On the Uniqueness of Nash Equilibria in Multiagent Matrix Games

TL;DR

This paper characterizes when Nash equilibria are unique in multiagent matrix games, revealing conditions for uniqueness in different game types and analyzing the impact on learning dynamics.

Contribution

It provides a complete characterization of equilibrium uniqueness in unconstrained polymatrix games, especially highlighting conditions in zero-sum games.

Findings

Uniqueness is typical in coordination and general polymatrix games.

Zero-sum polymatrix games often have non-unique equilibria due to dimensionality constraints.

Classical learning dynamics still produce unique estimates despite multiple equilibria.

Abstract

We provide a complete characterization for uniqueness of equilibria in unconstrained polymatrix games. We show that while uniqueness is natural for coordination and general polymatrix games, zero-sum games require that the dimension of the combined strategy space is even. Therefore, non-uniqueness is common in zero-sum polymatrix games. In addition, we study the impact of non-uniqueness on classical learning dynamics for multiagent systems and show that the classical methods still yield unique estimates even when there is not a unique equilibrium.