we propose to realize along originally calculated plan phN (0) = (phN (t0 ), . . . , phN (tm )) the following iterative coalition induction algorithm (ICIA). Iterative coalition induction algorithm. Step 1. Put k = 0. Step 2. Assume that plan phN (k) has been implemented within the period tk . We exclude nodes visited in the period tk from the set of customerâĂŹs nodes and consider current CVRG(tk+1 ) under new conditions for the set of customers to be served in periods tk+1 , . . . , t( m) and new depots location taking into account current positions of vehicles at the end of routs executed in period tk . If heuristic algorithm proposes new routing plan (pN (k + 1)) = (pN (tk+1 ), . . . , pN (tm )) in current CVRG(tk ) which gives less total costs for the grand coalition N for periods tk+1 , . . . , tm than the plan phN (k + 1) = (phN (tk+1 ), . . . , phN (tm )) we make the following substitution to improve the plan considered for implementation before period tk+1 :

phN (0)

=

(

(phN (t0 ), . . . , phN (tk )) – within the periods t0 , t1 , . . . , tk (pN (tk+1 ), . . . , pN (tm ) – within the periods tk+1 , . . . , tm

(7)

And move to step 2 putting k = k + 1 and, if k < m. If plan ((pN (k + 1))) in current CVRG(tk ) proposed by heuristic algorithm coincides with phN (k + 1) or gives bigger value of total costs for grand coalition put k = k + 1, we do not make substitution (7) and move to step 2, if k < m. In any case, if k = m go to step 3. Step 3. Stop the procedure and use for implementation plan h pN (0) = (phN (t0 ), . . . , phN (tm )) which has been gotten on the last iteration. Let phN (0) = (phN (t0 ), . . . , phN (tm )) be the optimal routing plan obtained by adjustment of the initial optimal plan with the help of the ICIA. For each period t1 , . . . , tm along optimal routing plan phN (0) = (phN (t0 ), . . . , phN (tm )) we can calculate values of characteristic function c(S, k, phN (t0 ), . . . , phN (tk−1 )) for current CVRG(tk ) using DCIA. Characteristic function for CVRG(t0 ) is c(S, 0). We can represent value of the characteristic function for the grand coalition in CVRG(t0 ) c(N, 0) =

n X m X

cost(i, phN,i (tk )) = c(N, phN (0))

i=1 k=0

When all players form grand coalition N , for optimal routing plan phN (0) the set of imputations in the cooperative game c(S, 0, phN (0)) = c(S, 0) will be determined as follows I(0, phN (0))

= {α = (α1 , α2 , . . . , αn ) : αi ≤ c({i}, 0), i = 1, . . . , n,

n X

αi = c(N, 0)}

i=1

In this paper the Sub-Core was used as a solution of the cooperative game (Zakharov and Kwon, 1999; Zakharov and Dementieva, 2004). Definition 1. Sub-Core of the cooperative game c(S, 0, phN (0)) is called the set [ SC(c(S, 0, phN (0))) = SC(c(S, 0, phN (0)), c0 (0)) (8) c0 (0)∈C0 (0)

356


where SC(c(S, 0, phN (0)), c0 (0)) =

! n n X 0 0 h = α=c −λ ci (0) − C(N, 0, pN (0)) , i=1

λ = (λ1 , λ2 , . . . , λn ) :

n X i=1

λi = 1, λi ≥ 0, i = 1, 2, . . . , n

o

and C0 is the set of solutions of the following maximization problem max

n X

ci

i=1

provided that

X i∈S

ci ≤ c(S, 0, phN (0)), S ⊂ N

We shall call the set C0 (0) as the basis of Sub-Core, any vector c0 (0) = (c01 (0), c02 (0), . . . , c0n (0)) ∈ C0 (0) — as the basis imputation of the cooperative game c(S, 0, phN (0)). By the structure the Sub-Core is not empty if and only if the Core of cooperative game with the characteristic function c(S, 0, phN (0)) is not empty, and necessary and sufficient condition for the Sub-Core (and hence the Core) to be not empty is fulfillment the following inequality X c0i (0) ≥ c(N, 0, phN ) (9) i∈N

Sub-Core in current CVRG(tk ) is determined by the same way. Presume that the Sub-Core SC(c(S, k, phN (t0 ), . . . , phN (tk−1 )), c0 (k)) is not empty for k = 1, . . . , m. Let αk = (αk1 , αk2 , . . . , αkn ) ∈ SC(c(S, k, phN (t0 ), . . . , phN (tk−1 ))), k = 0, 1, . . . , m, be vectors of cost sharing in the current games c(S, k, phN (t0 ), . . . , phN (tk−1 )). In this case costs of any coalition determined in the current game in accordance with the vector αk will not exceed the values of the characteristic function for this coalition for any value k = 0, 1, . . . , m. Thus, there is no coalition interested in leaving the agreement at any stage of the game, which means strong dynamic stability of the Sub-Core. By analogy with the imputation distribution procedures (IDP) discussed e.g. in paper (Petrosyan and Zenkevich, 2009), the cost sharing procedure (CSP) βk = (β1k , β2k , . . . , βnk ) can be considered in the multistage cooperative game, where βik = αki − αk+1 , k = 0, 1, . . . , m − 1, i ∈ N i

(10)

The crucial property of such procedure is fulfillment for any player i at any stage of the game of the condition m X

βij = αki , k = 0, 1, . . . , m,

j=k

that we call condition of individual costs balance of the player i ∈ N .

357


According to the definition of the Sub-Core, the following equation is valid for the vectors αk = (αk1 , αk2 , . . . , αkn ) of cost sharing in the current games n X

αki = c(N, k, phN (0)), k = 0, 1, . . . , m

i=1

and taking into account (10), the following equation can be obtained n X i=1

βik = c(N, k, phN (t0 ), . . . , phN (tk−1 )) − c(N, k + 1, phN (t0 ), . . ., phN (tk )), k = 0, 1, . . . , m − 1

This condition will be called as condition of collective balance of coalition costs in the multistage cooperative game. Presume that the numerical value βik determines the size of payoff of the player i within the period tk to a Costs Clearing Center (CCC), which accumulates funds for covering the costs of all players in the process of implementation of the routing plan phN (0) selected for implementation by coalition N . Then the economic meaning of the condition of individual costs balance will be, that the sum of payoffs of any player to Costs Clearing Center during the entire game will be equal to the size of costs, which player have to pay in accordance with the selected optimal distribution α0 = (α01 , α02 , . . . , α0n ). And collective balance of coalitional costs will provide the possibility of covering the costs of participants of the coalition N within the same period, when these costs are made. 6.

Example of Multi-Period CVRP

As illustration of the dynamic case, consider the static problem described earlier, but assume now that the entire servicing time is divided into 3 equal periods. All vehicles that are maintained in previous period by the grand coalition, begin their movement in current period from the last serviced customer’s node. Each company participating in one or another coalition may use additional vehicles which begin their movement from the depot belonging to the company. The same heuristic algorithm as in the static case (Ropke and Pisinger, 2006) is used for finding efficient routes for CVRG in each period. To calculate characteristic function values given in Table 3 we apply algorithms DCIA and ICIA. Using the obtained values of the characteristic function find the basis of SubCore for each period. In this case all three maximization tasks have the unique solution, and thus the set C0 for each period contains only of one element. It should be noted that Sub-Core will not be empty within each period due to fulfillment of the condition (9). In order to find certain imputation belonging to Sub-Core within each period, the value of each component of the vector λ was set to 0.25. After that, values of vectors βk using the obtained sharing vectors were calculated. The calculation results are given in Table 5. Negativity of payment values means that a company does not make payment to CCC within the respective period, but receives in this period compensation from CCC. Analyzing the data of Table 5, it is become clear that conditions of individual costs balance and collective costs balance in the three-period CVRG have been fulfilled.

358


Table 3: Values of characteristic function for three periods Coalition

The characteristic function c(S, 0) c(S, 1) c(S, 2)

(D1) (D2) (D3) (D4) (D1, D2) (D1, D3) (D1, D4) (D2, D3) (D2, D4) (D3, D4) (D1, D2, D3) (D1, D2, D4) (D1, D3, D4) (D2, D3, D4) (D1, D2, D3, D4)

25217,05 25066,46 19263,80 21229,39 34398,07 33754,39 32866,71 29745,90 30653,36 32511,41 40637,45 40830,99 42848,97 37853,79 47878,11

22268,11 15418,88 20099,82 22199,77 29613,92 33180,69 34810,43 32592,22 33699,61 37683,95 40119,31 36572,44 37133,72 44562,17 38552,26

13367,79 12544,99 12327,52 12707,63 19046,92 24390,32 24110,87 19223,78 19324,51 23736,35 24834,71 20308,93 29854,66 24723,40 30364,98

Table 4: Basis of Sub-Core for three periods Sub-Core basis Period 1 Period 2 Period 3 Company 1 Company 2 Company 3 Company 4 All companies

16203 11208 13226 13420 54057

8720 15419 15980 12433 52552

9725 2782 12328 7802 32637

Table 5: Values of imputations and vectors βk Period 1 α0 β0 Company Company Company Company

1 2 3 4

14658 9663 11681 11875

9438 -2256 -799 2942

Period 2 α1 β1

Period 3 α2 β2

5220 -3937 9157 9157 11919 9705 2214 2214 12480 720 11760 11760 8933 1699 7234 7234

Multi-period Cooperative Vehicle Routing Games 7.

359

Conclusions

The main result of this paper is the method of constructing the characteristic function of cooperative routing game of freight carriers, which guarantees its subadditive property. A new algorithm is proposed for solving this problem, which is called direct coalition induction algorithm (DCIA). To upgrade optimal routing plan and values of characteristic function of grand coalition we develop iterative coalition induction algorithm (ICIA) for dynamic CVRP. Both algorithms were built on the basis of the combination of various heuristic algorithms which are appropriate for solving large-scale VRP. For implementation of algorithms a special software has been developed and used for solving sample examples. Proposed cost sharing method allow to obtain sharing distribution procedure which provide strong dynamic stability of cooperative agreement based on this Sub-Core optimality principle and time consistency of the Sub-Core in multi-period CVRG. References Agarwal, R., Ö. Ergun, L. Houghtalen and O. O. Ozener (2009). Collaboration in Cargo Transportation. In Optimization and Logistics Challenges in the Enterprise. Springer Optimization and Its Applications, 30, 373–409. Baldacci, R., A. Mingozzi and R. Roberti (2012).Recent exact algorithms for solvingthe vehicle routing problem under capacity and time window constraints. European Journal of Operational Research, 218, 1–6. Ergun, Ö., G. Kuyzu and M. W. P. Savelsbergh (2007). Shipper collaboration. Computers & Operations Research, 34, 1551–1560. Kallehauge B. (2008). Formulations and exact algorithms for the vehicle routing problem with time windows. Computers & Operations Research, 35, 2307–2330. Krajewska M. A., H. Kopfer, G. Laporte, S. Ropke and G. Zaccour (2009). Horizontal cooperation among freight carriers: request allocation and profit sharing. Journal of the Operational Research Society, 59, 1483–1491. Petrosyan, L. A. and N. A. Zenkevich (2009). Principles of dynamic stability, Mat. Teor. Igr Pril.„ 1:1, 106–123 (in Russian). Ropke S. and D. Pisinger (2006). An adaptive large neighbourhood search heuristic for the pickup and delivery problem with time windows. Transportation Science, 40, 455–472. Zakharov, V., O-Hun Kwon (1999). Selectors of the core and consistency properties. Game Theory and Applications, 4, 237–250. Zakharov V., M. Dementieva (2004). Multistage cooperative games and problem of timeconsistency. International Game Theory Review 6, 1, 1–14.

Mechanisms of Endogenous Allocation of Firms and Workers in Urban Area: from Monocentric to Polycentric City⋆ Alexandr P. Sidorov1 National Research University the Higher School of Economics, Center for Market Studies and Spatial Economics, Room K-203, b.20, Myasnitskaya st., Moscow, 101000 , Russia, and Sobolev Institute of Mathematics, 4 Acad. Koptyug avenue, 630090, Novosibirsk, Russia, E-mail: [email protected]

Abstract The purpose of paper is to investigate how the interplay of trade, commuting and communication costs shapes economy at both inter-regional and intra-urban level. Specifically, we study how trade affects the internal structure of cities and how decentralizing the production and consumption of goods in secondary employment centers allows firms located in a large city to maintain their predominance. The feature of approach is using of two-dimensional city pattern instead of the “long narrow city” model. Keywords: city structure, secondary business center, commuting cost, trade cost, communication cost.

1.

Introduction

Spatial economics has acquired new life since publication of Krugman’s (1991) pioneering paper. Combined increasing returns, imperfect competition, commodity trade and the mobility of production factors Krugman has formed his now famous “core-periphery” model. Such a combination contradicts to the mainstream paradigm of constant returns and perfect competition, which has dominated in economic theory for a long time. Furthermore, to the trade-off between increasing returns and transport costs Krugman (1980) has added a third factor: the size of spatially separated markets. The main achievement of New Economic Geography (NEG) was to show how market size interacts with scale economies internal to firms and transport costs to shape the space-economy. In NEG, the market outcome arises from the interplay between a dispersion force and an agglomeration force operating within a general equilibrium model. In Krugman (1991) and Fujita et al. (1999), the dispersion force ensures from the spatial immobility of farmers. As for the agglomeration force, Krugman (1991, p.486) noticed that circular causation a la Myrdal (1957) takes place because the following two effects reinforce each other: “manufactures production will tend to concentrate where there is a large market, but the market will be large where manufactures production is concentrated.” In this framework, however, the internal structure of regions was not accounted for. In the present paper we consider NEG models which allows for the internal structure of urban agglomerations through the introduction of a land market. To be ⋆

This work was supported by the Russian Foundation for Fundamental Researches under grants No.99-01-00146 and 96-15-96245.

Mechanisms of Endogenous Allocation of Firms and Workers in Urban Area

361

precise, we start by focusing on the causes and consequences of the internal structure of cities, because the way they are organized has a major impact of the well-being of people. In particular, housing and commuting costs, which we call urban costs, account for a large share of consumers’ expenditures. At this point we are agree with Helpman (1998) for whom urban costs are the main dispersion force at work in modern urbanized economies. In our setting, an agglomeration is structured as a monocentric city in which firms gather in a central business district. Competition for land among consumers gives rise to land rent and commuting costs that both increase with population size. In other words, our approach endows regions with an urban structure which is absent in standard NEG models. As a result, the space-economy is the outcome of the interaction between two types of mobility costs: the transport costs of commodities and the commuting costs borne by workers. Evolution of commuting costs within cities, instead of transport costs between cities, becomes the key-factor explaining how the space-economy is organized. Moreover, despite the many advantages provided by the inner city through an easy access to highly specialized services, the significant fall in communication costs has led firms or developers to form enterprise zones or edge cities (Henderson and Mitra 1996). We then go one step further by allowing firms to form secondary business centers. This analysis shows how polycentricity alleviates the urban of urban costs, which allows a big city to retain its dominant position by accommodating a large share of activities. Creation of subcenters within a city, i.e. the formation of a polycentric city, appears to be a natural way to alleviate the burden of urban costs. It is, therefore, no surprise that Anas et al. (1998) observe that “polycentricity is an increasingly prominent feature of the landscape.” Thus, the escalation of urban costs in large cities seems to prompt a redeployment of activities in a polycentric pattern, while smaller cities retain their monocentric shape. However, for this to happen, firms set up in the secondary centers must maintain a very good access to the main urban center, which requires low communication costs. Trying to explain the emergence of cities with various sizes, our framework, unlike Helpman (1998), Tabuchi (1998) and others, allows cities to be polycentric. Moreover, in contrast to Sullivan (1986), Wieand (1987), and (Helsley and Sullivan, 1991), in our treatment, there are no pre-specified locations or numbers of subcenters, and our model is a fully closed general equilibrium spatial economy. As mentioned above, emergence of additional job centers is based on the urge towards decreasing of urban costs, rather than consumer’s “propensity to big malls”, as suggested by Anas and Kim (1996). Our approach, that takes into account various types of costs (trade, commuting, and communication) is similar to Cavailhès et al. (2007) with one important exception. We drop very convenient (yet non-realistic) assumption on “long narrow city.” Our analysis is extended to the two-dimension because the geographical space in the real world is better approximated by a twodimensional space. 2. 2.1.

Model overview Spatial structure

Consider an economy with G ≥ 1 regions, separated with physical distance, one sector and two primary goods, labor and land. Each region can be urbanized by accommodating firms and workers within a city, and is formally described by a two-

362

Alexandr P. Sidorov

dimensional space X = R2 . Whenever a city exists, it has a central business district (in short CBD) located at the origin 0 ∈ X.

