On the nonemptiness of the α-core of discontinuous games: Transferable and nontransferable utilities

The nonemptiness of the α-core of games with continuous payoff functions was proved by Scarf (1971) for nontransferable utilities and by Zhao (1999) for transferable utilities. In this paper we present generalizations of their results to games with possibly discontinuous payoff functions. Our handling of discontinuity is based on Reny's (1999) better-reply-security concept. We present examples to show that our generalizations are nonvacuous.