A Unified Implementation Theory

This paper unifies the theories of Nash implementation and Bayesian implementation in a single framework. Environments considered are such that each agent's characteristics include, in addition to a specification of his private information, a commonly known type parameter, while both attributes are unknown to the designer. Each social choice correspondence (SCC) assigns a commonly known type vector to a social choice set, a collection of functions mapping private type vectors to allocations. Conditions that fully characterize an implementable SCC in economic environments where agents are not satiated generalize and merge respective conditions in the complete information model of Danilov (1992) and the incomplete information model of Jackson (1991). In noneconomic environments there remains to exist a gap between the necessary and sufficient conditions, like in Jackson (1991). In order to narrow down this gap, we employ Danilov's notion of essential elements and develop a stronger necessary condition, termed essential-generalized-Bayesian monotonicity (EGBM).