Firms are free to locate in the CBD or to set up in the suburbs of the metro where they form secondary business districts, SBD in short. Both the CBD and SBDs are assumed to be dimensionless. In what follows, the superscript C is used to describe variables related to the CBD, whereas S describes the variables associated with a SBDs. We consider the case where the CBD of urbanized region g is surrounded by mg ≥ 0 SBDs; mg = 0 corresponds to the case of monocentric city. Without loss of generality, we focus on the only one of SBDs, because all SBDs are assumed to be identical. Even though firms consume services supplied in each SBD, the higher-order functions (specific local public goods and non-tradable business-to-business services such as marketing, banking, insurance) are still located in the CBDs. Hence, for using such services, firms set up in a SBD must incur a communication cost, K > 0. In paper of Cavailhès et al. (2007) more general communication cost function K(xS ) = K + k · ||xS || was used, where k > 0, and ||xS || is a distance between CBD and SBD. This generalization does not change the nature of our results, though analytical calculation became more tedious. Both the CBD and the SBD are surrounded by residential areas occupied by workers. There is no overlapping between residence zones. Furthermore, as the distance between the CBD and SBD is small compared to the intercity distance, we disregard the intra-urban transport cost of goods. Note that using the more general type of communication cost with k > 0 leads to consequence that in equilibrium Central and any Secondary residence zones should be adjacent to each other. This condition is non-necessary for fixed communication cost, although the real SBD can not be placed too far from City Center. Under those various assumptions, the location, size and number of the SBDs as well as the size of the CBD will be endogenously determined. In other words, apart from the assumed existence of the CBD, the internal structure of each city is endogenous. 2.2.

Workers/Consumers

The economy is endowed with L workers, distributed across the regions, where G X population of city g is lg , i.e., lg = L. In this paper our primary focus is on g=1

the intra-city cost effects and on the trade, therefore the distribution of labor is considered as exogenous. The welfare of a worker depends on her consumption of the following three goods. The first good is unproduced and homogeneous. It is assumed to be costlessly tradable and chosen as the numéraire. The second good is produced as a continuum n of varieties of a horizontally differentiated good under monopolistic competition and increasing returns, using labor as the only input. Any variety of this good can be shipped from one city to the other at a unit cost of τ > 0 units of the numéraire. The third good is land; without loss of generality, we set the opportunity cost of land to zero. Each worker living in city 1 ≤ g ≤ G consumes a residential plot of fixed size chosen as the unit of area. The worker also chooses a quantity q(i) of variety i ∈ [0, n], and a quantity q0 of the numéraire. She is endowed with one unit of labor, which is supplied absolutely inelastically.


363

Preferences over the differentiated product and the numéraire are identical across workers and cities and represented by Ottaviano’s quasi-linear utility function

U (q0 ; q(i), i ∈ [0, n]) = α

Zn 0

q(i)di −

β 2

Zn 0

 n 2 Z γ [q(i)]2 di −  q(i)di + q0 2

(1)

0

where α, β, γ > 0. Demand for these products (provided that job and location are already chosen) is determined by maximizing of utility subject to the budget constraint Zn ALRg p(i)q(i)di + q0 + Rg (x) + Tg (x) = wg (x) + , (2) lg 0

where Rg (x) is the land rent prevailing at location x, Tg (x) is commuting cost, Z wg (x) is the wage, and ARLg = Rg (x)dx is an aggregated land rent in the city x∈X

g. This form of the budget constraint suggests that there are no landlords, who appropriate the land rent, moving it out of city budget. In other words, land is in a joint ownership of all citizen. Each worker commutes to her employment center – without cross-commuting – and bears a unit commuting cost given by t > 0, so that for the worker located at x the commuting cost, Tg (x), is either t||x|| or t||x−xSg || according to the employment center. Moreover, the wage wg (x) depends only on type of employment center and takes one of two possible values: wage in CBD, wgC , or wage in SBD, wgS , which is uniform across all SBDs. Thus, the budget constraint of an individual working in the CBD is as follows Zn 0

p(i)q(i)di + q0g + RgC (x) + t||x|| = wgC +

ALRg , lg

(3)

while for individuals working in the SBD, located at xSg , it takes the form Zn 0

2.3.

p(i)q(i)di + q0g + RgS (x) + t||x-xSg || = wgS +

ALRg . lg

(4)

Firms

Our basic assumption on the manufacturing technology is that producing q(i) units of variety i requires a given number ϕ of labor units. One may assume that producing one unit of variety i requires additionally c ≥ 0 units of numéraire. This is not significant generalization, however, because this model is technically equivalent to one with c = 0 (see Ottaviano et al., 2002). There is no scope economy so that, due to increasing returns to scale, there is a one-to-one relationship between firms and varieties. Thus, the total number of firms is given by n = L/ϕ. Labor market clearing implies that the number of firms located (or varieties produced) in city g is such that ng = λg n, where λg = lg /L stands for the share of workers residing in g. Denote by ΠgC (respectively ΠgS ) the

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Alexandr P. Sidorov

profit of a firm set up in the CBD of city g (respectively the SBD). When the firm producing variety i is located in the CBD, its profit function is given by: ΠgC (i) = Ig (i) − ϕ · wgC , where Ig (i) = pgg (i) · Qgg (i) +

X

f 6=g

(5)

(pgf (i) − τ ) · Qgf (i)

stands for the firm’s revenue earned from local sales Qgg (i) and from exports Qgf (i) from city g to various cities f . When the firm sets up in the SBD of the same city, its profit function becomes: ΠgS (i) = Ig (i) − ϕ · wgS − K.

(6)

The firm’s revenue is the same as in the CBD because shipping varieties within the city is costless, so that prices and outputs do not depend on firm’s location in the city. 3.

Urban Costs and Decentralization within a City

A city equilibrium is such that each individual maximizes her utility subject to her budget constraint, each firm maximizes its profits, and markets clear. Individuals choose their workplace (CBD or SBD) and their residential location with respect to given wages and land rents. In each workplace, the equilibrium wages are determined by a bidding process in which firms compete for workers by offering them higher wages until no firm can profitably enter the market. Given such equilibrium wages and the location of workers, firms choose to locate either in the CBD or in the SBD. At the city equilibrium, no firm has an incentive to change place within the city. To ease the burden of notation, we drop the subscript g. 3.1. Land rents and Wage wedge Let Ψ C (x) and Ψ S (x) be the bid rent at x ∈ X of an individual working, respectively, in the CBD and in the representative SBD. Land is allocated to the highest bidder. An opportunity cost of land (e.g., for agricultural use) is assumed to be zero. Urban costs (commuting and communication) increase with Euclidean distance, thus “efficient” shapes of both Central and Secondary residence zones are circles. All locations with the same distance to the corresponding Business District (Central or Secondary) are equivalent with respect to urban cots. Because there is only one type of labor, at the city equilibrium it must be that the housing rent R(x) = max Ψ C (x), Ψ S (x), 0 . Within each city, a worker chooses her location so as to maximize her utility U (q0 , q(i); i ∈ [0, n]) under the corresponding budget constraint, (3) or (4). Because of the fixed lot size assumption, at the city equilibrium the value of the equilibrium consumption of the nonspatial goods Zn

p(i)q(i)di + q0 = E

(7)

0

is the same regardless of the worker’s location: wC +

ALR ALR −RC (x′ )−t||x′ || = E C (x′ ) ≡ E S (x′′ ) = wS + −RS (x′′ )−t||x′′ −xS || l l


365

for all x′ , x′′ , belonging to CBD and SBD residence zones, respectively. To ensure this, we assume for now, that the share of firms located in the CBD, θ, is given, then (1 − θ)/m is the share of firms in each SBD. Proposition 1. For any given city population l, SBD number m, and CBD share of firms θ: i) Central zone radius rC and SBD zone radius rS are as follows: r r θl (1 − θ)l C S r = , r = . (8) π mπ

ii) The following land rent function equalizes the disposable income E for all central and suburb residence locations x: ) ( r r θl (1 − θ)l S − ||x||, − ||xk − x|| , (9) R(x) = t · max 0, 1≤k≤m π mπ m where xSk k=1 is a set of all SBD locations. iii) Redistributed aggregated land rent: r Z ALR 1 t l 3/2 (1 − θ)3/2 √ = R(x)dx = · θ + . l l 3 π m

(10)

X

iv) In equilibrium there exists the positive wage wedge between CBD and SBD ! r r θl (1 − θ)l C S w −w = t· − (11) π mπ which is non-negative for all θ ∈

1 ,1 . 1+m

For analytical proof see Appendix. Figure 1 presents the plot of function R(x) for m = 4. 3.2. Urban Costs Let’s define urban cost function as a sum of rent and commuting costs minus the ALR 1 individual share of aggregated land rent . Due to (9) and (10) these urban l costs are as follows r r ALR t θl l 3/2 (1 − θ)3/2 C C √ Cu = Ψ (x) + t||x|| − =t − · θ + , l 3 π r m rπ ALR (1 − θ)l t l 3/2 (1 − θ)3/2 S S S √ Cu = Ψ (x) + t||x − x || − =t − · θ + . l π 3 π m (12) The city equilibrium implies that the identity wC − CuC = wS − CuS holds. In these terms, the wage wedge identity may be rewritten as a difference between urban costs in CBD and SBD: wC − wS = CuC − CuS . 1

For technical reasons it is convenient to treat ALR as some kind of rent compensation, l subtracting it from costs rather adding to wage.

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Alexandr P. Sidorov

Fig. 1: Rent function R(x)

3.3.

Equilibrium city structure

Regarding the labor markets, the equilibrium wages of workers are determined by the zero-profit condition. In other words, operating profits are completely absorbed by the wage bill. Hence, the equilibrium wage rates in the CBD and in the SBDs must satisfy the conditions Π C (wC∗ ) = 0 and Π S (wS∗ ) = 0, respectively. Thus, setting (5) (respectively (6)) equal to zero, solving for wC∗ (respectively wS∗ ), we get: I −K I (13) wC∗ = , wS∗ = ϕ ϕ K > 0, due to (8). Comparing the previous formula with (11) ϕ we obtain that CBD share of firms, θ satisfies the identity p √ √ ϕt mθl = K mπ + ϕt (1 − θ)l. (14)

Hence wC∗ − wS∗ =

Admissible solution θ∗ of equation (14) will be referred as equilibrium CBD share.

πK 2 then the unique solution of equation (14) is θ∗ = 1 ϕ2 t2 with m = 0, i.e. city is monocentric; πK 2 ii) Let l > 2 2 then for each m ≥ 1 equation (14) has unique solution θ∗ ∈ ϕ t 1 , 1 , i.e. there exists a unique equilibrium SBD share of firms. 1+m iii) The CBD share of firms θ∗ decreases with respect to population l, number of SBDs m and commuting costs t. Moreover, θ∗ increases with respect to communication cost K and Proposition 2. i) Let l ≤

lim θ∗ = lim θ∗ = lim θ∗ =

l→∞

t→∞

For analytical proof see Appendix.

K→0

1 . 1+m

(15)


367

r

l and note that it is in fact a radius of monocentric π πK 2 city with population l. Inequality l < 2 2 holds if and only if ϕt · rM (l) > K. The ϕ t left-hand side of this inequality is total commuting costs of firm’s workers, residing at periphery of monocentric city in case of firm’s location at CBD. To hire ϕ workers from periphery, firm should compensate their commuting costs in wage. On the other hand, locating the firm at the periphery causes the lesser communication cost K. Thus, producing on periphery (in SBD) is more efficient for new firm entering the industry. For any given K we obtain minimum polycentric city population: πK 2 lP = 2 2 . If city population l ≤ lP the corresponding central share θ∗ ≡ 1, i.e. ϕ t city pattern is monocentric. It is not surprising that increasing in commuting costs t leads to lager dispersion of firms and workers. For very large magnitude of t, communication costs K become negligible and the distribution of production across all business centers is almost uniform. Remark 1. Let r (l) = M

Substituting equilibrium SBD share θ∗ (m, l, t) into the urban cost function # r r " r θl t l 1−θ C 3/2 Cu = t − · θ + (1 − θ) π 3 π m and taking into account that r

√ √ 1 − θ∗ K π ∗ √ , = θ − m ϕ·t l

which follows from equation (14), we obtain that the urban cost function r 2t θ∗ (l, m, t) · l K C Cu (l, m, t) = + · (1 − θ∗ (l, m, t)). 3 π 3ϕ

(16)

In particular,

r 2 l = t for m = 0 and l ≥ 0 π r3 2 l πK 2 CuC (l, m, t) = t for all m > 0 and l ≤ 2 2 , 3 π ϕ t because in these cases θ∗ ≡ 1. CuC (l, 0, t)

Proposition 3. Function CuC (l, m, t) is continuous for all m ≥ 0, l ≥ 0, t ≥ 0 and continuously differentiable function for m > 0, l > 0, t > 0. Moreover, CuC (l, m, t) strictly increases with respect to l and t, strictly decreases with respect to m for all l > lP . For analytical proof see Appendix. Figure 2 represents results of Proposition 3 in visual way as simulation in Wolfram’s Mathematica 8.0. Remark 2. Note that urban cost function Cu is concave with respect to l. It may reflect the fact that the housing price at periphery of residence zone increases with l sufficiently slow. The newcomers reside at the periphery, where the housing rent is very small. Moreover, unlike the linear model, in two-dimensional case this periphery enlarges as the city population grows. Though immigration increases competition for housing, an increment of the per capita urban costs Cu is less than before.

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Alexandr P. Sidorov

Fig. 2: Comparative statics of urban costs

4.

Inter-City Equilibrium

Until now we studied equilibrium decentralization within the city, or Intra-City equilibrium. Let’s turn to Inter-City equilibrium assuming that the city populations lg and numbers of SBD mg are given for each city g. This paper focuses mainly on trade aspects, putting aside labor migration, therefore this assumption is consistent. Some considerations on endogenezation of SBD number are discussed at the end of this Section. Equilibrium shares of firms, θg∗ , located at CBD, may be obtained independently, as solutions of equation (14) for each city g. These shares, in turn, allow to determine the urban costs Cug , which do not depend on inter-city trade (and even on existence of other cities). On the contrary, wage   X 1 wgC = pgg (i) · Qgg (i) + (pgf (i) − τ ) · Qgf (i) , ϕ f 6=g

substantially depends on trade, as well as consumer’s utility U (q0 ; q(i), i ∈ [0, n]). Moreover, if trade costs are too large, e.g., τ ≥ pgf (i), export is non-profitable and firms choose the domestic sales only, which implies wgC =

pgg (i) · Qgg (i) . ϕ

Now we split the study of equilibrium into two sub-cases: Equilibrium under Autarchy and Equilibrium with Bilateral Trade. 4.1. Equilibrium under Autarchy This case suggests that equilibrium is separately established for each city, hence we may drop subscript g and consider the city with population l and SBD number m. Moreover, assume that the number of firms n is given and condition wC − CuC > 0 holds. What determines n and how to provide this consumers’ “surviving condition” will be discussed at the end of this subsection. Representative consumer maximizes utility

U (q0 ; q(i), i ∈ [0, n]) = α

Zn 0

β q(i)di − 2

Zn 0

 n 2 Z γ [q(i)]2 di −  q(i)di + q0 2 0

Mechanisms of Endogenous Allocation of Firms and Workers in Urban Area subject to

Zn 0

369

p(i)q(i)di + q0 = wC − CuC ,

First of all, recall some well-known results concerning consumer’s problem with this form of utility. Lemma 1. Consumer’s demand is linear function q(i) = where P =

Rn

α 1 γ − p(i) + · P, β + γn β (β + γn)β

p(i)di is price index. Equilibrium prices and demand of representative

0

are uniform by goods p∗ (i) ≡ p∗ =

αβ α , q ∗ (i) ≡ q ∗ = . 2β + γn 2β + γn

Consumer’s surplus at equilibrium is equal to CS =

α2 n(β + γn) . 2(2β + γn)2

For analytical proof see Ottaviano et al. (2002). Using this lemma and taking into l account that n = we obtain the terms of equilibrium wage at CBD ϕ wC∗ =

l · p∗ · q ∗ α2 βϕl = 2 ϕ (2βϕ + γl)

and consumer’s surplus

α2 (βϕ + γl)l , 2(2βϕ + γl)2 which does not depend on consumer residence. Moreover sum of wage and consumer surplus (urban gains, for short) is CS =

C∗ GC = u = CS + w

α2 (3βϕ + γl)l . 2(2βϕ + γl)2

Finally, consumer’s welfare in CBD is a difference of urban gains and urban costs V C = CS + wC∗ − CuC .

Similar to CBD we may calculate the corresponding SBD’s characteristics: wage ! r r α2 βϕl θ∗ l (1 − θ∗ )l S∗ w = − , 2 −t π mπ (2βϕ + γl) urban gains GSu = CS + wS∗

α2 (3βϕ + γl)l = −t 2(2βϕ + γl)

r

θ∗ l − π

r

(1 − θ∗ )l mπ

where θ∗ is solution of equation (14). Note that indirect utility V S = CS + wS∗ − CuS ≡ CS + wC∗ − CuC = V C .

