The Shapley value for games on matroids: The static model

In the classical model of cooperative games, it is considered that each coalition of players can form and cooperate to obtain its worth. However, we can think that in some situations this assumption is not real, that is, all the coalitions are not feasible. This suggests that it is necessary to rise the whole question of generalizing the concept of cooperative game, and therefore to introduce appropriate solution concepts. We propose a model for games on a matroid, based in several important properties of this combinatorial structure and we introduce the probabilistic Shapley value for games on matroids. Journal of Economic Literature Classi…cation Number: C71. 1. Introduction A cooperative game is a pair (N;v) of a …nite set N of players and a characteristic function v : 2N ! R; such that v (;) = 0. A subset S of N is called a coalition. This paper is concerned with cooperative games in which the cooperation among players is partial. We will consider that there are two rules of cooperation between players: 2 If a coalition may form, then every subset is also feasible, since if the players that take part in the formation of a coalition have common interests, then every subset of these players has at least the same common interests. 2 Given two feasible coalitions with di¤erent number of players, there is a player of the largest that he can join with the smallest making a feasible coalition. For this reason, we will de…ne the feasible coalitions by using combinatorial geometries called matroids. The set systems called matroids were introduced by Whitney (1935) as an abstraction of linear independence and the cyclic structure of graphs. The origin of the actual matroid theory is the work of Tutte (1959) and it has numerous applications in combinatorics 1 2 J. M. BILBAO, T. S. H. DRIESSEN, A. JIMÉNEZ LOSADA, AND E. LEBRÓN and optimization theory. We refer the reader to Welsh (1976) and Korte, Lovász and Schrader (1991) for a detailed treatment of matroids. Let us outline the contents. Section 2 treats the essential notions on matroids, such as its properties, its rank function and its basic coalitions (being maximal feasible coalitions with respect to inclusion of sets). For the sake of the game theoretic approach, the rank function of a matroid is interpreted as a classical cooperative game and next, the game theoretic solution concept called core is de…ned as the set of optimal solutions of a certain linear programming problem in which the rank function of the matroid is involved. Edmonds (1970) showed that the core coincides with the convex hull of the incidence vectors corresponding to basic coalitions of the matroid. Section 3 introduces the concept of a cooperative game on a matroid as a real-valued function on the matroid itself. In other words, the characteristic function of this type of a cooperative game is de…ned only for feasible coalitions arising from the matroid. Similar, but di¤erent versions already do exist, see Faigle (1989) and Nagamochi, Zeng, Kabutoya and Ibaraki (1997). The main part of Section 3 deals with the axiomatic development of the solution theory for games on matroids. As a matter of fact, we are concerned with the linearity axiom (in the variable being the characteristic function of the game), the monotonicity axiom (solutions should allocate nonnegative payo¤s to players whenever the utility of coalitions increases in accordance with the inclusion of coalitions), as well as the dummy player axiom (nonimportant players receive their natural solutions). The solutions that satisfy these three axioms are characterized as the so-called quasi-probabilistic values. Such a solution for any individual player may be interpreted as some expected outcome based on the player’s marginal contributions for joining the feasible coalitions of the induced contraction matroid. Another equivalence theorem (with no reference to axioms anymore; see Theorem 3.2) states that an individual solution is a quasi-probabilistic value if and only if the solution is decomposable as the weighted sum of certain solutions for induced subgames de…ned on power sets associated with the basic coalitions of the matroid. The relevant weights are interpreted as a probabilty distribution over the set of basic coalitions of the matroid. Section 4 introduces the solution concept called Shapley value for games on matroids, meant to be a generalization of the known Shapley value for classical cooperative games. The axiomatic approach taken here involves, besides the linearity and substitution axioms, a probabilistic version of the e¢ciency and dummy player property. In this framework, it is supposed that basic coalitions are formed randomly according to a …xed probability distribution over the set of basic coalitions of the matroid. As a result of this axiomatic approach, two explicit formulas for the probabilistic Shapley value of games on matroids are presented and discussed. THE SHAPLEY VALUE FOR GAMES ON MATROIDS: THE STATIC MODEL 3 2. Essential notions on matroids A matroid is a pair (N;M) consisting of a …nite set N and a set M of subsets of N with ; 2 M and satisfying the following two properties: (M1) If S 2M and T μ S; then T 2M. (M2) If S; T 2 M with jSj = jT j+ 1, then there exists i 2 S n T such that T [ fig 2 