We deﬁne and characterize a notion of correlated equilibrium for games with incomplete information, which we call Bayes correlated equilibrium: The set of outcomes that can arise in Bayes Nash equilibria of an incomplete information game where players may have access to additional signals beyond the information structure is characterized and shown to be equivalent to the set of Bayes correlated equilibria. A game of incomplete information can be decomposed into a basic game, given by actions sets and payoﬀ functions, and an information structure. We introduce a partial order on many player information structures — which we call individual suﬀiciency — under which more information shrinks the set of Bayes correlated equilibria. We discuss the relation of the solution concept to alternative deﬁnitions of correlated equilibrium in incomplete information games and of the partial order on information structures to others, including Blackwell’s for the single player case.
Bergemann, Dirk and Morris, Stephen, "The Comparison of Information Structures in Games: Bayes Correlated Equilibrium and Individual Sufficiency" (2013). Cowles Foundation Discussion Papers. 2294.