The theory of implementation in Nash equilibrium: A survey. In: Hurwicz,

The theory of implementation concerns the problem of designing game forms (sometimes called "mechanisms" or "outcome functions") the equilibria of which have properties that are desirable according to a specified criterion of social welfare called a social choice rule . A game form, in effect, decentralizes decision-making. The social alternative is selected by the joint actions of all individuals in society rather than by a central planner. Formally, a social choice rule assigns a set of alternatives to each profile of preferences (or other characteristics) that individuals in society might have; the set consists of the "welfare optima" relative to the preference profile. A game form is a rule that specifies an alternative (or outcome ) for each configuration of actions that individuals take. A game form implements (technically, fully implements) a social choice rule if, for each possible profile of preferences, the equilibrium outcomes of the game form coincide with the welfare optima of the social choice rule. Of course, the equilibrium set depends on the particular solution concept being used. Implementation theory has considered a variety of solution concepts, including equilibrium in dominant strategies, Bayesian equilibrium, and Hash equilibrium. Other chapters of this volume treat the first two equilibrium concepts. In the .main, this article is confined to implementation in Nash equilibrium, although it relates this theory to those of other solution concepts, dominant strategies in particular. Nash equilibrium is the noncooperative solution concept par excellence , and so it is not surprising that implementation theory should have employed it extensively. Nonetheless, one reason often advanced for the desirabilit