The theory of implementation of social choice rules

Suppose that the goals of a society can be summarized in a social choice rule, i.e., a mapping from relevant underlying parameters to final outcomes. Typically, the underlying parameters (e.g., individual preferences) are unknown to the public authority. The implementation problem is then formulated: Under what circumstances can one design a mechanism so that the unknown information is truthfully elicited and the social optimum ends up being implemented? In designing such a mechanism, appropriate incentives must be given to the agents so that they do not wish to misrepresent their information. The theory of implementation or mechanism design formalizes this "social engineering" problem and provides answers to the question just posed. I survey the theory of implementation in this article, emphasizing the results under two different benchmarks (that agents have dominant strategies and that they playa Nash equilibrium). Examples discussed include voting; and the allocation of private and public goods under complete and incomplete information.