The ¿-value for games on matroids - CiteSeerX

The ¿ -value for games on matroids J. M. Bilbao¤ , A. Jiménez-Losada¤ , E. Lebrón¤ , S. H. Tijsy ¤

Departamento de Matemática Aplicada II Escuela Superior de Ingenieros, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain ([email protected]) y CentER and Department of Econometrics & OR, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Netherlands ([email protected])

Abstract In the classical model of games with transferable utility one assumes that each subgroup of players can form and cooperate to obtain its value. However, we can think that in some situations this assumption is not realistic, that is, not all coalitions are feasible. This suggests that it is necessary to raise the whole question of generalizing the concept of transferable utility game, and therefore to introduce new solution concepts. In this paper we de…ne games on matroids and extend the ¿-value as a compromise value for these games. Keywords: ¿-value, matroid, cooperative game Mathematics Subject Classi…cation 2000: 91A12

1

Introduction

Faigle and Kern [4] proposed a model in which cooperation among players is restricted to a certain family of subsets of players, the feasible coalitions of the game. Their idea is to restrict the allowable coalitions by using underlying partially ordered sets and they obtained an extension of the Shapley value for games on distributive lattices. Their model is a special case of the games on convex geometries studied by Bilbao and Edelman [1]. In the present paper, we will de…ne the feasible coalitions by using combinatorial geometries called matroids. Games on matroids are introduced by Bilbao, Driessen, JiménezLosada and Lebrón [2]. Let us assume that there are two rules of cooperation between players: ² If a coalition can form, then any subcoalition is also feasible because if the players that take part in the formation of a coalition have common interests, a subset of these players has at least the same common interests. ² If there are two coalitions where the cardinality di¤ers one, there is a player of the largest coalition which can join the smallest making a feasible coalition. 1

The characteristic function of a game on a matroid is only de…ned on the elements of the matroid. We propose a generalization of the ¿ -value introduced by Tijs [6, 8] as a concept of compromise value for games de…ned on matroids. Let us brie‡y outline the content. In the next section, we de…ne matroids, describe some of their properties and introduce games on matroids. In the last section we extend the concept of the ¿ -value to games on matroids and obtain a family of axioms that characterize uniquely this value. Now, we introduce several concepts from the theory of cooperative games. De…nition 1.1 A transferable utility game is a pair (N; v), where N is a …nite set and v : 2N ! R; is a function with v(;) = 0: The elements of N = f1; : : : ; ng are called players, the subsets S 2 2N coalitions and v(S) is the worth of S. The worth of a coalition is interpreted as the maximal pro…t or minimal cost for the players in that coalition. The function v is called the characteristic function. Given a game (N; v) and a coalition S µ N, the subgame (S; v) is obtained by restricting v to 2S . By ¡N we denote the set of all games (N; v) where N is the set of players. We will use a shorthand notation and write S [ i for S [ fig, and S n i for S n fig: n is called e¢cient if it distributes the In a game (N; v); a vector x 2 RP worth v(N) among the players, i.e., i2N xi = v(N): The set of all e¢cient vectors is called the preimputation set and is denoted by I ¤ (v): The imputation set is de…ned by I (v) := fx 2 I ¤ (v) : xi ¸ v (i) for all i 2 Ng : P Note that I (v) 6= ; if and only if v(N) ¸ i2N v(i): Assuming that the coalition N of all players will be formed, a one-point solution concept will prescribe a distribution of the worth v(N) among P the players. Given a vector x 2 Rn and a coalition S; we de…ne x (S) := i2S xi and x(;) = 0.