!

,

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Alexandr P. Sidorov

Proposition 4. Wage function wC∗ (l) strictly increases for all 0 ≤ l < strictly convex for l >

2βϕ . Moreover, γ

lim wC∗ (l) = 0, wC∗ (0) = 0,

l→+∞

2βϕ and γ

∂wC∗ α2 (0) = < +∞. ∂l 2

Urban gains GC u (l) strictly increase for all l ≥ 0, lim GC u (l) =

l→+∞

α2 , GC u (0) = 0. 2γ

Proof of this proposition is straightforward from the formulas of wC∗ (l) and GC u (l). “Surviving” condition It is obvious that city equilibrium is consistent only if disposable income wC∗ (l) − CuC (l, m, t) ≥ 0, which is called Surviving condition. Feasibility of this condition depends on magnitude of commuting cost t: wage function wC∗ is bounded and does not depend on t, while urban cost CuC (l, m, t) increases unrestrictedly with t. As result, very large commuting cost makes the city formation impossible. K 3α2 Proposition 5. Let inequality < holds, then for any commuting cost t ∈ ϕ 16γ r K πγ 0, and any given SBD number m ≥ 0 there exist numbers 0 < lmin (m, t) < ϕ 2βϕ lmax (m, t) < ∞, such that inequality wC (l) − CuC (l, m, t) ≥ 0 holds if and only if lmin (m, t) ≤ l ≤ lmax (m, t). Moreover, if m′ > m, then lmin (m′ , t) ≡ lmin (m, t) < lmax (m, t) ≤ lmax (m′ , t) and lP < l∗ ⇒ lmax (m, t) ≤ lmax (m′ , t). For analytical proof see Appendix.

K 3α2 < is equivalent to ϕ 16γ r α2 2t lP 2K C∗ C P = max w (l) > Cu (l , m, t) = ≡ , l≥0 8γ 3 π 3ϕ

Remark 3. Note that inequality

which implies that the maximum possible wage exceeds the urban costs in the city with minimum polycentric city population lP . The lack of this condition means that the production transfer to SBD is ineffective, because per employee communication K cost is too large. ϕ Increasing of m broadens interval [lmin (m, t), lmax (m, t)] (to be more precise, lmin is not affected by changes in SBD number, while lmax increases with respect to m). Moreover, disposable income wC (l) − CuC (l, m, t) and welfare V C = GC u (l) − CuC (l, m, t) both increases with respect to m for all l > lP . Figure 3 illustrates the equilibrium existence under autarchy and comparative statics of lmax with respect to m using simulation in Wolfram’s Mathematica 8.0. Remark 4. Previous considerations show that autarchy may be very restrictive to the city sizes: city survives only if its size exceeds the lower threshold lmin > 0


371

Fig. 3: Comparative statics of the population limits

and does not exceed the upper one lmax . It is not surprising, because self-sufficient settlement of industrial type may exists only if its population is sufficiently large. Moreover, unrestrictedly growing urban costs (in particular, commuting cost) eventually stop the city growth. Developing of the city infrastructure (i.e. increasing in m) shifts up the upper bound lmax , but cannot affect the lower critical point lmin . 4.2. Endogenous SBD number The concluding remark concerns the question: How to endogenize SBD number? There is no simple and unambiguous answer, because in practice it depends on many factors. One of the main questions is “Who can afford the building of additional suburb?” If answer is “None”, we find ourself in setting with predefined number of SBDs (like model of Cavailhès et al., 2007). Otherwise, we assume that decision is up to ‘City Developer’, who takes into account the social welfare considerations. For example, when city population reaches the upper bound lmax , an increasing the number of subcenters is urgently needed. Let’s determine the following “compelled” SBD number for given population l and commuting cost t: m∗ (l, t) = min {m | l ≤ lmax (m, t)} . Proposition 6. SBD number m∗ is non-decreasing function with respect to the city population l and commuting costs t, i.e., for all l′ > l, t′ > t the following inequalities hold: m∗ (l′ , t) ≥ m∗ (l, t), m∗ (l, t′ ) ≥ m∗ (l, t). Proof. The statement concerning city population l is obvious: city is monocentric (m∗ = 0) until population l exceeds lmax (0, t). By Proposition 5 upper bound lmax (1, t) > lmax (0, t), thus while l ≤ lmax (1, m) the current SBD number m∗ = 1, until l exceeds this upper bound, e.t.c. Increasing in commuting cost leads to decreasing of lmax (m, t) = sup l | wC∗ (l) ≥ CuC (l, m, t) , because CuC (l, m, t) increases with respect to t by Proposition 3. Therefore, if l > lmax (m, t′ ) for t′ > t then to recover surviving condition we need to increase SBD number until lmax (m′ , t′ ) ≥ l.

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Alexandr P. Sidorov

Remark 5. Although this mechanism of endogenezation is not perfect, that theoretical comparative statics is fully supported by empirical evidences (see MacMillen and Smith, 2003). Anyway, it determines rather the endogenous minimum of SBD, which may be increased by some another reason, for example, to increase social welfare, i.e., total indirect utility of the city population.

Fig. 4: Disposable income and Welfare

Parametric Example Consider the numerical example of how may change the inner structure of city under increasing of population size. Parameter values are chosen as follows: ϕ = 5, K = 4, t = 1, α = 6, β = 4, γ = 1. Under these assumptions the lower bound of the city population lmin ≈ 0.75 is very small and once the city is grounded it starts to attract people, e.g., from rural neighborhood. Moreover, disposable income w − Cu increases very quickly with respect to city size


373

l at early stage, then it reaches the maximum and go down to zero when population size is close to upper population bound lmax (m). Its magnitude depends on city structure, i.e., number of SBD. For example, monicentric city reaches its maximum at lmax (0) ≈ 97.5, while for m = 3 the upper bound (or city capacity) is much larger, lmax (3) ≈ 156.5. The plots of disposable income for m = 0, 1, 2, 3 are presented at Figure 4a. However, taking into account Consumer’s Surplus along with Disposable Income we obtain that the resulting Consumer’s Welfare (i.e., Indirect utility) V = CS + w − Cu tends to grow further with respect to population size. It implies that there is a strong incentive for City Developer to increase the SBD number m, which in turn raises the city capacity. Of course, this expansion could be done “in advance”, i.e., before the population size reaches the maximum. It is not so easy to predict, however, when it happens, thus the “cautious strategy” of City Developer is presented at Figure 4b by bold line, i.e., an additional SBD appears only if capacity of the city is exhausted. Moreover, we have assumed that the building of new SBD is costless, but this is not the case in real world. Thus, the expansion m → m + 1 will be well-grounded when per capita effect (welfare gap) reaches the maximum, i.e., at current lmax (m). It can be easily observed that this welfare leap quickly decreases with any next expansion of city structure, which eventually stops the increasing of the city population. 4.3. Bilateral Trade Equilibrium The current subsection tell us what changes if trade comes to the place. To simplify description, assume that there are two cities, Home and Foreign. Let λ be the share of workers residing in Home city, then populations of both cities are lH = λL and lF = (1 − λ)L, respectively. Moreover, the equilibrium masses of firms are nH = lH /ϕ = λ · n, nF = lF /ϕ = (1 − λ) · n, where n = L/ϕ is a total mass of firms in the world. Demands of Home representative consumer for domestic and imported differentiated goods, qHH (i) and qF H (i) respectively, are determined as solution of consumer problem max U (q0 ; q(i), i ∈ [0, nH + nF ]) subject to ZnH

pHH (i)qHH (i)di +

nH Z+nF nH

0

C C pF H (i)qF H (i)di + q0 = EH = wH − CH .

(17)

Similarly demands of Foreign representative consumer, qF F (i) and qHF (i), are determined as solution of

subject to ZnF 0

max U (q0 ; q(i), i ∈ [0, nH + nF ])

pF F (i)qF F (i)di +

nH Z+nF nF

pHF (i)qHF (i)di + q0 = EF = wFC − CFC .

Facing these demands, firms maximize profits IH (i) = λL · pHH (i) · qHH (i) + (1 − λ)L · [pHF (i) − τ ] · qHF (i) IF (i) = (1 − λ)L · pF F (i) · qF F (i) + λL · [pF H (i) − τ ] · qF H (i)

(18)

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Alexandr P. Sidorov

and obtain optimal (equilibrium) prices and quantities. Zero-profit condition (13) determines equilibrium wages. It should be mentioned that bilateral trade is profitable only if trade costs τ are sufficiently small: pHF (i) > τ and pF H (i) > τ . The following results are well-known, see, for example, original papers of Ottaviano et al. (2002) and Cavailhès et al. (2007). Lemma 2. Trade equilibrium prices are uniform by goods p∗HH (i) ≡ p∗HH =

2αβ + τ γnF 2αβ + τ γnH , p∗F F (i) ≡ p∗F F = , 2(2β + γn) 2(2β + γn)

p∗HF = p∗F F +

τ τ , p∗F H = p∗HH + , 2 2

as well as equilibrium demands ∗ qHH (i)

≡

∗ qHH

qF∗ F (i) ≡ qF∗ F Consumer’s surplus

1 τγ τ ∗ ∗ α − pHH + nF , qF∗ H = qHH − = β + γn 2β 2β 1 τγ τ ∗ = α − p∗F F + nH , qHF = qF∗ F − β + γn 2β 2β

α2 n α − · [p∗HH · nH + p∗F H · nF ] + 2(β + γn) β + γn i 1 h ∗ 2 γ 2 2 + · (pHH ) · nH + (p∗F H ) · nF − · [p∗HH · nH + p∗F H · nF ] 2β 2β · (β + γn) CSH =

Bilateral trade is profitable if τ < τtrade =

2αβ . 2β + γn

For analytical proof see Appendix. λL (1 − λ)L L Substituting for nH , for nF and for n we obtain the equilibrium ϕ ϕ ϕ prices and quantities for the Bilateral Trade Equilibrium. We focus on the Home city only, considerations for Foreign city are similar, mutatis mutandis. Without loss of generality, we may assume that L ≤ lmax (mH ), which implies, in particular, C∗ wH (1) ≥ CuC (1). It allow us to consider the whole unit interval (0, 1) as a set of admissible values for λ instead of truncation (0, lmax (mH )/L). Bilateral trade changes magnitudes of wage, consumer’s surplus and indirect utility in comparison to autarchy case. To discriminate these cases, we add τ to notions of values, which are affected by trade. Recall that urban costs Cu (λ) does not depend on τ . The following results are well-known (see, for example, Ottaviano et al. (2002) and Cavailhès et al. (2007)). Lemma 3. Home Equilibrium wage C∗ wH (λ, τ ) =

βϕL (2βϕ + γL)2

"

# 2 2 τ γL τ γL α+ (1 − λ) · λ + (α − τ ) − (1 − λ) · (1 − λ) 2βϕ 2βϕ

(19)


375

is strictly concave function, increasing at λ = 0. Home Consumer’s Surplus CSH (λ, τ ) =

α2 L αL − · [p∗HH · λ + p∗F H · (1 − λ)] + 2(βϕ + γL) βϕ + γL

i L h ∗ 2 γL2 2 2 · (pHH ) · λ + (p∗F H ) · (1 − λ) − ·[p∗ · λ + p∗F H · (1 − λ)] 2βϕ 2βϕ · (βϕ + γL) HH (20) is strictly increasing and concave function of λ. +

Proof is straightforward (though tedious) from Lemma 2. Proposition 7. i) There exists 0 < τ ∗ < τtrade such that for all τ ∈ (0, τ ∗ ) inequality wC∗ (λ) > Cu (λ) holds for all λ ∈ (0, 1). ii) There exists 0 < τ ∗∗ < τtrade such that for all τ ∈ (0, τ ∗∗ ) indirect utility with trade C∗ C VH (λ, τ ) = CSH (λ, τ ) + wH (λ, τ ) − CuH (λ) C∗ exceeds the corresponding utility under autarchy VH (λ) = CSH (λ) + wH (λ) − C CuH (λ) for all λ ∈ (0, 1).

For analytical proof see Appendix. Typical results of simulation are presented at Figure 5.

Fig. 5: Autarchy and Trade

Remark 6. Proposition 7(i) implies that sufficiently free trade cancels the lower bound of city size lmin , i.e. small cities could survive, trading with the larger ones. It looks like small city became quasi-SBD for large one, replacing communication cost with trade cost. On the other hand, trade cannot cancel the upper bound, or maximum city capacity. Thus, all considerations endogenous SBD number from subsection 3.2 are still valid. This proposition cannot be generalized for all τ ∈ (0, τtrade ). Computer simulations show that for τ sufficiently close to τtrade both statements, (i) and (ii), are violated.

376 5.

Alexandr P. Sidorov Conclusion

Paradigm of linear city is well suited for both actual “long narrow cities” and monocentric “two-dimensional”, because in this case location may be characterized by scalar value – distance from Central Business District. In case of polycentricity – especially, with multiple Secondary Business Districts – linear model can’t include all range of possibilities, being limited at most by two SBDs. Two-dimensional polycentric model, presented in this paper, lacks this disadvantage, while it is still tractable and intuitive. The results obtained in presented paper are of two kinds: some of them are common for both linear and two-dimensional models, while other are specific for two-dimensional model with several Secondary Business Districts. We discuss here these results, focusing on the specific ones. Proposition 2 on Existence and Uniqueness of equilibrium CBD share implies that polycentric structure may exists only if population of city exceeds the certain threshold, i.e., too small city cannot bear the burden of polycentricity. This natural result is not 2D specific, nevertheless, it contains the statement that city with population beyond this threshold, could have any number of SBDs. Moreover, increasing in this number implies that per capita urban costs strictly decrease (see Proposition 3). It results in increasing (ceteris paribus) of disposable income and indirect utility of the city residents, therefore, developing of the inner city structure may be an important policy instrument. It is obvious, that positiveness of disposable income is necessary condition for city residents. One of results obtained in this paper is that disposable income is positive if and only if city population is not less than strictly positive lower threshold a do not exceeds the finite upper bound (see Proposition 5). It means that the effective production (with increasing return to scale) cannot be developed on the base of too small settlement, and, vice versa, very large city cannot survive because of too heavy burden of urban costs. Increasing in SBD number shifts up the upper threshold (i.e., increases city capacity), therefore, extensive development of the city structure can be an effective policy instrument for sufficiently large cities (see Proposition 5). It cannot help, however, small cities to survive as industrial settlements. Changes in city structure is mainly an instrument of inner policy, while change in trade openness may results outwards. Moreover, sufficiently high level of trade openness (i.e., sufficiently small trade costs) shifts down to zero the lower threshold of city population (see Proposition 6). It means that under condition of almost free trade, small cities could survive as satellites of large ones. Another benefit of sufficiently free trade is that real wage (indirect utility) increases for residents in all cities, not depending on their sizes (see Proposition 6), although this effect is more significant for small cities. It increases the relative attractiveness for the labor inflow. This inflow may result in overpopulation of city with given number of SBDs. To avoid this overpopulation, City Developer may increase the current SBD number, which increases city capacity. Mechanism of determining of endogenous minimum SBD number was suggested in Section 3.3, which is consistent with empirical evidences (see Proposition 7).


