When type is private information, the games become more complicated. This is because, for an equilibrium to exist, each player must be able to correctly predict others’ actions.
In a Bayesian game, player 1 can be of different types (assigned by mother Nature). He can observe his type, but player 2 knows only the type distribution. The issue becomes trickier when player 2 has to move in the ensuring information set: even after observing player 1’s action, player 2 is still uncertain about player 1’s type. Without further information, player 2 cannot decide his best response.
This technical difficulty is resolved by introducing the notion of belief. Specifically, for a given Bayesian Nash equilibrium (BNE), at each information set, each player must hold the correct belief that is consistent with the BNE; he then plays the best response to the belief. At the information set reached with positive probability, the belief is updated according to the Bayes’ rule. Off of the equilibrium path, however, the belief can take arbitrary form.
Formally, a Bayesian Nash equilibrium and a consistent belief constitute a perfect Bayesian equilibrium (PBE). Hence, PBE has four requirements: 1) every player holds a belief, 2) on the equilibrium path, the belief follows Bayes’ rule; 3) off the equilibrium path, the belief is arbitrary; 4) the BNE is each player’s best response to that belief.
In a sense, the belief is an intermediate concept that ensures the sequential rationality of BNE (Bayes’ rule and best response). The idea is similar to that of mean field theory.