The authors extend E. Maskin's results on Nash implementation. First, they establish a condition that is both necessary and sufficient for Nash implementability if there are three or more agents (the case covered by Maskin's sufficiency result). Second--and more important--they examine the two-agent case (for which there existed no general sufficiency results). The two-agent model is the leading case for applications to contracting and bargaining. For this case, too, they establish a condition that is both necessary and sufficient. The authors use their theorems to derive simpler sufficiency conditions that are applicable in a wide variety of economic environments.

This paper examines the use of stage mechanisms in implementation problems and provides a partial characterization of the set of subgam e perfect implementable choice rules. It is shown that, in many economic environments, virtually an y choice rule can be implemented. To illustrate the power of this approach, the paper discusses a number of models in which it is possible to implement the first-best (although it wouldn't have been possible to do so without using stage mechanisms). The diversity of these models suggests that subgame perfect implementation may find wide application.