Robustness of Bayesian Equilibria

The standard model of a Bayesian game used in most applications assumes that players' beliefs are derived from a common knowledge prior on preference parameters. I analyze the robustness of equilibria of such games to perturbations in the information structure. In a type space environment (Harsanyi, 1967-68), I embed types corresponding to this information structure into an appropriately defined larger type space. I then perturb the embedded set using the notion of common p-belief (Monderer and Samet, 1989) by considering all types for which it is common p-belief that all players derive their beliefs about preference parameters from similar priors. For types in the perturbed set, I define an $\varepsilon$-equilibrium in which every player's strategy is an equilibrium strategy for the game where his individual prior is a common knowledge prior. Hence, this strategy is only a function of such a player's prior and private information, and does not depend on the exact functional form of his higher order beliefs. Based on this definition of an $\varepsilon$-equilibrium, I propose a notion of robustness that is independent of the specification of the underlying type space. This independence significantly simplifies the characterization of robust equilibria, since they can be defined in relation to a selection from the equilibrium correspondence that maps common knowledge priors to corresponding equilibria. For finite games, I derive a precise characterization of such a selection. Specifically, this characterization implies that an admissible selection can be discontinuous and does not require lower hemicontinuity of the equilibrium correspondence for existence. The set of robust equilibria includes the set of ex post equilibria as a proper subset