Perfect Bayesian implementation

We study Social Choice Sets (SCS) implementable as perfect Bayesian equilibria of some incomplete information extensive form game. We provide a necessary condition which we call condition β. The condition is analogous to condition C that Moore and Repullo [1988] show to be necessary for subgame perfect implementation in games of complete information, and it is weaker than the Bayesian Monotonicity condition stated in Jackson [1991]. Our first theorem establishes that Incentive Compatibility, Closure and Condition β are necessary for implementation. Our second theorem establishes sufficient conditions. We show that any SCS which satisfies Incentive Compatibility, Closure and a condition called Sequential Monotonicity No Veto (SMNV) is implementable. SMNV is similar in spirit but weaker than the Monotonicity No Veto condition stated in Jackson [1991]. It is also similar to a combination of condition α and No Veto Power, which Abreau and Sen show to be sufficient for implementation in subgame perfect equilibrium.