  This paper studies Bayesian games with general action spaces, correlated
types and interdependent payoffs. We introduce the condition of ``decomposable
coarser payoff-relevant information'', and show that this condition is both
sufficient and necessary for the existence of pure-strategy equilibria and
purification from behavioral strategies. As a consequence of our purification
method, a new existence result on pure-strategy equilibria is also obtained for
discontinuous Bayesian games. Illustrative applications of our results to
oligopolistic competitions and all-pay auctions are provided.
