The Shapley Value on Convex Geometries

A game on a convex geometry is a real-valued function dened on the family L of the closed sets of a closure operator which satises the nite Minkowski-Krein-Milman property. If L is the boolean algebra 2 N then we obtain a n-person cooperative game. Faigle and Kern investigated games where L is the distributive lattice of the order ideals of the poset of players. We obtain two classes of axioms that give rise to a unique Shapley value for games on convex geometries. Key words: Cooperative game, Convex geometry, Shapley value. 1 Games on convex geometries The goal of this paper is to develop a theoretical framework in which to analyze cooperative games in which only certain coalitions are allowed to form. We will axiomatize the structure of such allowable coalitions using the theory of convex geometries, a notion developed to combinatorially abstract geometric convexity. In this sense our model acts as a bridge between traditional cooperative game theory and spatial games, in which ..

The Shapley Value on Convex Geometries

A game on a convex geometry is a real-valued function defined on the family L of the closed sets of a closure operator which satisfies the finite Minkowski-Krein-Milman property. If L is the Boolean algebra 2 N then we obtain a n-person cooperative game. Faigle and Kern investigated games where L is the distributive lattice of the order ideals of the poset of players. We obtain two classes of axioms that give rise a unique Shapley value for games on convex geometries. Finally, we consider the problem of computing the Shapley value for this games when the cardinal of the player set is small