Epsilon-Equilibria of Perturbed Games

Abstract:     
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 ε-equilibria converging to the given equilibrium of the original game. We strengthen the conclusion to show that the approaching equilibria are interim ε-equilibria (ε- 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.

Number of Pages in PDF File: 36

Keywords: epsilon-equilibrium, epsilon-Nash equilibrium, electronic mail game, global games, Bayesian games, trembling hand perfection, Nash equilibrium, lower hemi-continuity

JEL Classification: C72, D82

Accepted Paper Series