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 canone 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 play a Nash equilibrium). Examples discussed include voting, and the allocation of private and public goods under complete and incomplete information.