De…nition 1.2 The core of a game (N; v) is the set

C (v) := fx 2 I ¤ (v) : x (S) ¸ v (S) for all S µ Ng : Notice that no coalition S µ N is able to improve upon x 2 C (v) by splitting o¤. Games with a nonempty core are called balanced games. The ¿ -value of a game is a compromise between the upper and the lower vectors for the game introduced by Tijs [6] (see the survey [8]). The upper vector of the game (N; v) is the n-vector M v ; where Miv := v(N) ¡ v(N n i); for all i 2 N. The component Miv is called utopia payo¤ for player i in the grand coalition: The remainder of i 2 S if the coalition S forms and all other players in S obtain their utopia payo¤ is Rv (S; i) := v(S) ¡ M v (S n i). The lower vector is the vector mv 2 Rn , de…ned by mvi = max Rv (S; i): fS:i2Sg

2

De…nition 1.3 A game (N; v) is called quasi-balanced if it satis…es: (QB1) mv · M v ; and (QB2) mv (N) · v(N) · M v (N): The family of quasi-balanced games is a full-dimensional cone in the (2n ¡ 1)-dimensional vector space ¡N and contains the family of balanced games as a subset [6]. For a quasi-balanced game v the ¿ -value, denoted by ¿ v , is the unique preimputation (e¢cient vector) on the closed interval v n: [mv ; M Then we have ¿ v = mv + ¸ (M v ¡ mv ) ; where ¸ 2 R is such P ] in R v that i2N ¿ i = v(N).

2

Matroids and games

Let N = f1; : : : ; ng be a …nite set. A set system over N is a pair (N; F), where F µ 2N is a collection of subsets of N. De…nition 2.1 A matroid is a set system (N; M) with the following properties: (M1) ; 2 M. (M2) If S 2 M and T µ S, then T 2 M. (M3) If T; S 2 M and jSj = jT j + 1; then there exists an i 2 S n T such that T [ fig 2 M. Throughout this paper we will often denote the matroid (N; M) by M. We present some examples of matroids. Example 2.1 For k 2 N, 0 < k · n, the family fS µ N : jSj · kg ; of all subsets of N of size at most k is the uniform matroid Unk . In particular, 2N is called the free matroid. Example 2.2 If i; j 2 N; i 6= j; the family Mn (ikj) = fS µ N : fi; jg " Sg, is a matroid. Example 2.3 Let E be the set of edges of a graph G and let M = M(G) be a family that consists of those subsets of E that do not contain a circuit of G. The matroid (E; M) is called a graphic matroid. De…nition 2.2 A game on a matroid M is a function v : M ! R with v (;) = 0. We denote by ¡ (M) the set of all games on the matroid M 6= f;g. The set ¡ (M) is a vector space over R. We also need to introduce the following de…nitions and results. 3

De…nition 2.3 For a given set system (N; M) and S µ N; a subset B µ S; B 2 M is called a basis of S if B [ fig 2 = M for all i 2 S n B: In a matroid, B is a basis of S when B is a maximal feasible subset of S. The above de…nition can be used to characterize a matroid (see Welsh [9]). Proposition 2.1 A set system (N; M) is a matroid if and only if it satis…es (M1), (M2), and the following property: For every S µ N, all the basis of S have the same cardinality. Let (N; M) be a matroid. A basis of the ground set N is called basic coalition. We denote by B the set of basic coalitions of (N; M) and its cardinality by b = jBj. The rank of the matroid (N; M) is the cardinality of the basic coalitions. Then we have B = fB 2 M : jBj = rg ; where r is the rank of M. Remark 2.1 Note that the length of all the maximal chains of a matroid M is the rank of M. We can consider a matroid M 6= 2N split into in‡uence zones, where the basic coalitions are the summits of the di¤erent in‡uence zones. Note that these in‡uence zones are not disjoint. De…nition 2.4 The in‡uence set of S 2 M is the set BS = fB 2 B : S µ Bg ; of the basic coalitions that contain S and its in‡uence worth is the quotient bS =b, where bS = jBS j. We denote bfig by bi when S = fig. Note that M = 2N if and only if b = 1.

