It is well-known that a transferable utility game has a non-empty core if and only if it is balanced. In the class of non-transferable utility games balancedness or the more general p-balancedness due to billera (siam j. Appl. Math. 18 (1970) 567) is a sufficient, but not a necessary condition for the core to be non-empty. This paper gives a natural extension of the p-balancedness condition that is both necessary and sufficient for non-emptiness of the core.