Individually rational pure strategies in large games

Abstract

In “randomly selected” large finite normal form games, pure strategy “strongly individually rational” outcomes (all payoffs exceeding pure strategy minimax payoffs) exist with high probability. As a corollary, pure strategy Nash equilibria also exist with high probability in certain infinitely repeated games. Specifically, the pure strategy Folk Theorem, which can be true in a vacuous sense, has nontrivial substance with probability approaching one as pure strategy sets increase in cardinality, provided strategy set growth rates are not too different. To prove this, and noting a possible “bounded rationality” interpretation, we begin with a study of “approximate equilibria” in finite normal form games. Given a positive integer l, these are pure strategy outcomes where each player's payoff is among the l largest available given strategy choices of opponents. When pure strategy sets are large, such outcomes are highly probable and usually have payoffs dominating pure minimax payoffs.