A hierarchical model for cooperative games

Abstract

Classically, a cooperative game is given by a normalized real-valued function v on the collection of all subsets of the set N of players. Shapley has observed that the core of the game is non-empty if v is a non-negative convex (a.k.a. supermodular) set function. In particular, the Shapley value of a convex game is a member of the core. We generalize the classical model of games such that not all subsets of N need to form feasible coalitions. We introduce a model for ranking individual players which yields natural notions of Weber sets and Shapley values in a very general context. We establish Shapley's theorem on the nonemptyness of the core of monotone convex games in this framework. The proof follows from a greedy algorithm that, in particular, generalizes Edmonds' polymatroid greedy algorithm. © 2008 Springer-Verlag Berlin Heidelberg.