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International Journal of Game Theory (1990) 19:45-57

Monotonicity of the Core and Value in Dynamic Cooperative Games By E. C. Rosenthal I

Abstract: We examine behavior of the core and value of certain classes of cooperative games in which a dynamic aspect is introduced. New players are added to the games while the underlying structure is held constant. This is done by considering games that satisfy properties like convexity, or games that are derived from optimization problems in which a player's addition can be defined naturally. For such games we give conditions regarding monotonicity of the core and value.

Key Words: Cooperative games, core, Shapley value, monotonicity.

1

Introduction

Since their introduction by von Neumann and Morgenstern [31] in 1944, cooperative games have received attention from a wide variety of theorists. The fundamental modeling technique in these investigations is the use of the characteristic function form for games, also pioneered in [31]. But use of the characteristic function form for games is double-edged: it gives us Understandable and tractable models and yet it is open to criticism on account of its enormous simplification of the possibly complex economic or social issues it is applied to. Responses to this criticism have gone in several directions. One conceptual advance has been the development of cooperative games without side payments (see Owen [18] or Shubik [28]). Another well-studied arena has been axiomatization of solution concepts, for example, in the work of Shapley [23], Harsanyi [5], Hart [6], Kalai and Samet [9], and Aumann [1], to name but a few papers. A different response has been not to refine either the characteristic function form or the prevailing solution concepts, but to attempt to improve the modeling technique. One way to better incorporate aspects of the "real-world" situations being studied is to describe and include the communication structures of the players, which may inhibit or enhance their economic possibilities. Such work began with Myerson [16] and continued in Grofman and Owen [4], Owen [19], and Rosenthal [21], [22].

1

Edward C. Rosenthal, Department of Management Science and Operations Management, School of Business and Management, Temple University, Philadelphia, PA 19122, USA.

0020-7276/90/1/45-57 $2.50 9 1990 Physica-Verlag, Heidelberg

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E.C. Rosenthal

The present paper illustrates a different approach to improving the modeling capabilities inherent in characteristic function form games. We wish to concentrate on a deficiency common to most, although not all, work in games both with and without transferable utility. This deficiency has involved static models with respect to the player sets in the games. In other words, the set of players is fixed as the characteristic function is constructed from the original problem. With this approach one is not easily able to model dynamic considerations, that is, those in which the player set changes over time. One reason we feel that this dynamic aspect has been missing in cooperative game theory is that only fairly recently has literature appeared involving a mathematical framework within which dynamic cooperative games may be studied. However, before we mention a useful set of structures that will provide a foundation for the study of dynamic cooperative games, we should mention that the idea of examining how a cooperative game changes (over time) is not novel. For example, Young [32] studied monotonicity and its relationship with solutions for side payment games. His sense of monotonicity is that if a game changes so that a player's contribution to all coalitions does not decrease, neither should this player's allocation decrease. Ichiishi [8] also considered conditions under which payoffs in two different games can be compared regarding various welfare criteria. In examining these results from the point of view of a dynamic framework, we see indeed that the games are changing - the characteristic functions are perturbed but theplayers are the same. It is our intention to turn this around and augment the games by adding players while maintaining an underlying structure. One way to keep an underlying structure in these games is simply to restrict the characteristic function, as in convex games, where v(S) is a supermodular set function (Shapley [24]). Another way is to let the characteristic function arise from the optimal solution to a mathematical programming problem for any set S of players, S c_ N; one instance of the use of this approach is by Mo [15] who analyzes entry effects of a new player on equilibria in assignment games. Background literature in this field, which we will call mathematical programming games, has grown steadily since its inception in Shapley and Shubik [25]. Other examples include Owen [17], Tamir [29], Granot and Huberman [3], Kalai and Zemel [11], [12], Topkis [30], Dubey and Shapley [2], Sharkey [27], and Rosenthal [20]. The aim, then, of the present paper is to describe how dynamic cooperative games arise and to consider the behavior of two solution concepts, the core and the Shapley value, for games of this type. Our criterion in evaluating the solution concepts will be whether they are monotonic, that is, if they yield allocations for the augmented games that are not inferior, for any player, to the allocation generated for the original game. The organization of this paper will be as follows: Section 2 treats preliminaries. In Section 3 we show that the Shapley value is monotonic for convex games. Then Section 4 treats flow games and develops conditions under which monotonicity of solutions is assured. Finally, in Section 5 we briefly turn to an NTU extension of the result in Section 3. -

Monotonicity of the Core and Value in Dynamic Cooperative Games

2

47

Preliminaries

Let N = {1,2,...,n] be a finite set of elements called players. We define a cooperative game with side payments (a transferable utility (TU) game, or a game) to be a pair (N;v) where v:2 N - R satisfies the following two conditions: v(~b) = 0, and v(S) + v(T) < v(S U T) for S,T c_ N such that S M T = ff (superadditivity).

