Epsilon-equilibria of perturbed games

We prove that for any equilibrium of a (Bayesian) game, and any sequence of perturbations of that game, there exists a corresponding sequence of ex-ante epsilon-equilibria converging to the given equilibrium of the original game. We strengthen the conclusion to show that the approaching equilibria are interim epsilon-equilibria (epsilon-best responses for almost all types) if beliefs in the perturbed games converge in a strong-enough sense to the limit beliefs. Therefore, equilibrium selection arguments that are based on perturbations to a game are not robust to slight perturbations in best reply behavior (or to underlying preferences). This applies to many standard equilibrium selections, including Selten's (1975) definition of trembling-hand perfect equilibrium, Rubinstein's (1989) analysis of the electronic mail game, and Carlsson and van Damme's (1993) global games analysis, among others. (C) 2011 Elsevier Inc. All rights reserved.