377

APPENDIX Proof of Proposition 1 The land supply in equilibrium should equalize (inelastic) land demand π · rC

2

+ m · π · rS

2

= l · 1,

where rC is radius of central zone, rS is radius of single suburb. On the other hand, for given CBD’s share of firms, θ, the labor market clearing in CBD (without cross2 commuting) implies π · rC = θl. Therefore, r r r r θl (1 − θ)l θl (1 − θ)l C S S C S , r = , ||x || = r + r = + . y=r = π mπ π mπ The budget constraint of an individual residing at point x and working in the CBD implies that ALR E C (x) = wC + − Ψ C (x) − t||x||, l whereas the budget constraint of an individual working in the SBD is E S (x) = wS +

ALR − Ψ S (x) − t||x − xS ||. l

Note that equalizing condition E C (x) ≡ E S (x) ≡ const implies Ψ C (x) = A1 − t||x||, Ψ S (x) = A2 − t||x − xS ||, where A1 , A2 do not depend on x. On the other hand, worker living at the border of the CBD residential area (i.e., at the point y = rC of the SBD residential area closest to CBD, see Figure 1b) is indifferent to the decisions of working in the CBD or in the SBD. Moreover, for the border location y an identities Ψ C (y) = Ψ S (y) = 0 hold, because there is no difference for landlord where to rent out this plot of land: to Central city, to Suburb or for agricultural use. Therefore, r r θl (1 − θ)l S A1 = ty = t , A2 = t · (x − y) = t . π mπ As result, we obtain ALR 1 = l l

Z

X

t R(x)dx = · 3

r

l 3/2 (1 − θ)3/2 √ θ + . π m

Note that no need to integrate actually this function. We may simply apply the 1 well-known formula of the cone volume V = πh · r2 , where h is a hight and r is a 3 radius of the base of cone. C S C S Moreover, ! E (y) − E (y) = 0 implies w − w = A1 − A2 = ran identity r θl (1 − θ)l t· − . It means that the difference in the wages paid in the π mπ CBD and in the SBD compensates exactly the worker for the difference in the corresponding commuting costs. The wage wedge wC − wS is positive as long as 1 θ> , thus implying that the size of the CBD exceeds the size of each SBD. 1+m

378

Alexandr P. Sidorov

Proof of Proposition 2 There is one-to-one correspondence between θ ∈ [0, 1] and α ∈ [0, θ = cos2 α. Substituting it into equation r r θl (1 − θ)l = K + ϕt ϕt π mπ

π ] given by 2

we obtain, after simple transformations, the following one √ sin α K π F (α, l, m, t) := cos α − √ − √ = 0. m ϕt l Note that,

and

(21)

√ K π πK 2 √ < 1 ⇐⇒ l > lP = 2 2 ϕ t ϕt l ∂F cos α = − sin α − √ < 0. ∂α m

Consider three possible cases: √ α < 0 and equation (21) has no i) l < lP then F (α, l, m, t) < (cos α − 1) − sin m roots. √ α = 0 if and only if cos α = 1, ii) l = lP then F (α, l, m, t) = (cos α − 1) − sin m ∗ which implies θ = 1. √ √ K π 1 K π π P iii) l > l then F (0, l, m, t) = 1−0− √ > 0 and F ( , m, l) = 0 − √ − √ < 2 m ϕt l ϕt π l ∗ ∗ 0. Thus there exists unique root α ∈ 0, 2 of equation(21) and θ = cos2 α∗ . Accordingly to Theorem on Implicit Function Derivative, we obtain √ 3 ∂α∗ ∂F/∂l K π · l− 2 > 0. =− = √ α) ∂l ∂F/∂α 2ϕt(sin α + cos m

It implies that θ∗ (l) = cos2 (α∗ (l)) is decreasing function. Similarly, 3

∂F ∂α∗ sin α · m− 2 = = − ∂m > 0, ∂F √ α) ∂m 2(sin α + cos m ∂α

thus θ∗ (m) = cos2 (α∗ (m)) is also decreasing function. Furthermore, √ 1 ∂α∗ ∂F/∂t K π · l− 2 > 0, =− = ∂t ∂F/∂α √ α ϕt2 sin α + cos m

thus θ∗ (t) = cos2 (α∗ (t)) is also decreasing function with respect to t. Finally, √ ∂α∗ ∂F/∂t π < 0, =− =− √ ∂K ∂F/∂α √ α ϕt l · sin α + cos m thus θ∗ (t) = cos2 (α∗ (t)) increases with respect to t.


379

To obtain the limit value of θ∗ is sufficient to note that equation (21) for l → ∞, t → ∞, K → 0 transforms into sin α cos α − √ = 0 m which is equivalent to m · cos2 α = sin2 α = 1 − cos2 α, 1 . On the other hand, K → ∞ implies lP → ∞, 1+m therefore m = 0 and θ∗ = 1 is a unique outcome. implying θ∗ = cos2 α∗ =

Proof of Proposition 3 Let y(l) = θ∗ (l) · l, then y is an implicit function defined by equation √ √ 1 p K π =0 G(y, l) = y − √ l−y− m ϕt which is equivalent to equation (15). Thus √ y ∂(θ∗ (l) · l) ∂G ∂G =− =p √ > 0, ∂l ∂l ∂y m(l − y) + y

moreover

∂θ∗ < 0 by Proposition 2. It implies that function ∂l r 2t θ∗ (l, m, t) · l K C Cu (l, m, t) = + · (1 − θ∗ (l, m, t)). 3 π 3ϕ

increases with respect to l. Let’s prove that CuC (l) is continuously differentiable at r 2 2t l πK l = lP = 2 2 . Indeed, for all l < lP the urban cost function CuC (l) = , hence ϕ t 3 π ∂CuC P ϕt2 (l − 0) = . ∂l 3πK Note that θ∗ (lP ) = 1 and √ ∂(θ∗ (l) · l) P lP √ = 1, p (l + 0) = ∂l m(lP − lP ) + lP

on the other hand,

∂(θ∗ (l) · l) P ∂θ∗ P (l + 0) = lP (l + 0) + θ∗ (lP ). ∂l ∂l ∂θ∗ P (l + 0) = 0, therefore ∂l r ! θl ∂ C π ∂Cu P 2t ϕt2 ∂CuC P (l + 0) = · (lP + 0) = = (l − 0). ∂l 3 ∂l 3πK ∂l

It implies that

380

Alexandr P. Sidorov

Recall that for l ≤ l

P

the urban costs

CuC

Moreover, for l > lP

2t = 3

r

θl ∂CuC , therefore ≡ 0. π ∂m

∂CuC ∂CuC ∂θ∗ = , ∂m ∂θ ∂m where

∂θ∗ < 0 by Proposition 2 and ∂m r √ ∂CuC l πK √ t = √ 1− √ θ >0 ∂θ 3 θ π ϕt l

because l > lP =

πK 2 and θ < 1. Therefore ϕ2 t2 ∂CuC ∂CuC ∂θ∗ = · < 0. ∂m ∂θ ∂m

Moreover, θ∗ (lP ) = 1, hence ∂CuC P t (l + 0) = ∂θ 3

r

lP π

√ πK = 0, 1− √ ϕt lP

which implies that ∂CuC P ∂θ∗ ∂CuC P (l + 0) = · (l + 0) = 0, ∂m ∂m ∂θ i.e. the urban cost function is continuously differentiable with respect to m. Let y(t) = θ∗ (t) · t2 , then y is an implicit function defined by equation H(y, t) =

√ p K mπ √ √ =0 my − t2 − y − ϕ l

which is equivalent to equation (15). Moreover,

therefore

√ 1 ∂H t ∂H m = √ + p > 0, =− p < 0, 2 ∂y 2 y 2 t −y ∂t 2 t2 − y

∂y ∂H ∂H =− > 0. ∂t ∂t ∂y p p It implies that function t · θ∗ (t) = y(t) increases with respect to t, as well as 1 − θ∗ (l, m, t). Therefore, urban costs function 2t CuC (l, m, t) = 3

r

θ∗ (l, m, t) · l K + · (1 − θ∗ (l, m, t)) π 3ϕ

also increases with respect to t, increase with respect to t.


381

Proof of Proposition 5 Note that wage wC∗ (l) is bounded function, while urban costs increase unrestrictedly with respect to l, hence, wC∗ (l) − CuC (l, m) < 0 for all sufficiently large l. ∂wC∗ α2 ∂CuC (0) = < (0, m) = +∞, thus Moreover, wC (0) = CuC (0, m) = 0, while ∂l 2 ∂l C∗ C w (l) − Cu (l, m) < 0 for all sufficiently small l > 0. It implies that set of l guaranteeing “the surviving condition” wC∗ (l) − CuC (l, m, t) ≥ 0 is subsetC of some interval [lmin (m, t), lmax (m, t)], where lmin (m, t) = inf l > 0 | wC∗ (l) − Cu (l, m, t) ≥ 0 > 0 and lmax (m, t) = sup l > 0 | wC∗ (l) − CuC (l, m, t) ≥ 0 < ∞. It remains to prove C∗ that this subset is nonempty and inequality (l) − CuC (l, m, t) ≥ 0 holds for all w r K πγ l ∈ [lmin (m, t), lmax (m, t)], at least for t ∈ 0, . ϕ 2βϕ Note that r r K πγ πγ 3α2 t< ⇒t< ⇐⇒ ϕ 2βϕ 16γ 2βϕ r πγ α2 2t C∗ ∗ > = CuC (l∗ , 0, t) ≥ CuC (l∗ , m, t) w (l ) = 8γ 3 2βϕ 2βϕ is the “maximum wage” population size. It implies γ that “surviving” set of city population is non-empty and lmax (m, t) > l∗ . Moreover, K 3α2 inequality < ensures that equation ϕ 16γ for all m ≥ 0, where l∗ =

wC∗ (l) =

α2 βϕl 2

(2βϕ + γl)

=

2K 3ϕ

has two real positive roots

l1,2 =

3α2 ϕ2 β 2K

r 2 2 2 3α ϕ β − 4βγϕ ∓ − 4βγϕ − 16β 2 γ 2 ϕ2 2K 2γ 2

2K 2βϕ if and only if l1 < l < l2 . In particular, l∗ = ∈ (l1 , l2 ) 3ϕ γ because wC∗ (l∗ ) = max wC∗ . Note that r r r πγ K π K π πK 2 K P t< = ⇒ t < ⇐⇒ l = > l1 . ϕ 2βϕ ϕ l∗ ϕ l1 ϕ2 t2 and wC∗ (l) >

2K = CuC (lP , m) holds, which implies 3ϕ 2K = that lmin (m, t) < lP . On the other hand, if lP ≥ l2 > l∗ then wC∗ (l∗ ) > 3ϕ CuC (lP , m) > CuC (l∗ , m), which also implies that lmin (m, t) < lP . Assume at first that m = 0 and consider set of positive roots of equation r α2 βϕ · l 2t l C∗ C w (l) = = Cu (l, 0, t) = . 3 π (2βϕ + γl)2

Let l1 < lP < l2 then inequality wC∗ (lP ) >

382

Alexandr P. Sidorov

√ √ Dividing both sides by l and substituting x = l we obtain the equivalent equation 2 √ √ 3α2 βϕ π · x = 2t 2βϕ + γx2 ⇐⇒ 8tβ 2 ϕ2 − 3α2 βϕ π · x + 8tβγϕx2 + γ 2 x4 = 0.

Sign of coefficients changes twice, hence, this equation has either 2, or 0 positive roots, due to Descartes’ rule of signs. On the other hand, wC∗ (l∗ ) > CuC (l∗ , 0, t) and wC∗ (l) < CuC (l, 0, t) for sufficiently large t, i.e., there is at least one positive root. It implies that these roots are lmin (0, t) and lmax (0, t), respectively, and wC∗ (l) − CuC (l, 0, t) ≥ 0 if and only if l ∈ [lmin (0, t), lmax (0, t)]. Moreover, it was proved that lmax (0, t) > l∗ and lmin (0, t) < lP . Now let m > 0, then CuC (l, m) ≡ CuC (l, 0) for all l ∈ [0, lP ] and CuC (l, m) > C Cu (l, 0) for all l > lM by Proposition 2. Let’s show that equation wC∗ (l) = CuC (l, m, t) also has two positive roots lmin (m, t) and lmax (m, t), such that wC∗ (l) ≥ CuC (l, m, t) if and only if l ∈ [lmin (m, t), lmax (m, t)]. Indeed, CuC (l, m, t) ≡ CuC (l, 0, t) for all l ≤ lP , thus lmin (m, t) ≡ lmin (0, t) ∈ (0, lP ). There is no roots in interval (lmin (0, t), lmax (0, t)), because CuC (l, m, t) ≤ CuC (l, 0, t) < wC∗ (l). Therefore, there is a unique root of equation wC∗ (l) = CuC (l, m, t) on interval (lmax (0, t), +∞), because wC∗ (l) strictly decreases for all l > lmax (0, t) > l∗ , while CuC (l, m, t) strictly increases on (0, +∞). This completes the proof of proposition. Proof of Proposition 7 Note that C∗ (0) wH

α2 βϕL = (2βϕ + γL)2

for all τ < τtrade =

2 2αβϕ + γL 1−τ · > 0 = CuC (0) 2αβϕ

2αβϕ . Moreover, substituting τ = 0 into (19) we obtain 2βϕ + γL

C∗ wH (λ) ≡

α2 βϕL C∗ C C = wH (1) ≥ CuH (1) > CuH (λ) (2βϕ + γL)2

for all λ ∈ (0, 1), because L < lmax (mH ). Thus, for all sufficiently small τ < τ ∗ C C∗ inequality wH (λ) > CuH (λ) holds for all λ ∈ (0, 1). Let C∗ C∗ ∆(λ, τ ) = VH (λ, τ ) − VH (λ) = wH (λ, τ ) + CSH (λ, τ ) − wH (λ) + CSH (λ) .

We are about to prove that ∆(λ, τ ) > 0 for all λ ∈ (0, 1) and sufficiently small τ > 0. Note that ∆(1, τ ) = 0 and L 2ϕLβγ(α − 3τ )(α − τ ) + 6ϕ2 β 2 (α − τ )2 + L2 γ 2 τ 2 ∆(0, τ ) = . 4ϕβ(2ϕβ + Lγ)2 Quadratic equation

2ϕLβγ(α − 3τ )(α − τ ) + 6ϕ2 β 2 (α − τ )2 + L2 γ 2 τ 2 = 0

has no real solutions with respect to τ , while ∆(0, 0) > 0. It implies that ∆(0, τ ) > 0 = ∆(1, τ ) for all τ . Now we are about to prove that ∆(λ, τ ) is decreasing function. Note that ϕLα2 β(6ϕβ + Lγλ) ∂∆ (λ, 0) = − k0 such that for some i: Jiα (k, xα (k), uα ) < Vi (k, xα (k)), then time-inconsistency of the individual rationality condition is appear. To overcome the time inconsistency problem in the game with nontransferable payoffs the notion of Payoff Distribution Procedure (PDP) was introduced by L.A. Petrosyan (1997). In this paper the PDP and time-consistency of Pareto-optimal solution are detailed for linear-quadratic discrete-time dynamic games.

On The Irrational Behavior Proof Condition for Linear-Quadratic Games Definition 2. Vector β(k) = (β1 (k), ..., βn (k)) is a PDP if X ∞ ∞ X T α (xα (k))T Pi (k)xα (k) + (uα (k)) R (k)u (k) = βi (k), i i i k=k0

387

i = 1, . . . , n.

k=k0

Definition 3. Pareto-optimal solution is called time-consistent if there exists a PDP such that the condition of individual rationality is satisfied ∞ X k=l

βi (k) ≥ Vi (l, xα (l)),

∀l ≥ k0 ,

(13)

i = 1, . . . , n,

where Vi (l, xα (l)) – is Nash outcome of player i in subgame Γ (l, xα (l)). Let for some Pareto-optimal solution the condition (12) is satisfied. Then there exist such functions ηi (k) ≥ 0, that Jiα (k0 , x0 , uα ) − Vi (k0 , x0 ) =

∞ X

(14)

ηi (k).

k=k0

In (Petrosyan, 1997) the formula for PDP, which guarantees a time-consistency in cooperative differential game with nontransferable payoffs, is considered. The following theorem gives an analog of this formula. Theorem 1. Let inequalities Jiα (k0 , x0 , uα ) ≥ Vi (k0 , x0 ),

i = 1, . . . , n,

are satisfied for some Pareto-optimal solution. Then PDP β(k) computed by formula βi (k) = ηi (k) − Vi (k + 1, xα (k + 1)) + Vi (k, xα (k))

i = 1, . . . , n,

k > k0

(15)

guarantees time-consistency of this Pareto-optimal solution along the cooperative trajectory xα (k) for k > k0 . Here ηi (k) ≥ 0 – are functions satisfying (14).

Proof. Show that β(k) is a PDP: ∞ X

βi (k) =

k=k0

∞ X

k=k0

ηi (k) − Vi (∞, xα (∞)) + Vi (k0 , x0 ) =

= Jiα (k0 , x0 , uα ) − Vi (k0 , x0 ) + Vi (k0 , x0 ) = Jiα (k0 , x0 , uα ). (16)

Here Vi (∞, xα (∞)) = lim Vi (k, xα (k)) = 0. So β(k) satisfies definition 2. k→∞

Now show that the condition of individual rationality is satisfied. Using (15) we obtain ∞ X k=l

βi (k) =

∞ X k=l

ηi (k) − Vi (∞, xα (∞)) + Vi (l, xα (l)) = =

∞ X k=l

ηi (k) + Vi (l, xα (l)) ≥ Vi (l, xα (l)). (17) ⊔ ⊓

388

Anna V. Tur

2.1. Irrational Behavior Proof Condition The condition under which even if irrational behaviors appear later in the game the concerned player would still be performing better under the cooperative scheme was considered in (Yeung, 2006). The irrational behavior proof condition for differential games with nontransferable payoffs is proposed in (Belitskaia, 2012). In this paper the irrational behavior proof condition is concretized for linear-quadratic discretetime dynamic games with nontransferable payoffs. Definition 4. Pareto-optimal solution (J1α (k0 , x0 , uα ), . . . , Jnα (k0 , x0 , uα )) satisfies the irrational behavior proof condition (Yeung, 2006) in the game Γ (k0 , x0 ), if the following inequalities hold l X

k=k0

βi (k) + Vi (l + 1, xα (l + 1)) ≥ Vi (k0 , x0 ),

i = 1, . . . , n

(18)

for all l ≥ k0 , where β(k) = (β1 (k), . . . , βn (k)) is time-consistent PDP of (J1α (k0 , x0 , uα ), . . . , Jnα (k0 , x0 , uα )). So if for all i = 1, . . . , n the following inequalities holds βi (k) + Vi (k + 1, xα (k + 1)) − Vi (k, xα (k)) ≥ 0,

k ≥ k0 ,

then the Pareto-optimal solution satisfies the irrational behavior proof condition. Rewrite these inequalities using (8) βi (k) + (xα (k))T (A(k) + B(k)M α (k))T Θi (k + 1)(A(k) + B(k)M α (k))− !

