Maskin Meets Abreu and Matsushima

TL;DR

This paper unifies Maskin's and Abreu-Matsushima's approaches to full implementation, showing that finite mechanisms with mixed strategies and small transfers can achieve exact Nash implementation under Maskin monotonicity.

Contribution

It introduces a finite mechanism framework that explicitly handles mixed strategies and small transfers, extending implementation theory beyond previous integer-based methods.

Findings

Maskin monotonicity is necessary and sufficient for mixed-strategy Nash implementation.

Finite mechanisms without integer or modulo games can implement social choice rules.

Transfers can be made arbitrarily small and the mechanism is robust to information perturbations.

Abstract

The theory of full implementation has been criticized for using integer/modulo games which admit no equilibrium (Jackson (1992)). To address the critique, we revisit the classical Nash implementation problem due to Maskin (1999) but allow for the use of lotteries and monetary transfers as in Abreu and Matsushima (1992, 1994). We unify the two well-established but somewhat orthogonal approaches in full implementation theory. We show that Maskin monotonicity is a necessary and sufficient condition for (exact) mixed-strategy Nash implementation by a finite mechanism. In contrast to previous papers, our approach possesses the following features: finite mechanisms (with no integer or modulo game) are used; mixed strategies are handled explicitly; neither undesirable outcomes nor transfers occur in equilibrium; the size of transfers can be made arbitrarily small; and our mechanism is robust to information perturbations. Finally, our result can be extended to infinite/continuous settings and ordinal settings.