Perfect Bayesian Equilibria and Cross-Pair Independence from Common Actions

This paperdevelopsasimplenotionofperfectBayesianequilibriumforarbitraryfiniteextensive form games with perfect recall. Its key ingredient is cross-pair independence from common actions, a belief restriction that compares the likelihoods of two pairs of histories, possibly in different information sets, after canceling actions common to both pairs. The condition extends Bayes’ rule whenever possible and no signaling what you don’t know beyond sequentially ordered information sets. In multi-stage games with observable actions and independent types, it implies the belief-reasonableness requirements of Fudenberg and Tirole (1991). Extending the same cancellation logic to arbitrary multisets is equivalent to consistency. The construction therefore provides a tractable, assessment-based notion of reasonableness without non standard probabilities, plausibility orders, or auxiliary belief-revision structures.