(2.1) (2.2)

Note that v(S), S c N, called the characteristic function, is what S may obtain regardless of what the players in N-S do. We call S c N a coalition, N t h e grand coalition and the amount v(S) the worth of S. Let R N signify the space of real-valued vectors x = (xi), i E N, and let RN+ = {X E R N : x > 0} and RN++ = {x E R N : x > 0}. We define an allocation rule to be a method that generates an allocation x E R N s u c h that ~ x i = v(N), for some game (N;v). I f x satisfies x i > v({i}) for all iEN i E N, we say that x is an imputation. W h a t follows are two well-known methods of obtaining an imputation for a T U game. We define the core of a game as the set of all x C R N that satisfy

Xi >_ v(S)

for all S c_ N, and

(2.3)

iES

iEN

x i = v(N).

(2.4)

We will occasionally denote the core of a game (N;v) by Core (N;v). Note that the core of a game may be empty. Now let s = [S I, the cardinality of S __. N. We define the value q~[v] = (Oi[v]), i=1 .... ,n, also called the Shapley value (c.f. Shapley [23]) of a game (N;v) by

c)i[v] =

]~ (s-l) ! (n-s) ! [v(S) - v(S-{i})]. SC_N n! iES

(2.5)

At this point we observe that if the core of a game is large, then different core allocation rules may yield different core points. One natural goal would be to show, given any core allocation in the game (N;v), that a core allocation in the augmented game ( N ' ;w) exists which Pareto dominates the original allocation and is individually rational for the new player. Obtaining necessary and sufficient conditions for this quite general form of monotonicity remains an open problem. At present, then,

48

E.C. Rosenthal

what we must content ourselves with is to show monotonicity of the core with respect to a particular core rule. In case of checking the Shapley value, uniqueness is assured, so monotonicity is well-defined.

3

Convex Games, Monotonicity, and Dynamic Cooperative Games

3.1 Dynamic Cooperative Games Consider a game (N;v) to which we want to add an additional player k, k ~ N, to the player set. We then define a dynamic cooperative game (iV' ;w) by letting N ' = N O {k} and letting w satisfy (2.2) with respect to the augmented set N ' , provided that, for all S ___N, w(S) = v(S). Thus we may say that the game (N;v) is a subgame of (N' ;w). To give an example of the simplest case of a dynamic cooperative game, given a T U game (N;v), we first define a dummy player as one who contributes nothing to any coalition joined, that is, a player i such that v(S U {i}) - v(S) = 0 for all S c N-{i}. Note that we call a coalition T such that for all S c N, v(S) = v(S O T), the carrier of a game (N;v), so we see that d u m m y players do not belong to the carrier. A dynamic cooperative game (N' ;w) could be established by letting N ' = N U {k} where k is a dummy player and for which

w(T) : v(S), T = S o r T = S U {k} for T c_ N ' . In this instance it can be easily shown that the addition of a dummy player k will not affect nonemptiness of the core of a game, and will not affect the Shapley value allocation to the other players, i.e. 4~i[w] = r for all i ~: k, and ~k[w] = 0.

3.2 ConvexGames An important class of T U games is that of convex games. These games are such that all players contribute a non-decreasing marginal worth as the coalitions they join increase in size. We denote this " b a n d - w a g o n " effect by the "increment property,"

v(S U {i}) - v(S) < v ( T U {i]) - v(T)

(3.1)

for all i E N a n d S c T c N-{i}. Equivalently, we may write this behavior of the characteristic function as

v(S) + v(T) < v(S UT) + v(S A T )

(3.2)

Monotonicity of the Core and Value in Dynamic Cooperative Games

49

for all S , T c_ N , as in Shapley [24] and Ichiishi [7]. Real-valued set functions v that satisfy (3.2) are called supermodular. Shapley showed that cores of convex games are nonempty (they are specially structured polytopes) and that the value is the centroid of the core. We now define d y n a m i c c o n v e x g a m e s as dynamic cooperative games (N';w) whose characteristic function w is supermodular, and we proceed to a result in the next subsection.

