The Face Lattice of Polyhedral Cones in the Theory of Cooperative Games

Whether or not a given cooperative game with transferable utility is balanced; i.e. possesses a nonempty core, is a central question in the literature. The answer was furnished, independently, by Bondareva (In Vestnik Leningradskii Universitet, in Russian, 13:141–142, 1962) and Shapley (Nav Res Logist Q 14:453–460, 1967), who provided necessary and sufficient conditions in the form of a set of linear inequalities involving the game’s characteristic function. The purpose of this paper is to show how these inequalities arise naturally from the representation of a certain polyhedral cone as the intersection of half spaces. In the course of doing so we also show how each balanced collection of subsets corresponds to the complement of a face of the cone and how the set of coalitional excesses of a game coincides with its set of combination vectors. Finally, we utilize our framework to prove a notable result of Keane (Ph.D. Dissertation, Field of Math, Northwestern University, Evanston) concerning the L1-center of a cooperative game.