M. The rank function r : 2N ! Z+ of a matroid M on N is de…ned by r(X) := max fjSj : S μ X; S 2Mg for all X μ N: (1) Notice that S 2M if and only if r(S) = jSj. The following two theorems (see Korte et al. (1991)) axiomatize matroids in terms of their rank functions. Theorem 2.1. A function r : 2N ! Z+ is the rank function of a matroid on N if and only if, for all X; Y μ N , the following holds: (R1) 0 · r(X) · jXj: (R2) r(X) · r(Y ) whenever X μ Y: (R3) r(X [ Y ) + r(X \ Y ) · r(X) + r(Y ): Theorem 2.2. A function r : 2N ! Z+ is the rank function of a matroid on N if and only if, for all X μ N and all i; j 2 N nX, the following holds: (R1’) r(;) = 0: (R2’) r(X) · r(X [ fig) · r(X) + 1: (R3’) If r(X [ fig) = r(X [ fjg) = r(X), then r(X [ fi; jg) = r(X): In the setting of the two theorems above, the function r determines uniquely the corresponding matroid through M = fS μ N : r(S) = jSjg. Elements of a given matroid are called feasible sets and further, a maximal feasible set (with respect to inclusion of sets) is called a basic set. Property (M1) implies that all the subsets of any basic set are feasible sets too and thus, 2B μ M for every basic set B of the matroid M. It is known that all the basic sets have the same cardinality and thus, jBj = r(N) for every basic set B of the matroidMμ 2N . Throughout this work we suppose that S S2M fi : i 2 Sg = N . In addition, the basic sets are of particular interest to determine the optimal solutions of a certain linear programming problem arising from the rank function of a matroid. Since this paper aims to develop the solution theory for cooperative games on matroids, we start to interpret the rank function r : 2N ! Z+ of a matroid as a classical cooperative game (N; r) with player set N . The rank function indicates the maximal feasible cooperation level between the players of a coalition. In this context, the solution set of the relevant LP-problem agrees with the well-known game-theoretic concept called core. The core of the game (N; r) is de…ned to be Core(N; r) := © x 2 R : x (N) = r(N); x (S) · r(S) for all S μ N a ; where x (S) := P i2S xi and x (;) = 0: For every set S μ N , we de…ne the incidence vector eS 2 RN such that ¡ eS ¢ i := 1 for all i 2 S and ¡ eS ¢ i := 0 4 J. M. BILBAO, T. S. H. DRIESSEN, A. JIMÉNEZ LOSADA, AND E. LEBRÓN otherwise. The following theorem has been showed by Edmonds (1970) and provides one interpretation of the core of a cooperative game induced by the rank function of a matroid. Theorem 2.3. Let r : 2N ! Z+ be the rank function of a matroidMμ 2N and B(M) the set of basic coalitions of M. Then Core(N; r) = conv © e : B 2 B(M) a : Proof. First, we establish that eB 2 Core(N; r) for all B 2 B(M). Indeed, for every B 2 B(M), it holds that P j2N (e )j = jBj = r(N) as well as P j2S ¡ eB ¢ j = jB \ Sj · r(S) for all S μ N: Since the core is a convex set, we deduce conv © eB : B 2 B(M) a μ Core(N; r). To prove the reverse inclusion, it su¢ces to show that the vertices of the core belong to the set © eB : B 2 B(M) a . In view of Theorem 2.1 (R3), the rank game (N; r) is submodular and hence, by Driessen (1988) (the greedy algorithm for LP-problems with a submodular objective-function), the vertices of the Core(N; r) are determined by the marginal worth vectors, the components of which are composed of the marginal contributions r(S [ fig) ¡ r(S), S μ Nnfig, of player i 2 N , in the rank game (N; r). Together with Theorem 2.2 (R2’), this implies that any marginal worth vector y = (yi)i2N of the rank game (N; r) satis…es yi 2 f0; 1g for all i 2 N . That is, y = eS for some S μ N . From y 2 Core(N; r) we deduce jSj = P j2S yj · r(S), whereas, by construction, r(S) · jSj. Thus, r(S) = jSj, or equivalently, S 2 M. Moreover, from the e¢ciency of y we deduce r(N) = P j2N yj = jSj. Finally, from S 2 M and jSj = r(N), we conclude that S 2 B(M) and hence y = eB for some B 2 B(M). Let S 2M: The contraction M=S of S from M is the new matroid M=S := fT 2M : T \ S = ; and T [ S 2Mg : Then the contraction of a feasible coalition S is a matroid formed with the feasible coalitions of the initial matroid that do not include any members of S, whereas its union with S is still a feasible coalition in the initial matroid. In the particular case of a individual coalition S = fig ; i 2 N , we use M=i instead of M= fig. Example 2.1. Given the set N = f1; : : : ; ng and any number k, 1 · k · n, we de…ne the uniform matroid Un k := fS μ N : jSj · kg. Coalitions of cardinality k are the basic coalitions and its rank function r : 2N ! Z+ is given by r(X) =minfk; jXjg for all X μ N . With reference to the uniform matroid U3 2 , as depicted in …gure 1, the core of the corresponding rank game is the convex hull of the vectors (1; 1; 0), (1; 0; 1), and (0; 1; 1). THE SHAPLEY VALUE FOR GAMES ON MATROIDS: THE STATIC MODEL 5 {} {1} {2} {3} {1, 2} {1, 3} {2, 3} Figure 1: The uniform matroid U3 2 Example 2.2. For any i; j 2 N , with i 6= j, we de…ne the opposition matroid Mn(ijjj) := fS μ N : fi; jg * Sg being the largest matroid which excludes coalitions containing both players i and j. There are two basic coalitions N n fig and N n fjg. Its rank function r : 2N ! Z+ is given by r(X) = jXj if X 2 Mn(ijjj) and r(X) = jXj ¡ 1 otherwise. Players 2 and 3 are called istmus players because they belong to every basic coalition of M4(1jj4).