3.3

Monotonicity of the Value for Dynamic Convex Games

Let (N' ;w) be a dynamic cooperative game, and let (N;v) be the associated game. Let y be an allocation, under allocation rule R, to (N;v). We say an allocation x = (xi), i= 1..... n , n + 1 to the dynamic cooperative game (N;w) is m o n o t o n i c (under R) if x i >_ Y i , i= 1 ..... n and if x k >_ w({k]). That is, the allocation x is not inferior, for any player, to the allocation for the original game. Now let (N';w) be a dynamic convex game as defined in w 3.2, and let q~[w] be the value of this game. T h e o r e m 1." The value ~b[w] is montonic for dynamic convex games. Proof" We want to show

]~ SC_N iES

(s-l)! (n-s)! [v(S)-v(S-[i})] _ O, associated with player k. By assumption, e k 6 C ' . We then obtain

w ( N ' ) = v ( N ) + cek,

(4.2)

as e k is saturated, and since C = C'-{e k } implies that the capacity of C equals v (N).

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E.C. Rosenthal

The core allocation under rule R 1 is trivially seen to be monotonic. All of the allocation of the added flow v (N') - v (N) goes to player k, while the other players associated with C receive the same allocations as before (which equal the edge capacities). To see that r will be monotonic, first note that for i = k, either w({k}) = 0 or e k = (s,t), in which case the value allocation is 4~k[w] = Cek while 4~i[w] = 4~i[v], i:~ k. In general, we observe that the addition of player k doubles the number of coalitions, so we consider coalition pairs consisting of an "old" coalition S c_ N and the same coalition augmented by k, S U {k}. Clearly, as Cek > 0, when e/~ E C' (in G'), no coalition S c_ N t h a t k joins has v(S) > w ( S U {k}). For ~[w] to be monotonic, we must show that no player's expected marginal "share" of flow decreases when N ' = N U {k}, e k E C ' . To do this we first need two facts. L e m m a 1: The edge set (C'-{ek}) is the unique minimum cut C for G. Proof" Shown by applying (4.2). Let x be a flow which maximizes v(N) and let Pi be an s,t-path in G ' that includes edgee/. We call Pi a f l o w augmenting path. I_emma 2: There does not exist a flow augmenting path Pk such that 0 for any edge e i E C.

(r

xei ) >

Proof" Suppose the lemma is false, i.e., there exists e i E Pk such that e i E C and (eel - Xei) > 0. Then edge e i is not saturated by flow x which maximized v ( N ) , or

else flow was not conserved, both contradictions. Lemma 2 indicates that all flow augmenting paths are those found in G or else paths that include edge e k but no other edges in C'. Now we may express 4~i[w] as the expected marginal contribution of flow by player i E N as i joins coalitions S and S U [k}, for all S _c N, using the partition of flow augmenting paths into those excluding e k and those including e k. The former set of augmenting paths is accounted for in the computation of 4~i[v] (in G), and the latter set will not decrease player i's overall expected marginal contribution. By this partition, (gi[w ] =

]~ (s-l) ! (n-s+l) t [w(S) - w(S-{i})] SC_N ' (n+l) ! iES ]~ (s-l) ! (n-s+1) ! [w(S) - w(S-{i})] SC_N' (n+l) ! iES k~S

Monotonicity of the Core and Value in Dynamic Cooperative Games + SC_N' iES kES = ~i[vl +

53

(s-1) l (n-s+l) t [w(S) - w(S-{i})] (n+l) !

~, (s-l) ! (n-s+l) ! [w(S) - w(S-{i})]. SC_N' (n+l) 1 iES kES

(4.3)

The second term in (4.3) is non-negative (in fact, it is strictly positive if and only if i E Pk for some Pk and no e E Pk is saturated). Therefore 0i[w] is monotonic under the conditions stated in the theorem. []

5

Treatment of NTU Games

In this section we give a natural extension to Theorem 1 for NTU games. To obtain a similar result we need to impose several restrictions, as will be seen below. After Sharkey [26], we define a convex game without side payments as follows: first, an NTU game, in general, is given by a pair (N;V) where N = {1,2..... n] is a finite set and V(S), S c N, satisfies (i)-(vi) below: (i)

V(S) is a closed, nonempty subset of R N and V(N) is convex; (x 1..... Xn) E V(S) a n d i ~ S t h e n x i = 0; I f S --/=4~, x E V ( S ) , y w. The following two results are useful, and are given without proof.

