A Necessary and Sufficient Condition for the Nonemptiness of the Cores of Partitioning Games

Let N be a finite set of players and let ρ be a class of coalitions of N. We consider games with and without sidepayments such that only the coalitions in pi play essential roles but not the others. For an arbitrary ρ, we get the class of all such games. The purpose of this note is to provide a necessary and sufficient condition with respect to pi for the nonemptiness of the cores of all games in the class.