Axiomatizations of the Shapley value for cooperative games on antimatroids

Abstract

Abstract. Cooperative games on antimatroids are cooperative games restricted by a combinatorial structure which generalize the permission structure. So, cooperative games on antimatroids group several well-known families of games which have important applications in economics and politics. Therefore, the study of the rectricted games by antimatroids allows to unify criteria of various lines of research. The current paper establishes axioms that determine the restricted Shapley value on antimatroids by conditions on the cooperative game v and the structure determined by the antimatroid. This axiomatization generalizes the axiomatizations of both the conjunctive and disjunctive permission value for games with a permission structure. We also provide an axiomatization of the Shapley value restricted to the smaller class of poset antimatroids. Finally, we apply our model to auction situations. Mathematics Subject Classification 2000: 91A12

Power Measures and Solutions for Games Under Precedence Constraints

“…For example, Algaba et al [24] consider it in games on antimatroids, Bilbao and Edelman [5] studied it on convex geometries, and Bilbao and Ordoñez [25] and Algaba et al [26] on augmenting systems. Convex geometries and augmenting systems have been shown to be contained in the class of so-called regular set systems considered by Honda and Grabisch [7] and Lange and Grabisch [8], who also consider an extension of the precedence Shapley value to games on regular set systems.…”

Axiomatizations of two types of Shapley values for games on union closed systems

“…Notice that when v is monotone, it holds that for every ∈ C N also the restricted game r v, is monotone, since S ⊆ T implies that σ (S) ⊆ σ (T ). Now, the first solution is the union rule, which is defined similar as the Myerson rule for games with limited communication in Myerson (1977) and the Shapley value for games on antimatroids in Algaba et al (2003). The union rule, denoted by U , is given by…”

“…The first five axioms are generalizations of axioms used to axiomatize the conjunctive permission value in van den Brink and Gilles (1996) and the Shapley value for games on poset antimatroids in Algaba et al (2003). First, efficiency states that the total sum of payoffs equals the worth of the 'grand' coalition.…”

“…2 An example of an antimatroid is an acyclic permission structure where players need permission from (some of) their superiors in a hierarchical structure when they want cooperate with others. Since the concept of union closed system is more general than the notion of antimatroid, games on union closed systems are more general than the games on antimatroids as considered in Algaba et al (2003Algaba et al ( , 2004, and, therefore, also more general than the games with acyclic permission structure, considered in Gilles et al (1992), van den Brink and Gilles (1996), Gilles and Owen (1994) and van den Brink (1997).…”

“…Both solutions generalize the Shapley value in the sense that they are equal to the Shapley value when the union closed system is the power set of player set N . First, we apply a method similar as Myerson (1977) to define a solution for games on union closed systems which generalizes the Shapley value for games on antimatroids as axiomatized in Algaba et al (2003). To do so, a modified or restricted game is defined.…”