Multiplicity of Mixed Bayesian Equilibria in Mechanisms Roberto Serrano and Rajiv Vohra The literature on implementation with incomplete information has often left out the consideration of mixed-strategy equilibria. This is particularly problematic for a research program that attempts to address the problem of multiplicity of equilibrium. We investigate the necessary and sufficient conditions for full implementation of social choice sets in mixed-strategy Bayesian equilibrium. We refer to this as mixed implementation. Our results characterize both exact and virtual mixed implementation. For exact implementation, we identify a strengthening of Bayesian monotonicity, which we refer to as mixed Bayesian monotonicity. It is shown that, in environments with at least three agents, mixed Bayesian implementation is equivalent to mixed Bayesian monotonicity, incentive compatibility and closure. For implementing a social choice function, the case of two-agents is also covered by these conditions. However, the case of two agents and social choice sets requires an extra condition concerning the non-empty intersection of suitably defined lower contour sets of preferences. Mixed virtual implementation is shown to be equivalent to mixed virtual monotonicity, incentive compatibility and closure. The key condition, mixed virtual monotonicity, is shown to be very weak. In particular, it is weaker than Abreu-Matsushima's measurability, thereby implying that: (1) virtual implementation in Bayesian equilibrium is more permissive than virtual implementation in iteratively undominated strategies, and (2) non-regular mechanisms may be important for virtual implementation in Bayesian equilibrium.