Θi (k) xα (k) ≥ 0,

k ≥ k0

(19)

If we use formala (15), then βi (k) + Vi (k + 1, xα (k + 1)) − Vi (k, xα (k)) = ηi (k),

k ≥ k0 ,

where ηi (k) ≥ 0 for all k ≥ k0 . It means that conditions (19) are always satisfied in this case. Let’s formulate these results. Theorem 2. If in linear-quadratic discrete-time dynamic games with nontransferable payoffs for some Pareto-optimal solutions and its PDP the following inequalities hold βi (k) + Vi (k + 1, xα (k + 1)) − Vi (k, xα (k)) ≥ 0,

k ≥ k0

i = 1, . . . , n.

where Vi (l, xα (l)) – is Nash outcome of player i in subgame Γ (l, xα (l)), then the irrational behavior proof condition for this Pareto-optimal solutions is satisfied. Proposition 1. If the PDP β(k) of Pareto-optimal solution in linear-quadratic discrete-time dynamic games with nontransferable payoffs is calculated using formula (15), then the irrational behavior proof condition for this Pareto-optimal solutions is satisfied.

On The Irrational Behavior Proof Condition for Linear-Quadratic Games 3.

389

Example

As an example consider the government debt stabilization game(van Aarle, Bovenberg and Raith, 1995). Pareto solution of this game is considered in (Engwerda, 2005). This paper shows the discrete-time case of this problem and time-consistency of cooperative solution. Assume that government debt accumulation, d(k), is the sum of interest payments on government debt, rd(k), and primary fiscal deficits, f (k), minus the seignorage (i.e. the issue of base money) m(k). So, d(k + 1) = rd(k) + f (t) − m(t),

d(0) = d0 ,

The objective of the fiscal authority is to minimize a sum of time profiles of the primary fiscal deficit, base-money growth and government debt J1 =

k ∞ X 1 ((f (k) − f )2 + η(m(k) − m)2 + λ(d(k) − d)2 ). 1+ρ

k=0

The monetary authorities are assumed to choose the growth of base money such that a sum of time profiles of base-money growth and government debt is minimized. That is k ∞ X 1 ((m(k) − m)2 + γ(d(k) − d)2 ). J2 = 1+ρ k=0

Let

x1 (k) =

1 1+ρ

k2

(d(k) − d),

x2 (k) = (f − m + (r − 1)d)

1 1+ρ

k+1 2

,

k2 1 (f (k) − f ), u1 (k) = 1+ρ k2 1 u2 (k) = (m(k) − m) 1+ρ

Then our system can be rewritten as

x(k + 1) = A(k)x(k) +

2 X

Bi (k)ui (k)

i=1

  12 1 r 1  1+ρ  A= 12  , 1 0 1+ρ

The payoff function of player i Ji =

∞ X

k=k0



B1 = 

(xT (k)Pi (k)x(k) +

2 X j=1

 12  12  1  , B2 =  − 1+ρ  , 0 0

1 1+ρ

uTj (k)Rij (k)ui (k)),

∀i = 1, 2

390

Anna V. Tur P1 =

λ0 γ0 , P2 = , 00 00

R11 = 1,

R12 = η,

R21 = 0,

R22 = 1.

Following (Basar and Olsder, 1999) to find the Nash equilibrium we solve the system  2 2 X X   NE T  (A(k) + B (k)M (k)) Θ (k + 1)(A(k) + Bi (k)MiN E (k))−  i i i    i=1 i=1 

− Θi (k) + Pi (k) + MjN E (k)T Rij (k)MjN E (k) + MiN E (k)T Rii (k)MiN E (k) = 0,     MiN E (k) = −(Rii (k) + BiT (k)Θi (k + 1)Bi (k))−1 BiT (k)Θi (k + 1)×     × (A(k) + Bj (k)MjN E (k)), i = 1, 2, j 6= i. Let λ = 12 , η = 1,

1 1+ρ

12

= 14 , s = 2, γ = 1. Then

E uN 1 (k, x) = −0.073193 −0.166311 x(k), E uN 2 (k, x) = 0.142083 0.318188 x(k),

0.656174 0.354202 J1 = x , 0.354202 0.844156 0 1.273766 0.613087 T J2 = x0 x . 0.613087 1.444844 0 0.656174 0.354202 T V (1, x(k)) = x (k) x(k), 0.354202 0.844156 1.273766 0.613087 T V (2, x(k)) = x (k) x(k), 0.613087 1.444843 xT0

According to (8) to find the Pareto Solution we solve the system  (A(k) + B1 M1α + B2 M2α )T Θα (k + 1)(A(k) + B1 M1α + B2 M2α )−      − Θα (k) + P α (k) + M α (k)T Rα (k)M α (k) = 0,  M α (k) = −(Rα (k) + B T (k)Θα (k + 1)B(k))−1 ×     × B T (k)Θα (k + 1)A(k). αR11 O Where P α (k) = αP1 (k) + (1 − α)P2 (k), Rα (k) = , O αR21 + (1 − α)R22 B(k) = B1 (k) B2 (k) .

For α = 0, 45

M1α = (−0.2272618408 M2α

− 0.5075099515)

= (0.1022678284 0.2283794781) 0.6808499028 0.4139353163 J1 (uα ) = xT0 x 0.4139353163 0.9409769084 0

On The Irrational Behavior Proof Condition for Linear-Quadratic Games 1.223914910 0.4917964794 J2 (uα ) = xT0 x0 0.4917964794 1.139011465 If, for example, x0 = −3 2 , then

391

J1α (k0 , x0 , uα ) − V1 (k0 , x0 ) = −0.107435164999999722 J2α (k0 , x0 , uα ) − V2 (k0 , x0 ) = −0.216497664600000528

So, conditions (12) are satisfied (we consider the minimization problem, that is why we have an opposite sign in (12)). But on the next step we have J1α (k1 , x1 , uα ) − V1 (k1 , x1 ) = 0.0504046297943969643 It means, that time-inconsistency of the individual rationality condition is appear. To avoid this problem, use PDP, calculated by formula (15) −0.107435164999999722 0.537430998 0.0789600736 α + xαT (k) x (k), 0.0789600736 0.16307449 k(k + 1) −0.216497664600000528 1.060309389 0.144646529 α β2 (k) = + xαT (k) x (k). 0.144646529 0.35798954 k(k + 1) (20) β1 (k) =

Note, that ηi (k) < 0, because we consider the minimization problem now. Sufficient condition for realization of irrational behavior proof condition has form: 0.537430998 0.0789600736 α x (k) ≤ 0 β1 (k) − xαT (k) 0.0789600736 0.16307449 1.060309389 0.144646529 α αT β2 (k) − x (k) x (k) ≤ 0. 0.144646529 0.35798954

And they are satisfied for β(k), computed by formula (20). References

Aarle, B. van, Bovenberg, L. and Raith, M. (1995). Monetary and fiscal policy interaction and government debt stabilization. Journal of Economics, 62(2), 111–140. Basar T. and Olsder G. J. (1999). Dynamic Noncooperative Game Theory, 2nd edition. Classics in Applied Mathematics, SIAM, Philadelphia. Belitskaia, A. V. The D.W.K. Yeung Condition for Cooperative Differential Games with Nontransferable Payoffs. Graduate School of Management, Contributions to game theory and management, 5, 45–50. Bertsekas D. P. (2007). Dynamic Programming and Optimal Control, Vol I and II, 3rd edition. Athena Scientific, Engwerda, J. C. (2005). LQ Dynamic Optimization and Differential Games. Chichester: John Wiley Sons, 497 p. Markovkin, M. V. (2006). D. W. K. Yeung’s Condition for Linear Quadratic Differential Games. In: Dynamic Games and Their Applications (L. A. Petrosyan and A. Y. Garnaev , eds.), St Petersburg State University, St Petersburg, 207–216. Markovkina, A. V. (2008). Dynamic game-theoretic model of production planning under competition. Graduate School of Management, Contributions to game theory and management, 2, 474–482.

392

Anna V. Tur

Petrosjan, L. A. (1997). The Time-Consistency Problem in Nonlinear Dynamics. RBCM J. of Brazilian Soc. of Mechanical Sciences, Vol. XIX, No 2. pp. 291–303. Petrosyan, L. A. and N. N. Danilov (1982). Cooperative differential games and their applications. (Izd. Tomskogo University, Tomsk). Yeung, D. W. K. (2006). An irrational-behavior-proofness condition in cooperative differential games. Intern. J. of Game Theory Rew., 8, 739–744. Yeung, D. W. K. and L. A. Petrosyan (2004). Subgame consistent cooperative solutions in stochastic differential games. Journal of Optimization Theory and Applications, 120(3), 651-666.

Von Neumann-Morgernstern Modified Generalized Raiffa Solution and its Application Radim Valenčík1 and Ondřej Černík2 The University of Finance and Administration Prague, Faculty of Economic Studies, Estonská 500, 101 00 Prague 10, Czech Republic E-mail: [email protected] WWW home page: http://www.vsfs.cz/en 2 The University of Finance and Administration Prague, Faculty of Economic Studies, Estonská 500, 101 00 Prague 10, Czech Republic E-mail: [email protected] WWW home page: http://www.vsfs.cz/en 1

Abstract In this paper we would like to discuss one of the possible modifications of Raiffa’s unique point solution which has applications in the analysis of social networks associated with investing in social position and creating the structures based on mutual covering of violations of the generally accepted principles. These structures are formed on the base of games of Tragedy of commons type when one player detects breaking the rules by another player. Hence the first player begins bribing the other player and simultaneously covering his back, one player is rejudiced in favour of another player. This gives a rise to social networks that significantly affect the formation of coalitions in various areas of the social system, including institutions whose mission is to protect society against violations of the generally accepted principles. We also consider an original theoretical concept. We show that this concept can be used to implement the NM-modified Raiffa’s solution for n = 3. Keywords: three-person game, bribing, Nash bargaining problem; NMmodified Raiffa sequential solution; redistribution system; social networks based on mutual covering violate the generally accepted principles.

1.

Introduction

Our approach comes from formal definition Nash bargaining problem for n players as a set B settled pairs (S, d), where S is compact convex subset Rn and point d belongs to S. The elements B of are called instance (examples) of the problem B, elements S are called variants or vector of utility, point d is called the point of disagreement, or status quo. Every example is called d-comprehensive. The theory suggests for the one-point solution several concepts. The term “solution” is understood as function f from B to Rn that each example (S, d) from B assigns value f(S, d) belonging to S. The most known concept of solution is Nash’s one (Nash, 1950), the other is Kalai-Smorodinsky’s one. The egalitarian approach suggested by Kalai (Kalai, 1977) can be also understood as the solution. All mentioned solutions can be expressed by axioms. Kalai-Smorodinsky’s solution (Kalai and Smorodinsky, 1975) is maximum point on the segment S connecting point and so called utopian point , whose coordinates are defined as Ui (S, d) = max{xi : x ∈ S a x ≥ d} From the point of view that we develop it is interesting Raiffa’s solution that was proposed in the early 1950’s. Raiffa (Raiffa, 1953) suggested dynamic procedures

394

Radim Valenčík, Ondřej Černík

for the cooperative bargaining in which the set S of possible alternatives is kept unchanged while the disagreement point d gradually changes. He considers two variants of such process – a discrete one and the continuous one. Discrete Raiffa’s solution is the limit of so called dictated revenues. Diskin, A., Koppel, M., Samet D. (Diskin et al., 2011) have provided an axiomatization of a family of generalized Raiffa’s discrete solutions. 2.

Experimental Section

Let S is a nonempty, closed, convex, comprehensive, and positively bounded subset of Rn whose boundary points are Pareto optimal. They propose a solution concept which is composed of two solution functions. One solution function specifies an interim agreement and the other specifies the terminal agreement. Such a step-bystep solution concept can formally be defined as follows. The pair (f, g) functions is called step-by-step solution, if as f (S, d) as g(S, d) belongs to for each example (S, d) from B. The set of generalized Raiffa’s solution is certain kind of step-bystep negotiation solution {(fp , gp )0 0.

410

David W.K. Yeung, Leon A. Petrosyan

Each agent gains from the existing level of public capital and the ith agent seeks to maximize its expected stream of monetary gains: Eϑ1 ,ϑ2 ,··· ,ϑT

T X [αi Ks − ci (Isi )2 ](1 + r)−(s−1) + (q1i KT +1 + q2i )(1 + r)−T , (4.2) s=1

subject to (4.1); where αi , ci , q1i and q2i are positive constants. In particular, αi gives the gain that agent i derives from the public capital, i i c (Is (s))2 is the cost of investing Isi in the public capital, and (q1i KT +1 + q2i ) is the terminal valuation of the public capital at stage T + 1. The noncooperative market outcome of the industry will be explored in the next subsection. 4.1.

Noncooperative Market Outcome

Invoking the analysis in (2.1)-(2.5) in section 2 we obtain the corresponding HamiltonJacobi-Bellman equations V i (t, K) = max Eϑt Iti

+V i

t + 1,

n X

j=1 j 6= i

[αi K − ci (Iti )2 ](1 + r)−(t−1)

φjt (K) + Iti − δK + ϑt

, for t ∈ {1, 2, · · · , T },

V i (T + 1, K) = (q1i KT +1 + q2i )(1 + r)−T , for i ∈ N.

(4.3)

(4.4)

Performing the maximization operator in (4.3) yields:

φit (K) =

ωt X

h=1

λht

n X 1 i V [ t + 1, φjt (K) − δK + ϑht ] (1 + r)(t−1) , for i ∈ N. (4.5) K 2ci t+1 j=1

To solve the game (4.1)-(4.2) we first obtain the value functions as follows. Proposition 4.1. The value function of agent i can be obtained as: V i (t, K) = (Ait K + Cti )(1 + r)−(t−1) , for t ∈ {1, 2, · · · , T + 1} and i ∈ N ; where AiT +1 = q1i and CTi +1 = q2i , Ait = (αi − Ait+1 δ) and Cti = −

for t ∈ {1, 2, · · · , T }. Proof. See Appendix A.

(Ait+1 )2 i 4ci +At+1

Pn

Ajt+1 ¯h j=1 2cj +ϑt

(4.6)

i +Ct+1 ,

⊔ ⊓

Using Proposition 4.1 and (4.5) the game equilibrium strategies can be obtained to characterize the market equilibrium. The asymmetry of agents brings about different payoffs and investment levels in public capital investments.

Subgame Consistent Cooperative Solution of Stochastic Dynamic Game

411

4.2. Cooperative Provision of Public Capital Now we consider the case when the agents agree to act cooperatively and seek higher gains. They agree to maximize their expected joint gain and distribute the cooperative gain proportional to their non-cooperative expected gains. To maximize their expected joint gains the agents maximize Eϑ1 ,ϑ2 ,··· ,ϑT

n X T X j=1 s=1

+

n X

[αj Ks − cj (Isj )2 ](1 + r)−(s−1)

(q1j KT +1 + q2j )(1 + r)−T

j=1

,

(4.7)

subject to dynamics (4.1). Following the analysis in (3.2)-(3.3) in Section 3, the corresponding stochastic dynamic programming equation can be obtained as: W (t, K) =

+W

max

{Itj f or j∈N }

t + 1,

n X ℓ=1

Eϑt

n X j=1

Itℓ − δK + ϑt

W (T + 1, K) =

n X

[αj K − cj (Itj )2 ](1 + r)−(t−1)

, for t ∈ {1, 2, · · · , T },

(q1j KT +1 + q2j )(1 + r)−T .

(4.8)

(4.9)

j=1

Performing the maximization operator in (4.8) yields: ψti (K) =

ωt X

h=1

n

λht

X j 1 WKt+1 [ t+1, ψt (K)−δK +ϑht ] (1+r)(t−1) , for i ∈ N. (4.10) i 2c j=1

Proposition 4.2. The value function W (t, K) can be obtained as W (t, K) = (At K + Ct )(1 + r)−(t−1) ,

(4.11)

for t ∈ {1, 2, · · · , TP+ 1}; P where AT +1 = nj=1 q1j and CT +1 = nj=1 q2j , Pn Pn (At+1 )2 +At+1 ϑ¯ht +Ct+1 , At = j=1 αj − At+1 δand Ct = j=1 4c j for t ∈ {1, 2, · · · , T }. Proof. Follow the proof of Proposition 4.1.

