Core stability of the Shapley value for cooperative games

Our objective is to analyze the relationship between the Shapley value and the core of cooperative games with transferable utility. We first characterize balanced games, i.e., the set of games with a nonempty core, through geometric properties. We show that the set of balanced games generates a polyhedral cone and that a game is balanced if and only if it is a nonnegative linear combination of some simple games. Moreover, we show that the set of games whose Shapley value lies in the core also yields a polyhedral cone and that a game obeys this property if and only if it is a nonnegative linear combination of simple games satisfying certain properties. By-products, we also show that the number of games that correspond to the extreme rays of the polyhedron coincides with the number of minimal balanced collections.