In game theory, a Perfect Bayesian Equilibrium (PBE) is a solution with Bayesian probability to a turn-based game with incomplete information. More specifically, it is an equilibrium concept that uses Bayesian updating to describe player behavior in dynamic games with incomplete information. Perfect Bayesian equilibria are used to solve the outcome of games where players take turns but are unsure of the "type" of their opponent, which occurs when players don't know their opponent's preference between individual moves. A classic example of a dynamic game with types is a war game where the player is unsure whether their opponent is a risk-taking "hawk" type or a pacifistic "dove" type. Perfect Bayesian Equilibria are a refinement of Bayesian Nash equilibrium (BNE), which is a solution concept with Bayesian probability for non-turn-based games.

Any perfect Bayesian equilibrium has two components -- strategies and beliefs:

The strategies and beliefs also must satisfy the following conditions:

A perfect Bayesian equilibrium is always a Nash equilibrium.

Consider the following game:

For any value of ${\displaystyle p,}$ Equilibrium 1 exists, a pooling equilibrium in which both types of sender choose the same action:

The sender prefers the payoff of 0 from not giving to the payoff of -1 from sending and not being accepted. Thus, Give has zero probability in equilibrium and Bayes's Rule does not restrict the belief Prob(Friend|Give) at all. That belief must be pessimistic enough that the receiver prefers the payoff of 0 from rejecting a gift to the expected payoff of ${\displaystyle x(1)+(1-x)(-1)=2x-1,}$ from accepting, so the requirement that the receiver's strategy maximize his expected payoff given his beliefs necessitates that Prob(Friend|Give)${\displaystyle \leq .5.}$ On the other hand, Prob(Friend|Not give) = p is required by Bayes's Rule, since both types take that action and it is uninformative about the sender's type.

If ${\displaystyle p\geq 1/2}$, a second pooling equilibrium exists as well as Equilibrium 1, based on different beliefs: