Dual Games on Combinatorial Structures

In the classical model of cooperative games it is generally assumed that there are no restrictions on cooperation and hence, every subset of players is a feasible coalition. However, in many social and economic situations, this model does not apply. Examples are provided by local public goods which are supplied by local communities, social and sports clubs, labor unions, political parties, and other institutions. We will de…ne the feasible coalitions by using the dual combinatorial structures called convex geometries and antimatroids. We introduce the Shapley and Banzhaf values for these games and observe the axioms which characterize such values in detail. 1. Introduction Let N be a …nite set of players. A pair (N; v); where v : 2 ! R satis…es v(;) = 0; is a game in coalitional form. The subsets S 2 2 are called coalitions. The coalitions of the game are the subsets of the set N and ¡ 2 ;[;\ ¢ form a Boolean algebra. An important generalization of a Boolean algebra is a distributive lattice which has the join (_) and meet (^) operations with the same properties as in the case of Boolean algebras but without the complementation operation. It is no true that every member of the lattice is a join of single elements, but it is true that every member is a join of join-irreducible elements. A join-irreducible is an element of the lattice which cannot be represented as a join of elements distinct from itself. Birkho¤ [5] proved that every element of a …nite distributive lattice has a unique irredundant decomposition as a join of joinirreducible elements. Antimatroids seem to have been considered …rst by Dilworth [8] who investigated representations of an element a in a …nite and semimodular lattice as a meet a = V M of a set M of meet-irreducible elements. This paper is organized as follows. Convex geometries and antimatroids, such as their properties, are treated in section 2. We interpret their properties in the framework of partial cooperation and explain the duality relation between them. In section 3, this duality is translated to vector spaces of the corresponding games. The Shapley and Banzhaf values are studied in sections 4 and 5. These values have been investigated by Bilbao [1], Bilbao and Edeman [4], and Bilbao, Jiménez-Losada and López [2] on convex geometries. Key words and phrases. Antimatroid, convex geometry, cooperative game, Shapley value, Banzhaf value. 1 2 J. M. BILBAO, C. CHACÓN, A. JIMÉNEZ-LOSADA, AND E. LEBRÓN 2. Convex geometries and antimatroids We refer the reader to Korte, Lovász and Schrader [13] for a detailed treatment of combinatorial structures. Convex geometries are a combinatorial abstraction of convex sets introduced by Edelman and Jamison [10]. De…nition 2.1. A family L of subsets of N is a convex geometry if it satis…es the following properties: (C1) ; 2 L. (C2) L is closed under intersection. (C3) If S 2 L and S 6= N , then there exists j 2 N n S such that S [ j 2 L. Property (C2) implies that intersections of feasible coalitions should also be feasible, since the players agree on a pro…le of cooperation and (C3) is the augmentation property. We call the sets in a convex geometry convex sets. A maximal chain of L μ 2 is an ordered collection of convex sets that is not contained in any larger chain. Edelman and Jamison [10] showed that every maximal chain contains n+1 convex sets ; = S0 1⁄2 S1 1⁄2 ¢ ¢ ¢ 1⁄2 Sn¡1 1⁄2 Sn = N; and the cardinal jSij = i; for all i = 0; 1; : : : ; n: Moreover, the hierarchical situations by Moulin [14], when users pay their incremental costs according to an ordering of N; can be modeled by convex geometries. The map ¡ : 2 ! L de…ned by