Strategy Space Reduction in the Maskin - Williams Theorem: Sufficient Conditions for Nash Implementation

Abstract

Any social choice correspondence satisfying monotonicity and no veto power with at least three participants is Nash implementable. This theorem by Haskin, of which an extended version was proved by Williams, requires a rather large strategy space. Each participant announces every participant's preferences and an alternative. This paper presents a significantly smaller strategy space when the number of participants is large. Each participant announces his own preferences, his neighbor's preferences, an alternative, and an integer between zero and the number of participants less one. With this specification of the strategy spaces, the Haskin-Williams Theorem remains valid without imposing any restrictions on the size of the alternative set or the environment set. That is, a complete proof for the original Haskin theorem is provided.