Compellingness in Nash Implementation

A social choice function (SCF) is said to be Nash implementable (in pure strategies) if there exists a mechanism in which every pure-strategy Nash equilibrium induces outcomes specified by the SCF. The main objective of this paper is to assess the impact of considering mixed-strategy equilibria in Nash implementation. We define compelling Nash implementation as a case where the implementing mechanism possesses a pure-strategy equilibrium that strictly Pareto dominates any undesired mixed-strategy equilibrium. We show that if the finite environment and the SCF to be implemented jointly satisfy what we call Condition COM, then we can construct a finite mechanism which compellingly implements the SCF. We also identify a class of voting environments that satisfies Condition COM, extend Condition COM to accommodate social choice correspondences, and explore a preliminary stability-based justification for the implementing mechanism. Our mechanism has several desirable features: transfers are completely dispensable; only finite mechanisms are considered; integer games are not invoked; and agents’ attitudes toward risk do not affect implementation.