Title: Necessary and Sufficient Conditions for the Approximate Core Property
Abstract: Bondareva and Shapley showed that games have nonempty cores if and only if the games are balanced. In this paper, we show necessary and sufficient conditions for large games, ones with many players, to have nonempty approximate cores. We use a framework where, for any game with many players, most players have many substitutes, called the substitution property. We introduce the property of continuity of average payoffs, which is simply that small groups of players can have only small effects on per-capita payoffs of large groups. It is shown that all sufficiently large games have nonempty approximate cores if and only if the continuity of average payoffs condition is satisfied. The result is proven by showing that continuity of average payoffs is necessary and sufficient for efficacy of small groups, or, in other words, inessentiality of large groups, another condition equivalent to the approximate core property.
Creation-Date: February 1991
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