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On the Uniqueness of Nash Equilibria in Multiagent Matrix Games

James P. Bailey

TL;DR

It is shown 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.

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.

On the Uniqueness of Nash Equilibria in Multiagent Matrix Games

TL;DR

It is shown 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.

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.

Paper Structure

This paper contains 13 sections, 13 theorems, 11 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Notation
  4. A Roadmap for Determining Unique Equilibria.
  5. Conditions for the Invertibility of $A$
  6. Polymatrix Games
  7. Coordination Games and General Games
  8. Zero-Sum Games
  9. Games with Constrained Strategy Spaces
  10. Recovering Uniqueness with Learning Dynamics
  11. A Cursory Understanding of \ref{['eqn:ContGD']}
  12. Converging to the Closest Nash Equilibrium.
  13. Conclusions

Key Result

Lemma 1

$x^*$ is a Nash equilibrium for the eqn:BilinearGame$G=(n,k,A,b)$ if and only if it satisfies where $\vec{0}^{k_i}$ is a vector of $k_i$ 0's. Equivalently, $x^*$ is a Nash equilibrium if and only if $Ax^*=b$.

Figures (1)

  • Figure 1: Despite non-uniqueness, \ref{['eqn:ContGD']} contains strategies to a subspace with a unique equilibrium. For instance, when using \ref{['eqn:ContGD']} in $\mathbb{R}^3$, agents' strategies remain equidistant from distinct Nash strategies $x^*+d$ and ${x}^*-\lambda \cdot d$ causing agent strategies to cycle around a lower-dimensional circle in $\{x\in \mathbb{R}^3: d^\intercal x = d^\intercal x^*$} that uniquely contains the unique Nash equilibrium ${x}^*$ that is closest to the initial strategy $x(0)$.

Theorems & Definitions (23)

  • Lemma 1
  • proof
  • Lemma 2
  • Theorem 1
  • Lemma 3: Roots of an Analytic Function mityagin2015zero
  • proof : Proof of Theorem \ref{['thm:UniqueOrNone']}
  • Proposition 1
  • proof
  • Theorem 2
  • proof
  • ...and 13 more