⊔ ⊓

Using (4.10) and Proposition 4.2 the optimal investment strategy of public capital stock can be obtained as: ψti (K) =

At+1 , for i ∈ N and t ∈ {1, 2, · · · , T }. 2ci

(4.12)

Using (4.1) and (4.12) the optimal trajectory of public capital stock can be expressed as:

412


Kt+1 =

n X At+1 − δKt + ϑt , K 1 = K 0 , for t ∈ {1, 2, · · · , T }, j 2c j=1

(4.13)

We use Xs∗ to denote the set of realizable values of Ks generated by (4.13) at stage s. The term Ks∗ ∈ Xs∗ is used to denote and element in Xs∗ . 4.3.

Subgame Consistent Payoff Distribution

Next, we will derive the payoff distribution procedure that leads to a subgame consistent solution. With the agents agreeing to distribute their gains proportional to their non-cooperative gains, the imputation vector becomes V i (s, Ks∗ ) ξ i (s, Ks∗ ) = Pn W (s, Ks∗ ) j (s, K ∗ ) V s j=1 = Pn

Ais Ks∗ + Csi

j ∗ j j=1 (As Ks + Cs )

(As Ks∗ + Cs )(1 + r)−(s−1) ,

(4.14)

for i ∈ N and s ∈ {1, 2, · · · , T } if the public capital stock is Ks∗ ∈ Xs∗ . To guarantee dynamical stability in a dynamic cooperation scheme, the solution has to satisfy the property of subgame consistency which requires the satisfaction of (4.14) at all stages s ∈ {1, 2, · · · , T }. Invoking Theorem 3.1 we can obtain: Proposition 4.3. A PDP which would lead to the realization of the imputation ξ(s, Ks∗ ) in (4.14) includes a terminal payment (q1i KT∗ +1 + q2i ) to agent i ∈ N at stage T + 1 and an payment at stage s ∈ {1, 2, · · · , T }: Bsi (Ks∗ ) = Pn

Ais Ks∗ + Csi

j ∗ j j=1 (As Ks + Cs )

(As Ks∗ + Cs )

ωs X

i Ai K ∗ (ϑh ) + Cs+1 ∗ λhs Pn s+1j s+1∗ s [As+1 Ks+1 (ϑhs ) + Cs+1 ](1 + r)−1 , for i ∈ N, h) + C j ] [A K (ϑ s+1 s+1 s s+1 j=1 h=1 (4.15) Pn As+1 ∗ ∗ h (ϑhs ) = − δK +ϑ . ⊔ ⊓ where Ks+1 s s j=1 2cj Finally, when all agents are using the cooperative strategies, the payoff that agent i will directly receive at stage s is

−

αj Ks∗ −

(AS+1 )2 . 4cj

However, according to the agreed upon imputation, agent i is to receive Bsi (Ks∗ ) in Proposition 4.3. Therefore a transfer payment (which can be positive or negative) equalling (A )2 ̟ii (s, Ks∗ ) = Bsi (Ks∗ ) − αj Ks∗ − S+1 (4.16) 4cj will be imputed to agent i ∈ N at stage s ∈ {1, 2, · · · , T }.

413

Subgame Consistent Cooperative Solution of Stochastic Dynamic Game 5.

Concluding Remarks

This paper presented subgame consistent cooperative solutions for stochastic discretetime dynamic games in public goods provision. The solution scheme guarantees that the agreed-upon optimality principle can be maintained in any subgame and provides the basis for sustainable cooperation. A "payoff distribution procedure" (PDP) leading to subgame-consistent solutions is developed. Illustrative examples are presented to demonstrate the derivation of subgame consistent solution for public goods provision game. This is the first time that subgame consistent cooperative provision of public goods is analysed in discrete time. Various further research and applications, especially in the field of operations research, are expected. Appendix A. Proof of Proposition 4.1. Using the value functions in Proposition 4.1 the optimal strategies in (4.5) becomes: φit (K) =

Ait+1 , for i ∈ N and t ∈ {1, 2, · · · , T }. 2ci

(A.1)

Using (A.1) the Hamilton-Jacobi-Bellman equations (4.4)-(4.5) reduces to: ω

Ait K + Cti = αi K −

t (Ait+1 )2 X + λht Ait+1 i 4c

h=1

n X Ajt+1 j=1

2cj

− δK + ϑht

i + Ct+1

,

(A.2)

for i ∈ N and t ∈ {1, 2, · · · , T }, AiT +1 K + CTi +1 = q1i K + q2i , for i ∈ N.

(A.3)

For (A.3) to hold it requires AiT +1 = q1i and CTi +1 = q2i .

(A.4)

Re-arranging terms in (A.2) yields:

Ait K + Cti = (αi − Ait+1 δ)K −

(Ait+1 )2 + Ait+1 4ci

n X Ajt+1 j=1

2cj

+ ϑ¯ht

i + Ct+1 , (A.5)

for i ∈ N and t ∈ {1, 2, · · · , T }. For (A.5) to hold it requires

Ait = (αi − Ait+1 δ) and Cti = −

(Ait+1 )2 + Ait+1 4ci

n X Ajt+1 j=1

2cj

+ ϑ¯ht

i + Ct+1 . (A.6)

Note that Ait and Cti depend on the model parameters and the succeeding values i of Ait+1 andCt+1 . Using (A.4) all Ait and Cti , for i ∈ N and t ∈ {1, 2, · · · , T }, are explicitly obtained. Hence Proposition 4.1 follows. Q.E.D.

414


References Basar, T., Olsder, G. J. (1995). Dynamic noncooperative game theory. Academic Press, London/New York. Bergstrom, T., Blume, C., Varian, H. (1986). On the private provision of public goods. J Public Econ, 29, 25–49. Chamberlin, J. (1974). Provision of collective goods as a function of group size. Am Polit Sci Rev, 65, 707–716. Dockner, E., Jorgensen, S., Long, N. V., Sorger, G. (2000). Differential games in economics and management science. Cambridge University Press, Cambridge Fershtman, C., Nitzan, S. (1991). Dynamic voluntary provision of public goods. Eur Econ Rev, 35, 1057–1067. Gradstein, M., Nitzan, S. (1989). Binary participation and incremental provision of public goods. Soc Sci Choice Welf, 7, 171–192. McGuire, M. (1974). Group size, group homogeneity, and the aggregate provision of a pure public good under Cournot behavior. Public Choice, 18, 107-126. Wang, W-K, Ewald, C-O (2010). Dynamic voluntary provision of public goods with uncertainty: a stochastic differential game model. Decisions Econ Finan, 3, 97–116. Wirl, F. (1996). Dynamic voluntary provision of public goods: extension to nonlinear strategies. Eur J Polit Econ, 12, 555–560. Yeung, D. W. K., Petrosyan, L. A. (2004). Subgame consistent cooperative solution in stochastic differential games. J Optim Theory Appl, 120(2.3), 651-666. Yeung, D. W. K., Petrosyan, L. A. (2006). Cooperative stochastic differential games. Springer-Verlag, New York Yeung, D. W. K., Petrosyan, L. A. (2010) Subgame Consistent Solutions for cooperative Stochastic Dynamic Games. Journal of Optimization Theory and Applications 145(3): 579-596. Yeung, D. W. K., Petrosyan, L. A. (2012). Subgame consistent economic optimization: an advanced cooperative dynamic game analysis. Boston, Birkhauser Yeung, D. W. K., Petrosyan, L. A. (2013). Subgame Consistent Cooperative Provision of Public Goods. Dynamic Games and Applications, forthcoming in 2013. DOI: 10.1007/s13235-012-0062-7.

Joint Venture’s Dynamic Stability with Application to the Renault-Nissan Alliance Nikolay A. Zenkevich and Anastasia F. Koroleva St.Petersburg State University, Graduate School of Management, Volkhovskiy per. 3, St.Petersburg, 199004, Russia E-mail: [email protected] E-mail: [email protected]

Abstract The cooperative dynamic stochastic multistage game of joint venture is considered. We suggest a payoff distribution procedure (PDP), which defines a time consistent imputation. Based on the results obtained, we conduct a retrospective analysis of dynamic stability of the Renault-Nissan alliance. It is shown that partners within the alliance have divided their cooperative payoffs according to the suggested PDP. Keywords:strategic alliance, joint venture, dynamic stochastic cooperative games, dynamic stability, normalized share.

1.

Introduction

In the recent decades economic globalization continuously increases at a rapid pace. There are constant and strong changes in competitive environment and markets structure. Moreover, customers become more and more informed and seek better quality of products and services. Under such conditions companies are confronted with the increasing challenges of providing themselves with the resources, technologies, competences, skills and information, necessary for achieving competitive advantage. Thus, strategic alliances, and, in particular, joint ventures (JV), are considered to become a necessary condition for company to survive in a violent competitive world. For this reason during the recent decades a number of strategic alliances and JVs shows steadily growth (Meschi and Wassmer, 2013). Indeed, strategic alliances allow companies expanding their geography, entering new markets, getting access to new knowledge, information, technologies, skills and competencies rather quickly (Barringer and Harrison, 2000; Bucklin and Segupta, 1993; Inkpen and Beamish, 1997). Hence, it is not surprising that numerous companies across the world view strategic alliances and JVs as a source of competitive advantage that allows them managing challenges that arise under conditions of markets globalization (Kumar, 2011; Smith et al., 1995). Strategic alliances have attracted much academic attention in the recent decades. In particular, due to strategic alliances high failure rates (according statistics, more than 50% of strategic alliance and JV agreements dissolve (Kale and Singh, 2009), researchers are especially interested in the issues of strategic alliances and JVs stability (Das and Teng, 2000; Inkpen and Beamish, 1997; De Rond and Buchikhi, 2004). However, despite considerable interest in the academic community to the issue of strategic alliance and JV stability, common view on this topic has not yet been reached. Moreover, there are several major unresolved issues that require solutions, among which are:

416

Nikolay A. Zenkevich, Anastasia F. Koroleva

1. Problem of measuring stability of strategic alliances. The question of how to measure the degree of stability of alliance remains unanswered. In general, studies are focusing on identifying factors that may affect stability or instability of strategic alliance ( Deitz et al., 2010; Gill and Butler, 2003; Jiang et al., 2008). 2. Strategic alliance stability evaluation. Currently, existing research examining the stability of the strategic alliance has not offered a method of assessing the strategic alliance stability. This can be explained by the existence of various factors of different nature that can influence overall alliance stability in numerous ways. Hence, it becomes a challenge to assess all the components of alliance stability. For instance, there are external factors that affect stability, such as institutional and competitive environment, but there are internal factors as well - trust, opportunistic behaviour, distribution of cooperative benefits, etc. It is clear, that methods of stability evaluation of stability components (e.g. external and internal), probably, should differ due to the different nature of factors, that determine stability. The important problems associated with the concept of a strategic alliance and its’ stability warrant further theoretical and methodological research in this area. In this paper, we attempt to address this gap by implementing and testing game theory methodology for evaluating alliance and JV stability component, that is determined by cooperative benefits allocation factor. Despite the existence of different factors that affect alliance and JV stability, the factor of allocation of cooperative benefits between the partners during the whole period of alliance realization can be considered as one of the most important (Dyer et al., 2008). It is obvious that when one or several alliance participants do not agree on the distribution of cooperative benefits their motivation for participation decreases which affects stability. Hence, it would be highly useful for alliance partners to know in what way they should design the part of their cooperative agreement concerning benefits allocation for alliance to be stable and to have some instrument that will allow them assessing alliance stability during its realization phase. In this paper we make an attempt develop an approach for solving these tasks. The approach is based on the concept of dynamic stability in dynamic cooperative games (Petrosjan, 2006). The paper organized as follows: the first section presents the model of joint venture and suggests a way for cooperative benefits allocation among partners; in the second section the model is applied to a case of Renault-Nissan JV to analyze it’s stability; in the conclusions we summarize the main results of the article. 2.

Model of Joint Venture

In order to model a JV, a cooperative stochastic multistage dynamic game is considered (Petrosjan, 2006). In particular, a multistage game with the infinite duration and random time closure is used due to the fact that most of the agreements on strategic cooperation between the companies do not have a predetermined end date of the alliance. Dependent on the circumstances in which alliance partners are, they can only make assumptions on when strategic partnership will come to an end. Multistage principle of the game means that players make decisions at certain discrete points of time which correspond to the steps of the game. The stochastic multistage game with random duration G(xt0 ) = (N ; V (S, xt0 )) is considered:

417

Joint Venture’s Dynamic Stability

1. players together take a decision on their cooperative strategy in order to obtain the highest overall benefits; 2. players agree on the allocation mechanism of jointly received benefits between partners. The game G(xt0 ) that we are considering is described as follows: N = 1, ..., n – is a number of players (members of JV). Z = 0, ..., ∞ – is a set of steps in the game G(xt0 ). tm , m = 0, z − 1 – is time, during which the game evolves. X – is a set of all possible states in the game, such that: ∞ [

Xtk ∩Xtl = ∅ ,

Xtm = X ,

m=0

k 6= l ,

t0 < t1 < ... < tl 0 , X p(xtm , xtm+1 ; utm ) = 1 , xtm+1 ∈Ftm (x)

where p(xtm , xtm+1 ; utm ) is the probability that at the step m + 1 the xtm+1 state is realized , provided that at the step m was implemented control utm . In each possible

418


state x ∈ X is given a probability qm , 0 < qm 6 1, m = 0, z − 1, that the game will end at step m. Now, let us consider only those states in the game, that have positive probability of being reached by implementation of control vectors utm , m = 0, z − 1: CX = {xtm : p(xtm , xtm+1 ; utm > 0, ∀xtm ∈ X, m = 0, ∞}. CX ⊂ X.

Value function of the game G(N ; V (S, xt0 )) is defined as a lower value of a zerosum game between two players – coalition S and coalition N \ S, assuming that the players use only pure strategies. Details on the value function of a cooperative game can be found in (Zenkevich et al. 2009). Let us define it. Coalition N acts as one decision making center and will try to maximize their total benefits in the game. Suppose, that a sequence of control vectors ut0 , ut1 , · · · , utm , · · · , utz−1 was implemented. Then the payoff of player i will be determined by the formula:

Ki (xt0 ; ut0 , ut1 , · · · , utm , · · · , utz−1 ) = Ki (xt0 ) =   ! j ∞ X Y X tm   = qm (1 − qm ) Ki (utm . j=0

m0

k=0

Due to the fact that the game has a random nature, it is reasonable to consider the expected payoff of the alliance, that players try to maximize in the game G(xt0 )):

V (N, xt0 ) = maxutm

"

X

i∈N

#

Ei (xt0 ; utm , · · · , utz−1 )

(1)

.

Vector u ¯ = (¯ u1 , · · · , u¯n ) is called a cooperative solution.

Maximum of (1) is found by solving the corresponding Bellman equation

V (N, xt0 ) = maxui (xt0 )∈Ui (xt0 ), i∈N

"

X

i∈N

(1 − q0 ) =

X

i∈N

Kit0 (¯ ut0 ) + (1 − q0 )

X

xt1 ∈F (xt0 )

X

Kit0 (ut0 )+ 

p(xt0 , xt1 ; ut0 V (N, xt1 ) =

p(xt0 , xt1 ; u ¯t0 )V (N, xt1 )

(2)

xt1 ∈F (xt0 )

with the boundary condition V (N, xtm ) = maxui (xtm )∈Ui (xtm ),i∈N

X

i∈N

Kitm (utm ) ,

x ∈ {x : F (x) = ∅} . (3)

419

Joint Venture’s Dynamic Stability

In the case when coalition S 6= N and S 6= ∅, value function is described by the following equation

maxuS (xtm )∈US (xtm ) minuN \S (xtm )∈UN \S (xtm )

" X

V (S, xtm ) = Kitm (uS (xtm ), uN \S (xtm ))+

i∈S

(1 − qm )

X

xtm+1 ∈F (xtm )

 p xtm , xtm+1 ; uS (xtm ), uN \S (xtm ) V (S, xtm+1 )

(4)

with the boundary condition

maxuS (xtm )∈US (xtm ) minuN \S (xtm )∈UN \S (xtm )

X

V (S, xtm ) = Kitm (uS (xtm ), uN \S (xtm )),

i∈S

x ∈ {x : F (x) = ∅} ,

(5)

where i1 , · · · , ik ∈ S, ik+1 , · · · , in ∈ N \S and uS (xtm ) = (uit1m , · · · , uitkm );

i

uN \S (xtm ) = (utk+1 , · · · , uitnm ) . m

For the case when S = ∅ it is assumed that its’ payoff is 0 : V (∅, xtm ) = 0 .

