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.