Compellingness in Nash implementation - CHATTERJI, Shurojit (Institutional Knowledge at Singapore Management University, 2025)
A social choice function (SCF) is said to be Nash implementable if there exists a mechanism in which every Nash equilibrium outcome coincides with that speciﬁed by the SCF. The main objective of this paper is to assess the impact of considering mixed strategy equilibria in Nash implementation. To do this, we focus on environments with two agents and restrict attention to ﬁnite mechanisms. We call a mixed strategy equilibrium “compelling” if its outcome Pareto dominates any pure strategy equilibrium outcome. We show that if the ﬁnite environment and the SCF to be implemented jointly satisfy what we call Condition P+M, we construct a ﬁnite mechanism which Nash implements the SCF in pure strategies and possesses no compelling mixed strategy equilibria. This means that the mechanism might possess mixed strategy equilibria which are “not” compelling. Our mechanism has several desirable features: transfers can be completely dispensable; only ﬁ-nite mechanisms are considered; integer games are not invoked; and players’ attitudes toward risk do not matter.