Stability and largeness of core for symmetric games

Largeness of the core is sufficient for stability of the core. In general t he necessity is not known. In this paper we answer affirmatively the necess ity for symmetric games. We also prove its equivalence to n specified vecto rs being imputations and also to the convexity of the lower boundary of the set of all acceptable pay-off vectors of the game. In this paper we establ ish the equivalence of a condition given by Shapley to the newly evolved co ndition, thereby give an alternate proof to Shapley's condition.