Subgame Perfect Implementation

We examine the use of stage mechanisms in implementation problems, and we partially characterize the set of choice rules that are implementable in subgame perfect equilibria. The conditions we derive will usually be satisfied if there is at least one private good. Our conclusion is that in these "economic" environments, almost any choice rule can be implemented--even if the rule is nonmonotonic, even if it is single-valued, and, under quite reasonable circumstances, even if there are just two agents. Moreover, the mechanism need have no more than three stages. To illustrate the power of this approach, we discuss a number of models in which it is possible to implement the first-best (although it would have been impossible to do so without using stage mechanisms). The diversity of these models suggests that subgame perfect implementation may find wide application: public goods, contracts, agency theory, constitution design. In several cases, the mechanisms we present have the added attraction that agents' moves are sequential, not simultaneous. Finally, we show that the Walras correspondence is implementable in subgame perfect equilibria.