Epsilon-Equilibria of Perturbed Games
36 Pages Posted: 13 Aug 2010 Last revised: 3 Jan 2013
Date Written: August 26, 2010
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.
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
Suggested Citation: Suggested Citation