Perfect Bayesian implementation in economic environments

We study social choice functions implementable as perfect Bayesian equilibria (PBE) of incomplete information extensive form games. We restrict attention to two-stage mechanisms in which agents observe a common public signal after the first stage, and fully characterize the set of implementable social choice functions in economic environments. The key condition for implementation, called Extended Bayesian Monotonicity, provides for some kind of preference reversal either at the first or the second stage. The condition is satisfied by social choice functions that satisfy Bayesian Monotonicity (a necessary and sufficient condition for Bayesian implementation in economic environments) or the Posterior Reversal condition proposed by Bergin and Sen (J. Econ Theory 80 (1998) 222-256), a sufficient condition for implementation with extensive form games. The analysis is limited to two-stage games, but the techniques adopted can be applied more in general to k-stage games, with k > 2.