Correlated equilibrium and higher order beliefs about play

We study a refinement of correlated equilibrium in which playersʼ actions are driven by their beliefs and higher order beliefs about the play of the game (beliefs over what other players will do, over what other players believe others will do, etc.). For any finite, complete-information game, we characterize the behavioral implications of this refinement with and without a common prior, and up to any a priori fixed depth of reasoning. In every finite game “most” correlated equilibrium distributions are consistent with this refinement; as a consequence, this refinement gives a classification of “most” correlated equilibrium distributions based on the maximum order of beliefs used by players in the equilibrium. On the other hand, in a generic two-player game any non-degenerate mixed-strategy Nash equilibrium is not consistent with this refinement.