Theorem 3 (Sharkey): A convex N T U game has a nonempty core. Let x S represent an S-vector with components corresponding to i E S. Theorem 4 (Sharkey): Let (N;V) be a strong convex N T U game and suppose x S E Core (S;V). Then there exists x E Core (N;V) such that ~ = x i for all i E S. At this point, let a dynamic convex N T U game be defined as a convex N T U game with an extended player set in the manner of the preceding sections. We then see that Theorem 4 shows us that there exist core solutions that are monotonic in dynamic strong convex N T U games, since adding a player yields a game in a onedimensional-higher space in which a core point x E R N may be extended to a core point x ' E R N ' . Turning to the value, to obtain a result analogous to Theorem 1 we now restrict ourselves to the class of hyperplane games (Maschler and Owen [14]). We call a game (N;V) defined on finite set N a hyperplane game if V: 2 N - - R N i s such that

V(S) = {x E R N :

~ p S x i 0 for all i E S, for all S _c N. (Of course, certain values of r S might lead to a non-superadditive game, as formulated). When (5.2) is satisfied as an equation, the hyperplane itself is actually the Pareto boundary of V(S). Note also that if all p S = 1, this class of games reduces to the class of T U games. Now let 7r be a bijective mapping on N; in other words, let 7r be apermutation of N and let I I be the set of all permutations 7r. Maschler and Owen now define a vector y(lr) = (yi(a-)) as the vector, relative to some ordering 7r = (i1,i 2 ..... in) of N that allocates to player i I the most he can obtain in V({i 1 }), allocates to i2 the most he can obtain in V({i2 }) given what i 1 has already obtained, and so on. Heuristically, this is a "greedy algorithm" which, for some ordering 7r C II, will generate an extreme point in the core of (N;V). In [14], an expected marginal payoff vector r is now defined by r

=

1 ~ y(Tr). n! 7rEII

(5.3)

I f (N;V) is a TU game then r is precisely the Shapley value, and therefore r is a generalization of the value. We now call strong convex N T U games with feasible sets as in (5.2) convex hyperplane games. Our result follows.

Monotonicity of the Core and Value in Dynamic Cooperative Games

55

Theorem 5: The vector ~b[V] is monotonic and is a core allocation for convex hyperplane games. Proof" Recall that y(Tr) represent vectors that are constructed inductively by adding on a component at a time, relative to an ordering a-, and that their construction progresses monotonically. Let {a-1..... a-k}, k __. 1, be a set of permutations of N, and let y(Trj), j = l .... ,k be vectors as constructed above. Now let ym(Trj), m = l ..... n, represent the vector y(Trj) at the mth stage of construction. That is, all players i whose mapping i' under 7rj is such that i' > m => ym(Trj) = 0. Now define

r

= __1 Z

Ym(Trj) , m = l ..... n.

(5.4)

n! ~rfiII Note that ~n[v] = ~[V].

Lemma 3: The allocations ~/,m[v], m = 1..... n, are monotonic. Proof" By induction on m. For m = 1 ffm[v] weakly dominates (0 ..... 0). Suppose true for m _< q. For m = q + 1, we have added in one additional player to the q! orderings present. This player's addition to the game, by construction ofy(~r) a n d y m (Tr), will not decrease the allocation to the players already present. Therefore neither will the convex combination in (5.4). Now using L e m m a 3, we obtain the monotonicity o f ~b[V] since ~b[V]is the center of mass of the extreme points y (70. Finally, since all such points y (70 are in the core (by Theorem 4), so is ~b[V]. [] To extend Theorem 5 to N T U games whose Pareto boundaries are not hyperplanes, we note that the boundary of V(N) will, in general, dominate the hyperplane (5.2) (with S = N ) by invoking the conditions of convexity and comprehensiveness. Then selecting the point on the boundary could be accomplished much in the same manner as in the Kalai-Smorodinsky [10] approach to the Nash Bargaining Problem. The results in this paper serve to illustrate the development of a new approach to improving the modeling inherent in cooperative game theory, along with a few techniques appropriate in the various instances presented. Many restrictions resorted to were quite strong and it is hoped progress can be made regarding this. Another issue worth pursuing is obtaining conditions on the more general form of monotonicity described in Section 2. More generally, the author hopes that future research in characteristic function form games will be alert to the dynamic aspect of behavior in models in which new players join over time.

Acknowledgment: The author is very grateful to an anonymous referee for helpful remarks on earlier versions.

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E.C. Rosenthal

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[31] von Neumann J, Morgenstern O (1944) Theory of Games and Economic Behavior, Princeton University Press, Princeton [32] Young HP (1985) Monotonic Solutions of Cooperative Games, Int. J. Game Theory 14, 65-72

Received September 1988 Revised version April 1989 Final version September 1989