3

The ¿ -value for games on matroids

Let M be a matroid on N = f1; : : : ; ng such that fig 2 M for all i 2 N; and let v 2 ¡ (M). The upper vector M v 2 Rn is given by 1 X Miv = (v (B) ¡ v (B n i)) : bi B2Bi

If we consider the e¢ciency property on matroids then the payo¤ for the players must not exceed this value and therefore M v is the utopia vector. Let i 2 N and we consider S 2 M such that i 2 S. If the other players in S obtain their utopia vectors, then the number Rv (S; i) = v (S) ¡ M v (S n i) ; is called the remainder for player i in the coalition S. The lower vector mv 2 Rn is de…ned by mvi =

max

fS2M: i2Sg

fRv (S; i)g :

This value is the minimal value expected by the player i and then is called also the minimal right vector. 4

Remark 3.1 A di¤erent model of behavior of the players is given by the following utopia vector sup Miv = max fv (B) ¡ v (B n i)g : B2Bi

Proposition 3.1 If v 2 ¡ (M), d 2 ¡ (M) is an additive game and ® 2 R, then 1. M ®v+d = ®M v + d; 2. R®v+d (S; i) = ®Rv (S; i) +di ; where S 2 M and i 2 S. Proof. 1. Let i 2 N;

1 X ((®v + d) (B) ¡ (®v + d) (B n i)) bi B2Bi 1 X (®v (B) ¡ ®v (B n i)) + di = bi B2Bi ® X = (v (B) ¡ v (B n i)) + di bi

Mi®v+d =

=

B2Bi ®Miv + di :

2. Let S 2 M and i 2 S; R®v+d (S; i) = ®v (S) + d (S) ¡ M ®v+d (S n i) = ®v (S) + d (S) ¡ ®M v (S n i) ¡ d (S n i) = ®Rv (S; i) + di : ¤ In the context of games on matroids, we de…ne the gap function introduced by Driessen and Tijs [3]. De…nition 3.1 The gap function of v 2 ¡ (M) is the new game gv : M ! R given by g v (S) = M v (S) ¡ v (S) ; for all S 2 M. Note that the maximal concession which a player i can give is the number ¸vi =

min

fS2M:i2Sg

g v (S) ;

and therefore we call ¸v the concession vector. It is easy to prove that the lower vector is the di¤erence between the upper and the concession vector. Proposition 3.2 If v 2 ¡ (M) then mv = M v ¡ ¸v . 5

Proposition 3.3 Let v 2 ¡ (M), d an additive game on M and ® 2 R. Then 1. g ®v+d (S) = ®gv (S) for all S 2 M, 2. ¸®v+d = ®¸v , if ® ¸ 0. 3. m®v+d = ®mv + d; if ® ¸ 0: Proof. 1. Let S 2 M; then g®v+d (S) = M ®v+d (S) ¡ ®v (S) ¡ d (S)

= ®M v (S) + d (S) ¡ ®v (S) ¡ d (S) = ®g v (S) :

2. Let i 2 N; then ¸®v+d = i =

min

g ®v+d (S)

min

®gv (S)

fS2M: i2Sg fS2M: i2Sg

=®

min

fS2M: i2Sg

gv (S) = ®¸vi : ¤

3. This property follows from Propositions 3.1 and 3.2. De…nition 3.2 A game v 2 ¡ (M) is quasi-balanced if it satis…es: 1. mv · M v : P P P v v 2. B2B m (B) · B2B v (B) · B2B M (B) :

The class of quasi-balanced games on matroids is denoted by QB (M). We can also de…ne this class using the gap function and the concession vector, ) ( X X X v v v m (B) · v (B) · M (B) QB (M) = v 2 ¡ (M) : ¸ ¸ 0; =

(

B2B

v 2 ¡ (M) : g v (S) ¸ 0; 8S 2 M;

B2B

B2B

X

X

B2B

gv (B) ·

B2B

)

¸v (B) :

De…nition 3.3 The core of a game on a matroid v 2 ¡ (M) is C (v; M) = fx 2 Rn : x(B) = v(B) for all B 2 B; x (S) ¸ v (S) for all S 2 Mg : We prove now that the core elements of games on matroids are bounded by the upper and the lower vectors. 6

