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A game of incomplete information can be decomposed into a basic game and an information structure. The basic game defines the set of actions, the set of payoff states, the payoff functions, and the common prior over the payoff states. The information structure refers to the signals that the players receive in the game. We characterize the set of outcomes that can arise in Bayes Nash equilibrium if players observe the given information structure but may also observe additional signals. The characterization corresponds to the set of (a version of) incomplete information correlated equilibria, which we dub Bayes correlated equilibria. We identify a partial order on many-player information structures (individual sufficiency) under which more information shrinks the set of Bayes correlated equilibria. This order captures the role of information in imposing (incentive) constraints on behavior.
Bergemann et al. (Sun,) studied this question.