Perfect Bayesian equilibrium and sequential equilibrium

We introduce a formal definition of perfect Bayesian equilibrium (PBE) for multi-period games with observed actions. In a PBE, (P) the strategies form a Bayesian equilibrium for each continuation game, given the specified beliefs, and (B) beliefs are updated from period to period in accordance with Bayes rule whenever possible, and satisfy a "no-signaling-what-you-don't-know" condition. PBE is equivalent to sequential equilibrium if each player has only two types, or there are only two periods, but differs otherwise. Equivalence is restored by requiring that (B) apply to the relative probabilities of types with posterior probability zero.