Implementation with Incomplete Information in Exchange Economies

In this paper, we analyze the problem of designing incentive compatible mechanisms in pure exchange economic environments when agents have incomplete information. The equilibrium concept employed is Bayesian Nash equilibrium and the notion of implemantation is full implementation, which is stronger than the more commonly employed notion of truthful implementation. An allocation rule is truthfully implementable if there exists a direct mechanism to which truth telling is an equilibrium and which yields the allocation rule as its truthful equilibrium outcome. An allocation rule is fully implementable if there exists mechanism which yields the allocation rule as its unique equilibrium outcome. More generally, a set of allocation rules, or a social choice set, is fully implementable if there exist a mechanism whose equilibrium outcomes coincide with the set. This stronger notion of implemention avoids the well known problems of multiple equilibria which arise in direct revelation games. We develop a condition, termed Bayesian monotonicity, which we show is necessary for full implementation. An incentive compatibility condition is also necessary. We prove that Bayesian monotonicity and a slightly stronger incentive compatibility condition are sufficient for full implementation when there are at least three agents. We present several examples of allocation rules which do and do not satisfy our condition. One example is that of an allocation rule which is fully inplementable by an indirect mechanism, but for which every equivalent direct mechanism has multiple equilibrium outcomes