Theorem 3.4 Let v 2 ¡ (M) be a game such that C (v; M) 6= ;. Then for all x 2 C (v) we have mv · x · M v : Proof. If x 2 C (v) then v (B) = x (B) and v (B n i) · x (B n i) for all the basic coalitions. So 1 X (v (B) ¡ v (B n i)) Miv = bi B2Bi 1 X ¸ (x (B) ¡ x (B n i)) bi B2Bi 1 X xi = xi . = bi B2Bi

Further mvi =

max

fS2M:i2Sg

fRv (S; i)g

= Rv (T; i) = v (T ) ¡ M v (T n i) · x (T ) ¡ x (T n i) = xi , where T 2 arg

max

fS2M:i2Sg

fRv (S; i)g : ¤

Corollary 3.5 If v 2 ¡ (M) is a game with C (v; M) 6= ; then v 2 QB (M) : The ¿ -value for a game on a matroid is the unique e¢cient vector on the segment between mv and M v . De…nition 3.4 Let v 2 ¡ (M). The ¿ -value of v is the vector de…ned by ¿ v = mv + ® (M v ¡ mv ) ; P P where ® 2 R is such that i2N bi ¿ vi = B2B v (B) :

This value is given by 8 v if g v (B) = 0 for all B 2 B; 0. By Propositions 3.3 and 3.1 we obtain that the game ®v + d 2 QB (M) and the value ¿ ®v+d = ®¿ v + d: Restricted proportionality: This property follows from ¿ v = mv + ® (M v ¡ mv ) : ¤ Theorem 3.7 The ¿ -value is the unique value on QB (M) which satis…es e¢ciency, covariance and the restricted proportionality property. Proof. The ¿ -value satis…es the above properties by Theorem 3.6. We only prove that is the unique one-point solution with these properties. Let ª be a value that satis…es the axioms and let v 2 QB (M). We consider the game w = v ¡ mv : From the covariance we obtain that ª (w) = ª (v) ¡ mv and ¿ w = ¿ v ¡ mv : Thus we only need to prove that ª (w) = ¿ w . By Propositions 3.1 and 3.3, we have M w = M v ¡ mv and mw = 0. Then ¿ w = ¯M w and, by the restricted proportionality, ª (w) = ®M w . Since ª and ¿ are e¢cient we ¤ obtain that ® = ¯:

References [1] Bilbao, J. M. and P. H. Edelman (2000). The Shapley value on convex geometries. Discrete Appl. Math. 103, 33-40. [2] Bilbao, J. M., T. S. H. Driessen, A. Jiménez-Losada and E. Lebrón (2001). The Shapley value for games on matroids: The static model. Math. Method. Oper. Res. 53 (2). [3] Driessen, T. S. H. and S. H. Tijs (1983). The ¿ -value, the nucleolus and the core for a subclass of games. Methods of Operations Research 46, 395–406. [4] Faigle, U. and W. Kern (1992). The Shapley value for cooperative games under precedence constraints. Int. J. Game Theory 21, 249–266. 8

[5] Stanley, R. P. (1986). Enumerative Combinatorics I. Wadsworth, Monterey, CA. [6] Tijs, S. H. (1981). Bounds for the core and the ¿ -value. In O. Moeschlin and D. Pallaschke, Eds., Game Theory and Mathematical Economics, North-Holland, Amsterdam. [7] Tijs, S. H. (1987). An axiomatization of the ¿ -value. Math. Social Sciences 12, 9–20. [8] Tijs, S. H. and G.-J. Otten (1993). Compromise values in cooperative game theory. Top 1, 1–51. [9] Welsh, D. J. A. (1995). Matroids: Fundamental Concepts. In R. Graham, M. Grötschel and L. Lovász, Eds., Handbook of Combinatorics, Elsevier Science B. V., 481–526.

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