Epistemic conditions for Bayesian Nash equilibrium, and common knowledge of rationality

We show that, in games of complete information, the Aumann-Brandenburger (1995) sufficient conditions for Nash equilibrium in beliefs also imply common knowledge of rationality. We then consider beliefs held in games of incomplete information, especially beliefs the players hold, conditioning on other players' types, about other players' actions. We provide sufficient conditions for such beliefs to form a Bayesian Nash equilibrium, a solution concept more suited to such settings. Whether or not these new conditions imply common knowledge of rationality depends on which formal definition of a Bayesian game is used. Out equilibria do not concern beliefs held in some constructed ex ante or extended game, but rather the beliefs held by players of the original incomplete information game about other players' actual preferences, actions and beliefs.