Purification of Bayes Nash equilibrium with correlated types and interdependent payoffs

We establish purification results for Bayes–Nash equilibrium in a large class of Bayesian games with finite sets of pure actions. We allow for correlated types and interdependent payoffs and for type-dependent feasible action sets. The latter feature allows us to prove existence and purification results for pure Bayes–Nash equilibria in undominated strategies. We give applications to auctions, global games, and voting to illustrate the usefulness of our results.