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.
Consider an analyst who models a strategic situation in terms of an incomplete information game and makes a prediction about players’ behavior. He believes that his model correctly describes the probability distribution over the players’ Mertens-Zamir hierarchies of beliefs over payoff-relevant states (Mertens and Zamir, 1985). However, players may have observed signals generated by an individually uninformative correlating device (Liu, 2015), which allows the players to correlate their behavior. In other words, the true incomplete information game may have correlated duplicated belief hierarchies (Ely and Peski, 2006; Dekel et al., 2007). Then, a natural candidate for the analyst’s prediction is the set of outcomes that can arise in some Bayes Nash equilibrium (BNE) of some incomplete information game with the same distribution over belief hierarchies. Liu (2015) shows that this set of outcomes can be characterized as the set of belief-invariant Bayes correlated equilibria (BIBCE) of the analyst’s model. A BIBCE is a Bayes correlated equilibrium (BCE) in which a prescribed action does not reveal any additional information to the player about the opponents’ types and the payoff-relevant state, thus preserving the player’s belief hierarchy.