(6)

Thus, the game G(xt0 ) is defined by the pair (N ; V (S, x0 )), where 1. Value function V (S, xt0 ) is determined by the formula (2) with the boundary condition (3) for S = N ; 2. Value function V (S, xt0 ) is determined by the formula (4 ) with the boundary condition (5) with S 6= ∅ ; 3. Value function V (S, xt0 ) is determined by the formula (6) with S = ∅. The main objective of alliance members is a division of the benefits derived by joint efforts. In the game theory terminology, payoffs of players at the end of the game are called imputation. Definition 1 (Petrosjan et al., 2004). Vector ξ(xt0 ) = (ξ1 (xt0 ), · · · , ξn (xt0 )) is called imputation in a cooperative stochastic game with the random duration G(xt0 ), if : P 1. i∈N ξi (xt0 ) = V (N, xt0 ) ; 2. ξi (xt0 ) > V ({i}, xt0 ), for all i ∈ N ,

where V ({i}, xt0 ) is a winning coalition S in a zero-sum game against the coalition V ({i}, xt0 ) when coalition S consists of only one player i. The set of all possible imputations in the cooperative stochastic game G(xt0 ) is denoted as I(xt0 ).

Definition 2 (Petrosjan et al., 2004). Solution of a cooperative stochastic game is any fixed subset of C(xt0 ) ⊂ I(xt0 ).

420


Value function definition and definitions 1-2 are also valid for any subgame G(xtm ) of the original game G(xt0 ), that starts at time tm from the state xtm . Thus, having introduced the cooperative stochastic game G(xt0 ) and having defined the concept of sharing the benefits of cooperation, we defined the stochastic model of strategic alliance. The main issue of cooperative game theory is the study of the dynamic stability of the division of benefits from cooperation. So let us move to the results obtained in the game theory in the area of the stability of cooperative behaviour. Definition 3 (Petrosjan et al., 2004). Vector function β(xtm ) = (β1 (xtm ), · · · , βn (xtm )), where xtm ∈ CX, is called payoff distribution procedure (PDP) at a vertex xtm , if X X X βi (xtm ) = Kitm (¯ u1tm , · · · , u ¯ntm ) = Kitm (¯ utm ) , i∈N

i∈N

(¯ ut1m , · · ·

i∈N

where u¯tm = is the situation at the time tm in the game element G(xtm ) that was realized under cooperative solution u¯ = (¯ u1 , · · · , u ¯n ) in the game G(xt0 ). , u¯tnm )

Definition 4 (Zenkevich et al. 2009). Imputation ξ(xt0 ) ∈ C(xt0 ) is called time consistent in a cooperative stochastic game G(xt0 ), if for each vertex xtm ∈ CX ∩ (F (xt0 ))k there exists a nonnegative PDP β(xtm ) = (β1 (xtm ), · · · , βn (xtm )) such that X p(xtm , xtm+1 , u¯tm ξi (xtm+1 ) (7) ξi (xtm ) = βi (xtm ) + (1 − qm ) xtm+1 ∈F (xtm )

and ξi (xtm ) = βi (xtm ), xtm ∈ {xtm : F (xtm ) = ∅} ,

where xtm ∈ (F (xtm ))k , ξ(xtm+1 ) = (ξ1 (xtm+1 ), · · · , ξn (xtm+1 )) is some imputation, that belongs to a solution C(xtm+1 ) of cooperative subgame G(xtm+1 ). Definition 5 (Zenkevich et al. 2009). Cooperative stochastic game with random duration G(xt0 ) is a time consistent solution C(xt0 ), if all imputations ξ(xt0 ) ∈ C(xt0 ) are time consistent. Now, based on definitions 1-5, we introduce a normalized share. Consider normalized shares for imputation ξ(xtm ) in the subgame G(xtm ), where θi (xtm ) =

ξi (xtm ) , i∈N . V (N, xtm )

(8)

According to equation (7) θi (xtm ) = ai (xtm )+(1−qm )

X

p(xtm , xtm+1 , u ¯tm )

xtm+1 ∈F (xtm )

where ai (xtm ) ≡

θi (xtm+1 )V (N, xtm+1 ) , V (N, xtm ) (9)

βi (xtm ) , V (N, xtm )

(10)

421

Joint Venture’s Dynamic Stability X

ai (xtm ) =

i∈N

P

βi (xtm ) = V (N, xtm ) i∈N

P

Kitm (¯ utm ) 2, ν : 2N → R, ν(∅) = 0. So-called discrete game GD differs from GT that ν is integer-valued function and players payoffs must be integers (Azamkhuzhaev, 1991). In economic settings, the integer requirement reflects some forms of indivisibility. Both games summarizes the possible outcomes to a coalition

Symmetric Core of Cooperative Side Payments Game

429

by one number, i.e. side payments are allowed. GT and GD can be also described N as a games with nontransferable utility (NTU games). Let GTN and GD be the N N N sets of n-person TU and discrete games respectively, G = GT ∪ GD . Denote by Ω = 2N P \ {N, ⊘} the family of proper coalitions. Given x ∈ RN and ∅ 6= K ⊆ N : x(K) = i∈K xi , x(∅) = 0. The cardinality of coalition ∅ 6= K ⊆ N is denoted by |K|. When there is no ambiguity, we write ν(i), K \ i instead of ν({i}), K \ {i} and so on. Two players i, j ∈ N are called symmetric (substitutes, interchangeable) in a game G ∈ G N if ν(K ∪ i) = ν(K ∪ j) f or every K ∈ N \ {i, j}.

(1)

Player i ∈ N is veto player in a game G ∈ G N if ν(K) = 0 for all K 6∋ i. Denote by veto(G) the set of veto players of G ∈ G N . A game GT is called convex if ν(K) + ν(H) 6 ν(K ∪ H) + ν(K ∩ H) for K, H ⊆ N . A game GT is integer if N ν : 2N → Z, where Z denotes the set of integer numbers. The operator Ψ : GD → GTN will be used to compare TU and discrete game solutions, i.e. Ψ (GD ) is an integer TU game corresponding to GD . The set of feasible payoff vectors X ∗ (GT ) and pre-imputation set X(GT ) of TU game GT are defined by X ∗ (GT ) = {x ∈ RN |x(N ) 6 ν(N )}, X(GT ) = {x ∈ RN |x(N ) = ν(N )}.

The related sets of discrete game GD are

X ∗ (GD ) = X ∗ (Ψ (GD )) ∩ ZN , X(GD ) = X(Ψ (GD )) ∩ ZN . For any set GÑ ⊆ G N a set-valued solution (or multisolution) on GÑ is a mapping ϕ : GÑ →→ RN which assigns to every G ∈ GÑ a set of payoff vectors ϕ(G) ⊆

X ∗ (G). Notice that the solution set ϕ(G) is allowed to be empty. A value of game G is a function f : GÑ → X(G). The core of TU game and core of discrete game are the sets C(GT ) = {x ∈ X(GT )|x(K) > ν(K), K ∈ Ω}, C(GD ) = C(Ψ (GD )) ∩ ZN .

The formulas to obtain the CIS-value, ENSC-value, Shapey value and equal division solution of a game GT are P ν(N ) − j∈N ν(j) CISi (GT ) = ν(i) + , n P ν(N ) − j∈N ν ∗ (j) EN SCi (GT ) = ν ∗ (i) + , n X |K|!(n − |K| − 1)! ν(N ) Shi (GT ) = (ν(K ∪ i) − ν(K)), EDi (GT ) = , n! n K6∋i ∗

where i ∈ N , ν (K) = ν(N ) − ν(N \ K), K ⊆ N . The CIS-value is also called the equal surplus division solution. Notice that CIS-value, ENSC-value and equal division solution assign to every player some initial payoff and distribute the remainder of ν(N ) equally among all players. For CIS-value (the center of gravity of imputation set I(GT ) = {x ∈ X(GT )|xi > ν(i), i ∈ N }) initial payoff to player i ∈ N is equal to its individual worth ν(i). For ED-value and ENSC-value the initial payoffs are equal to zero and player’s marginal contribution ν ∗ (i) to grand coalition N , respectively. Thus, the ENSC-value assigns to any game GT the CIS-value of dual game (N, ν ∗ ).

430 3.

Alexandra B. Zinchenko Symmetric core properties

For a game G ∈ G N denote by ℑ(G) the family of coalitions each of which contains only symmetric players ℑ(G) = {K ∈ 2N ||K| > 2, every i, j ∈ K, i 6= j, are symmetric in G}. Definition 1. A game G ∈ G N is called semi-symmetric if at least two players are symmetric in G, i.e. ℑ(G) 6= ∅. A game G ∈ G N is (totally) symmetric if ℑ(G) = {{N }}. A game G ∈ G N is non-symmetric if ℑ(G) = ∅. N N Let SG N = SG N T ∪ SG D be the set of semi-symmetric games G ∈ G .

Definition 2. The symmetric core SC(G) of a game G ∈ G N is the set of core allocations for which the payoffs of symmetric players are equal SC(G) = {x ∈ C(G)|xi = xj f or all i, j ∈ K, i 6= j, K ∈ ℑ(G)}. Example 1. Let U H = (N, uH ) be n-person (n > 3) unanimity game for a coalition H ∈ Ω: uH (K) = 1 for K ⊇ H, uH (K) = 0 otherwise. Since  if |H| = n − 1,  {H} ℑ(U H ) = {N \ H} if |H| = 1,  {H, N \ H} else, then the game U H is semi-symmetric. Well known that any unanimity game is convex and C(U H ) = {x ∈ RN |xi = 0, i ∈ N \ H, x(H) = 1}. Therefore, the symmetric core SC(U H ) consists of one point which is the Shapey value: SC(U H ) = 1 for i ∈ H, Shi (U H ) = 0 otherwise. {Sh(U H )}, where Shi (U H ) = |H|

Example 2. Consider situation with four investors having the endowments 80, 60, 50, 50 units of money (m.u. for short). Assume the following investment projects are available: a bank deposit that yields 10 interest rate whatever the outlay, two production processes that require an initial investment of 100 ore 200 m.u. and yields 15 ore 20 rate of return, respectively. The related four-person investment game (de Waegenaere et al., 2005) GT ∈ GTN is given by  N = {1, 2, 3, 4}, ν(N ) = 284,     ν(1) = 88, ν(2) = 66, ν(3) = ν(4) = 55,  ν(1, 2) = 159, ν(1, 3) = ν(1, 4) = 148,   ν(2, 3) = ν(2, 4) = 126, ν(3, 4) = 115,    ν(1, 2, 3) = ν(1, 2, 4) = 214, ν(1, 3, 4) = 203, ν(2, 3, 4) = 181.

We obtain non-convex (ν(2, 4)+ν(3, 4) > ν(4)+ν(2, 3, 4)) balanced semi-symmetric game with symmetric players 3 and 4, ℑ(GT ) = {{3, 4}}. The core of game GT has 16 extreme points whereas symmetric core is the convex hull of 4 points SC(GT ) = co{x1 , x2 , x3 , x4 }, 1 1 1 1 x1 = (100 , 68 , 57 , 57 ), 2 2 2 2 x3 = (98, 66, 60, 60),

1 1 1 1 x2 = (90 , 78 , 57 , 57 ), 2 2 2 2 x4 = (88, 76, 60, 60).


431

Denote by G0T = (N, ν 0 ), where   5 if |K| ∈ {2, 3}, ν 0 (K) = 20 if K = N,  0 else,

the zero-normalization of game GT . All players are substitutes in G0T , ℑ(G0T ) = {{1, 2, 3, 4}}. The symmetric core of game G0T consists of one point SC(G0T ) = {x0 }, x0 = (5, 5, 5, 5) = Sh(G0T ) = CIS(G0T ) = EN SC(G0T ) = ED(G0T ). The payoff vector x0 corresponds to symmetric core allocation x6 = (93, 71, 60, 60) 3 4 , it is equal the Shapey value Sh(GT ) of original game GT . Notice, that x6 = x +x 2 of original game, but does not coincide with the barycenter (94 41 , 72 41 , 58 43 , 58 43 ) of the symmetric core of game GT . In game theory literature there exist two (equivalent) versions of TU game balancedness: a game GT ∈ GTN is called balanced if it has a nonempty core ore if it satisfies the Bondareva-Shapley condition X X λK ν(K) 6 ν(N ), λ : Ω → R+ , λK = 1, i ∈ N, (2) K∈Ω

K∈Ω, K∋i

see (Bondareva, 1963) and (Shapley, 1967). Since (2) is necessary but not sufficient condition for the nonemptiness of core of discrete game, the unified definition is required. Definition 3. A game G ∈ G N with nonempty core is called balanced. We need the following axiom to be satisfied by solution ϕ. Axiom 3.1 (equal treatment). For all G ∈ GÑ , all x ∈ ϕ(G) and every symmetric players i, j in G: xi = xj . Known that Sh(GT ), CIS(GT ), EN SC(GT ) and ED(GT ) satisfy equal treatment. From above definitions it straightforwardly follows that: • the symmetric core of a game G ∈ G N may be empty; • the symmetric core of TU game GT is a convex subset of its core; • the symmetric core of non-symmetric game G ∈ G N coincides with its core, therefore, apart from their different definitions the real difference is exposed for semi-symmetric balanced games; • the symmetric core of balanced symmetric TU game consists of one point which is the equal division solution SC(GT ) = {ED(GT )}; • the symmetric core of balanced semi-symmetric TU game contains all core selectors satisfying equal treatment, in particular, the nucleolus that realizes a fairness principle based on lexicographic minimization of maximum excess for all coalitions; • if the Shapley value of semi-symmetric TU game satisfies the core inequalities then it belongs to symmetric core, the Shapley value is always symmetric core allocation on the domain of convex TU games; • the CIS-value, the ENSC-value, the equal division solution which "have some egalitarian flavour" (Brink and Funaki, 2009) and any convex combination of these

432

Alexandra B. Zinchenko

solutions cannot belong to symmetric core of balanced semi-symmetric TU game. A nonempty core of NTU game (even 3-person) may contains no equal treatment outcomes (Aumann, 1987). The following two propositions show that balancedness of TU game is the necessary and sufficient condition for nonemptiness of its symmetric core, but the same is not true for balanced discrete game. Proposition 1. Let GT ∈ SG N T . Then SC(GT ) 6= ∅ iff C(GT ) 6= ∅.

Proof. If SC(GT ) 6= ∅ then C(GT ) 6= ∅ by inclusion SC(GT ) ⊆ C(GT ). Assume now that C(GT ) 6= ∅ and take x1 ∈ C(GT ). If x1 ∈ SC(GT ) then SC(GT ) 6= ∅. Otherwise, there exist a coalition K ∈ ℑ(GT ) and players i, j ∈ K such that x1i < x1j . Construct x2 ∈ RN as follows: x2i = x1j , x2j = x1i , x2l = x1l for l ∈ N \ {i, j}. 1

2

∈ C(GT ). So, Using (1) we see that x2 ∈ C(GT ). By core convexity, x3 = x +x 2 we get the core allocation x3 satisfying x3i = x3j , x3l = x1l for l ∈ N \ {i, j}. If x3 ∈ / SC(GT ) then by repeated application of above procedure one obtains the payoff vector belonging to SC(GT ). ⊔ ⊓ Proposition 2. There exist discrete games GD ∈ SG N D such that C(GD ) 6= ∅ but SC(GD ) = ∅.

Proof. Consider discrete games GsD , defined by set function ν s on N : ν s (K) ∈ {0, 1} for K ⊂ N and ν s (N ) = 1. The associated TU game Ψ (GsD ) = (N, ν s ) is simple. Assume |veto(Ψ (GsD ))| > 2. Then C(Ψ (GsD )) = co{ei ∈ ZN |i ∈ veto(Ψ (GsD ))} and C(GsD ) = {ei ∈ ZN |i ∈ veto(Ψ (GsD )), where eij = 0 for i 6= j, eii = 1. Obviously, veto players are substitutes in games Ψ (GsD ) and GsD . However xi 6= xj for all x ∈ C(GsD ) and every (i, j) ∈ veto(GsD ). Thus SC(GsD ) = ∅. ⊔ ⊓

The core of TU game has been intensely studied and axiomatized. We shall formulate some convenient properties of a solution concept ϕ on GÑ ⊆ G N which has been employed in the well-known core axiomatizations. The axiomatic characterizations of discrete game solutions are not yet provided. Axiom 3.2 (efficiency). x(N ) = ν(N ) for all x ∈ ϕ(G) and all G ∈ GÑ . Axiom 3.3 (symmetry). For all G ∈ GÑ and every symmetric players i, j in G: if x ∈ ϕ(G) then there exists y ∈ ϕ(G) such that xi = yj , xj = yi and xp = yp for p ∈ N \ {i, j}. Axiom 3.4 (modularity). For any modular game G ∈ GÑ generated by the vector x ∈ RN : ϕ(G) = {x} . Axiom 3.5 (antimonotonicity). For any pair of games G1 , G2 ∈ GÑ defined by set functions ν 1 , ν 2 on N such that ν 1 (N ) = ν 2 (N ) and ν 1 (K) 6 ν 2 (K) for all K ⊂ N , it holds that ϕ(G2 ) ⊆ ϕ(G1 ). Axiom 3.6 (reasonableness (from above)). For all G ∈ GÑ , all x ∈ ϕ(G) and every i ∈ N : xi 6 max {ν(K ∪ i) − ν(K)}. K⊆N \i

Axiom 3.7 (covariance). For any pair of games G1 , G2 ∈ GÑ defined by set functions ν 1 , ν 2 such that ν 2 = αν 1 + β for some α >0 and some β ∈ RN it holds that ϕ(G2 ) = αϕ(G1 ) + β.


