Incomplete Information Robustness

Consider an analyst who models a strategic situation in terms of an incomplete information game and makes a prediction about players’ behavior. The analyst’s model approximately describes each player’s hierarchies of beliefs over payoff-relevant states, but the true incomplete information game may have correlated duplicated belief hierarchies, and the analyst has no information about the correlation. Under these circumstances, a natural candidate for the analyst’s prediction is the set of belief-invariant Bayes correlated equilibria (BIBCE) of the analyst’s incomplete information game. We introduce the concept of robustness for BIBCE: a subset of BIBCE is robust if every nearby incomplete information game has a BIBCE that is close to some BIBCE in this set. Our main result provides a sufficient condition for robustness by introducing a generalized potential function of an incomplete information game. A generalized potential function is a function on the Cartesian product of the set of states and a covering of the action space which incorporates some information about players’ preferences. It is associated with a belief-invariant correlating device such that a signal sent to a player is a subset of the player’s actions, which can be interpreted as a vague prescription to choose some action from this subset. We show that, for every belief-invariant correlating device that maximizes the expected value of a generalized potential function, there exists a BIBCE in which every player chooses an action from a subset of actions prescribed by the device, and that the set of such BIBCE is robust, which can differ from the set of potential maximizing BNE.