In Chapter 1, I briefly survey the literature on Bayesian implementation, discuss its shortcomings, and summarize the contribution of this thesis. In Chapter 2, I formally state the implementation problem, making no assumptions about the agents' sets of types, preferences, or beliefs, and I prove Jackson's (1991) necessity and sufficiency results for environments satisfying two weak conditions called "invariance" and "independence." In short, incentive compatibility and Bayesian monotonicity are necessary for Bayesian implementability, and incentive compatibility and monotonicity-no-veto are sufficient. I prove Jackson's result that, for environments with conflict of interest, Bayesian monotonicity and monotonicity-no-veto are equivalent, but I show that conflict-of-interest places an unnatural restriction on agents' beliefs when the set of states is uncountable. I note that, when agents have uncountable sets of types, preferences over social choice functions derived from conditional expected utility calculations will generally be incomplete, and I show that this incompleteness sometimes leads to implausible Bayesian equilibrium predictions. I propose an extension of expected utility preferences that preserves the properties of invariance and independence.
In Chapter 3, I consider environments satisfying invariance and a condition called "interiority," and I show that incentive compatibility and an extension of Bayesian monotonicity are necessary and sufficient for Bayesian implementability. Using the extension of expected utility preferences proposed in Chapter 2 and assuming best-element-private values, I then show that interiority is satisfied in two important classes of environments: it holds in private and public good economies, and it holds in lottery environments, for which the set of outcomes is the set of probability measures over a measurable space of pure outcomes.
In Chapter 4, I consider lottery environments satisfying best-element-private values and a condition called "strict separability," and I use the results of Chapter 3 to show that incentive compatibility is necessary and sufficient for virtual Bayesian implement ability. I then show that strict separability is satisfied for a suitably large class of environments. It holds when private values and value-distinguished types are satisfied and the set of pure outcomes is finite, and it holds when private values and value-distinguished types are satisfied and the set of pure outcomes is a finite set crossed with an open set of allocations of a transferable private good.