433

Axiom 3.8 (projection consistency (or reduced game property)). Let G ∈ GÑ , ∅ 6= H ⊂ N and x ∈ ϕ(G), then RxH = (H, rxH ) ∈ G˜H and xH ∈ ϕ(RxH ), where xH = (xi )i∈H ∈ RH and  if K = ∅, 0 rxH (K) = ν(K) if ∅ 6= K ⊂ H,  ν(N ) − x(N \ H) if K = H, is the projected reduced game with respect to H and x.

Known (Llerena and Carles, 2005) that the core is the only solution on GTN satisfying projection consistency, reasonableness (from above), antimonotonicity and modularity. Notice that projection consistency is one of the fundamental principle used in this field. By summarizing the statements formulated above we can say that the symmetric core of balanced semi-symmetric TU and discrete games satisfies equal treatment, efficiency, symmetry, modularity, reasonableness (from above) and many other core axioms based on only the original game. Theorem 1 (below) shows that for the class of balanced semi-symmetric games the symmetric core is in conflict with antimonotonicity, covariance and projection consistency. All these properties involve the pairs of games. Lemma 1. Let G ∈ SG N is a balanced game and G0 is its zero-normalization. Then G0 ∈ SG N , SC(G0 ) ⊆ SC(G) and there exist games G ∈ SG N such that SC(G0 ) 6= SC(G). Proof. The zero-normalization G0 of any game G ∈ G N is uniquely determined by set function ν 0 on N , where X ν 0 (K) = ν(K) − ν(l), ∅ 6= K ⊆ N. (3) l∈K

Obviously, G0 ∈ SG N . Let i, j ∈ N , i 6= j, are symmetric players in G. The formulas (1) and (3) imply that ν 0 (K ∪i) = ν 0 (K ∪j) for all K ⊆ N \ {i, j}. Thus, symmetric players in G remain symmetric in G0 . Example 2 shows that non-symmetric in G players can become symmetric in G0 . If G = GT then a linear system defining SC(G0T ) contains the one for SC(GT ) and, perhaps, additional equality constraints. So SC(G0T ) ⊆ SC(GT ). In view of Example 2 this inclusion can be strict. For discrete game G = GD the final part of lemma is proved analogously. ⊔ ⊓ Theorem 1. Let G ∈ SG N is a balanced game. Then SC(G) does not satisfy (i) Axiom 3.5; (ii) Axiom 3.7 even for α = 1 and β = (ν(1), ..., ν(n)); (iii) Axiom 3.8. Proof. (i) Consider two balanced four-person TU games G1T , G2T defined by set functions ν 1 , ν 2 such that  2 |K| = 2,   1  4 |K| = 3, ν (K) + 1 = 5, K = {1, 3, 4}, 1 ν (K) = ν 2 (K) = 6 K = N, ν 1 (K), else.    0 else,

434


The games G1T and G2T are symmetric and semi-symmetric, respectively. ℑ(G1T ) = {{1, 2, 3, 4}}, ℑ(G2T ) = {{3, 4}}, ν 1 (N ) = ν 2 (N ) and ν 1 (K) 6 ν 2 (K) for all K ⊂ N . It holds that 1 1 1 1 1 1 SC(G2T ) = co{(2, 1, 1 , 1 ), (2, 0, 2, 2), (1, 1, 2, 2)} 6⊂ SC(G1T ) = {(1 , 1 , 1 , 1 )}. 2 2 2 2 2 2 Consider now discrete games G1D , G2D corresponding to given TU games. We have SC(G2D ) = {(2, 0, 2, 2), (1, 1, 2, 2)} 6⊂ SC(G1T ) = ∅. Thus, antimonotonicity is violated by SC(G). (ii) This statement follows from lemma 1. (iii) In four-person TU game GT defined by  N = {1, 2, 3, 4}, ν(N ) = 8, ν(i) = 0, i ∈ N,  ν(1, 2) = ν(1, 3) = ν(1, 4) = ν(2, 3) = ν(2, 4) = 2, ν(3, 4) = 3,  ν(1, 2, 3) = ν(1, 2, 4) = 6, ν(1, 3, 4) = 5, ν(2, 3, 4) = 4

players 3 and 4 are symmetric, ℑ(GT ) = {{3, 4}}. The symmetric core is the convex hull of four points SC(GT ) = co{x1 , x2 , x3 , x4 }, where x1 = (4, 0, 2, 2), x2 = (4, 1, 1 21 , 1 21 ), x3 = (1, 3, 2, 2) and x4 = (2, 3, 1 21 , 1 12 ). The projected reduced game RxH2 = (H, rxH2 ) relative to H = {1, 2, 3} at x2 is defined by: rxH2 (1, 2, 3) = 6 12 , rxH2 (i) = 0, i ∈ H, rxH2 (1, 2) = rxH2 (1, 3) = rxH2 (2, 3) = 2. The reduced game is symmetric. Its symmetric core consists of one point (2 61 , 2 16 , 2 61 ). The restriction of x2 to H, x2H = (4, 1, 1 21 ), does not belong to the symmetric core of reduced game. For discrete game GD corresponding to last TU game GT we have SC(GD ) = {x1 , x3 , x5 , x6 }, where x5 = (3, 1, 2, 2), x6 = (2, 2, 2, 2). The projected reduced game RxH1 relative to H = {1, 2, 3} at x1 is defined by: rxH1 (1, 2, 3) = 6, rxH1 (i) = 0, i ∈ H, rxH1 (1, 2) = rxH1 (1, 3) = rxH1 (2, 3) = 2. Since the reduced game is symmetric SC(RxH1 ) = {(2, 2, 2)}. The restriction of x1 to H, x1H = (4, 0, 2), does not belong to SC(RxH1 ). So SC(G) does not provide projection consistency. ⊔ ⊓ It has been interesting to study the interrelation between the symmetric core of a game G ∈ SG N and strongly egalitarian core allocations.

Definition 4. Let G ∈ G N , x ∈ C(G) and x ∈ RN is obtained from x by permuting its coordinates in a non-decreasing order: x1 6 x2 6 ... 6 xn . A core allocation x is Lorenz allocation (Lorenz maximal, strongly egalitarian ) iff itP is undominated in P the sense of Lorenz, i.e. there does not exist y ∈ C(G) such that pi=1 y i > pi=1 xi for all p ∈ {1, ..., n − 1} with at least one strict inequality. For a game G ∈ G N denote by LA(G) the set of its Lorenz allocations. Example 3. Consider balanced   7 ν(K) = 12  0

four-player TU game GT defined by if (K = {1, 2}) ∨ (K = {1, 3}), if K = N, else.

In was proved (Arin et al., 2008) that the set of Lorenz allocations is of the form LA(GT ) = {x ∈ C(GT )| x = (7 − µ, µ, µ, 5 − µ), 2

1 1) 6 µ 6 3 }. 2 2

435


Taking µ = 3 12 , µ = 2 12 and µ = 3 yield the lexmax solution Lmax(GT ) = (3 12 , 3 21 , 3 21 , 1 21 ), the lexmin solution Lmin(GT ) = (4 21 , 2 21 , 2 21 , 2 21 ) and least squares solution LS(GT ) = (4, 3, 3, 2), respectively ((Arin et al., 2008, p.571)). By the formulas in section 2 one obtains Sh(GT ) = (4 16 , 3, 3, 1 65 ) 6∈ LA(GT ), CIS(GT ) = EN SC(GT ) = ED(GT ) = (3, 3, 3, 3) 6∈ LA(GT ). The next theorem states that the symmetric core of balanced semi-symmetric TU game contains all Lorenz allocations. Besides, SC(GT ) is externally stabile with respect to Lorenz domination, but internal stability does not hold. Theorem 2. Let GT ∈ SG N T is a balanced game. Then (i) LA(GT ) ⊆ SC(GT ) and the inclusion can be strict; (ii) SC(GT ) Lorenz dominates every other core allocation. Proof. (i) LA(GT ) satisfies equal treatment and LA(GT ) ⊆ C(GT ). Therefore, LA(GT ) ⊆ SC(GT ). The four-person TU game in Example 3 is semi-symmetric ℑ(GT ) = {{2, 3}}, 1 1 1 1 1 1 1 1 LA(GT ) = co{(3 , 3 , 3 , 1 ), (4 , 2 , 2 , 2 )} 2 2 2 2 2 2 2 2

⊂ SC(GT ) = co{(2, 5, 5, 0), (7, 0, 0, 5), (12, 0, 0, 0)}.

(ii) If C(GT ) = SC(GT ) then the statement is straightforward. Let C(GT ) 6= SC(GT ) and take x0 ∈ C(GT )\SC(GT ). Then there exists K ∈ ℑ(GT ) and i, j ∈ K such that x0j > x0i . By symmetry there is y ∈ C(GT ) with yi = x0j , yj = x0i , yl = x0l 0

for l ∈ N \ {i, j}. Consider x1 = x 2+y . By core convexity x1 ∈ C(GT ). Vector x1 Lorenz dominates x0 (x1 ≻L x0 ) since x1j = x1i = x0j − δ = x0i + δ, x1l = x0l for l ∈ N \{i, j}, δ > 0. Repetition of this procedure gets the sequence x0 , x1 , ..., xp core allocations, where xk ≻L xk−1 for all k ∈ {1, ..., p}, x0 6∈ SC(GT ), xp ∈ SC(GT ). The transitive property of Lorenz domination completes the proof. ⊔ ⊓ 4.

Existence conditions

The balancedness condition (2) is derived by means of dual linear programming problems associated with a game GT ∈ GTN X X f (x) = xi → min, xi > ν(K), K ∈ Ω, (4) i∈N

g(λ) =

X

K∈Ω

i∈K

ν(K)λK → max,

X

K∈Ω,i∈K

n

λK = 1, i ∈ N, λ ∈ R2+ −2 .

(5)

The condition (2) can be as well written as X λK ν(K) 6 ν(N ), λ ∈ ext(M n ), K∈Ω

where ext(M ) is the set of extreme points of problem (5) constraint set M n . The number of extreme points and their explicit representation known only for small n n

|ext(M 3 )| = 5, |ext(M 4 )| = 41, |ext(M 5 )| = 1291, |ext(M 6 )| = 200213.

We concentrate now on n-person non-negative semi-symmetric TU games in zero0 normal form (SG N T )+ . The following example illustrates how the problem (4) is modified by replacing the core by symmetric core.

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0 Example 4. Consider two four-person games (G0T )1 , (G0T )2 ∈ (SG N T )+ with two and 0 1 0 2 three symmetric players, ℑ((GT ) ) = {{3, 4}}, ℑ((GT ) ) = {{2, 3, 4}}. The explicit representations of (4) and modified problems given in table 1. It is remarkable that the number of extreme points of modified dual problems constraint sets Ms4 , where s is the number of symmetric players, decreases as s increases: |ext(M24 )| = 21, |ext(M34 )| = 6.

Table 1. Original problem

Modified problem 1, Modified problem 2, ℑ((G0T )1 ) = {{3, 4}} ℑ((G0T )2 ) = {{2, 3, 4}} f (x) = x1 + x2 + x3 + x4 → min f (x) = x1 + x2 + 2x3 → min f (x) = x1 + 3x2 → min xi > 0, i ∈ {1, 2, 3, 4} xi > 0, i ∈ {1, 2, 3} xi > 0, i ∈ {1, 2} x1 + x2 > ν(1, 2) x1 + x2 > ν(1, 2) x1 + x2 > ν(1, 2) x1 + x3 > ν(1, 3) x1 + x3 > ν(1, 3) x1 + x4 > ν(1, 4) x2 + x3 > ν(2, 3) x2 + x3 > ν(2, 3) 2x2 > ν(2, 3) x2 + x4 > ν(2, 4) x3 + x4 > ν(3, 4) 2x3 > ν(3, 4) x1 + x2 + x3 > ν(1, 2, 3) x1 + x2 + x3 > ν(1, 2, 3) x1 + 2x2 > ν(1, 2, 3) x1 + x2 + x4 > ν(1, 2, 4) x1 + x3 + x4 > ν(1, 3, 4) x1 + 2x3 > ν(1, 3, 4) x2 + x3 + x4 > ν(2, 3, 4) x2 + 2x3 > ν(2, 3, 4) 3x2 > ν(2, 3, 4)

The symmetry of all players makes a game especially easy to handle. The criterion for existence of its core (and, by Proposition 1, for symmetric core too) contains (n − 1) inequalities only ν(K) ν(N ) 6 f or all K ∈ Ω. |K| n

It is then natural to focus the attention on games with (n − 1) symmetric players. Notice that any such game is determined by 2(n − 2) numbers ν(K), K ∈ Ω1 ∪ Ω2 , where Ω1 = {{2, 3}, {2, 3, 4}, ..., {2, ..., n}}, Ω2 = {{1, 2}, {1, 2, 3}, ..., {1, ..., n − 1}}. A few of their applications: • market with one seller and symmetric buyers; • games with a landlord and landless workers; • weighted majority game with one large party and (n − 1) equal sized smaller parties; • patent licensing game with the firms each producing an identical commodity and a licensor of a patented technology (Watanabe and Muto, 2008); • subclass of games related information collecting situations under uncertainty (Branzei et al., 2000) where an action taker can obtain more information from other agents; • big boss games (Muto et al.,1988) with symmetric powerless players. The characterization of such games and the sufficient conditions under which the symmetric core is a singleton have been provided in (Zinchenko, 2012). Let


437

0 0 0 G0T ∈ (SG N T )+ , ℑ(GT ) = {{2, ..., n}} and n > 3. The symmetric core of game GT is nonempty iff the system

ν 0 (T ) +

n − |T | 0 n−1 0 ν (H) 6 ν 0 (N ), ν (H) 6 ν 0 (N ), H ∈ Ω1 , T ∈ Ω2 |H| |H|

is consistent. Notice that system consists of (n − 1)(n − 2) inequalities. If G0T ∈ 0 0 0 (SG N T )+ is a balanced game, ℑ(GT ) = {{2, ..., n}}, n > 4 and ν satisfies at least one of three equalities n−1 0 n−2 0 ν (N \ {1, n}) = ν 0 (N ), ν (N \ 1) + ν 0 (1, 2) = ν 0 (N ), n−2 n−1 ν 0 (N \ 1) + ν 0 (N \ n) = ν 0 (N ) n−1

then SC(G0T ) consists of a unique allocation. References

Arin, J., J. Kuipers and D. Vermeulen (2008). An axiomatic approach to egalitarianism in TU-games. International Journal of Game Theory, 37, 565–580. Aumann, R. J. (1987). Value, symmetry and equal treatment: a comment on scafuri and yannelis. Econometrica, 55(6), 1461–1464. Azamkhuzhaev, M. Kh. (1991). Nonemptiness conditions for cores of discrete cooperative game. Computational Mathematics and Modeling, 2(4), 406–411. Bondareva, O. N. (1963). Certain applications of the methods of linear programing to the theory of cooperative games. Problemy Kibernetiki, 10, 119–139 (in Russian). Branzei, R., S. Tijs and J. Timmer (2000). Collecting information to improve decision making. International Game Theory Review, 3, 1–12. van den Brink, R. and Y. Funaki (2009). Axiomatizations of a class of equal surplus sharing solutions for cooperative games with transferable utility. Theory and Decision, 67, 303– 340. Dutta, B. and D. Ray (1989). A concept of egalitarianism under participation constraints. Econometrica, 57, 403–422. Hougaard, J. L., B. Peleg and L. Thorlund-Petersen (2001). On the set of Lorenz-maximal imputations in the core of a balanced game. International Journal of Game Theory, 30, 147–165. Llerena, F. and R. Carles (2005). On reasonable outcomes and the core. Barcelona Economics Working Paper Series, 160, 1–9. Muto, S., M. Nakayama, J. Potters and S. Tijs (1988). On big boss games. The Economic Studies Quarterly, 39, 303–321. Norde, H., V. Fragnelli, I. Garcia-Jurado, F. Patrone and S. Tijs (2002). Balancedness of infrastructure cost games. European Journal of Operational Research, 136, 635–654. Shapley, L. S. (1967). On balanced sets and cores. Naval Research Logistics Quarterly, 14, 453–460. de Waegenaere, A., J. Suijs and S. Tijs (2005). Stable profit sharing in cooperative investment. OR Spectrum, 27(1), 85–93. Watanabe, N. and S. Muto (2008). Stable profit sharing in a patent licensing game: general bargaining outcomes. International Journal of Game Theory, 37(4), 505–523. Zinchenko A. B. (2012). Semi-symmetric TU-games. Izvestiya vuzov. Severo-Kavkazckii region. Natural science, 5, 10–14 (in Russian).

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