Type diversity and virtual Bayesian implementation

It is well known that a social choice function is truthfully implementable in Bayesian Nash equilibrium if and only if it is incentive compatible. However, in general it is not possible to rule out other equilibrium outcomes, and additional conditions, e.g., Bayesian monotonicity, are needed to ensure full implementation. We argue that this multiplicity problem is not very severe for virtual Bayesian implementation. We show that any incentive compatible social choice function is virtually (and fully) Bayesian implementable in an environment satisfying a condition we term type diversity. If there are at least three elements in the set of social alternatives, this condition holds generically. Type diversity is a condition with a simple economic interpretation compared to Bayesian monotonicity, and the mechanism we construct is also simple. We also provide a necessary and sufficient condition – type diversity with respect to deceptions – for virtual Bayesian implementation in economic environments.