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<p>We prove necessary conditions for unique, fully-mixed equilibria in polymatrix zero-sum games; specifically, a unique, fully-mixed equilibrium cannot occur when the dimension of the combined strategy space is odd, e.g., there is no 3-agent 2-strategy zero-sum game where all agents have a unique fully mixed Nash equilibrium. We also provide necessary and sufficient conditions for unique equilibria in unconstrained polymatrix games, an important setting for motivating learning algorithms. Specifically, we show that (i) there is almost always a unique Nash equilibrium when no agent controls more than half of the strategies and the dimension of the combined strategy space is even, and (ii) there is never a unique Nash equilibrium if (i) is violated. As a result, multiple equilibria frequently occur in the mutliagent setting. Finally, we study the impact of multiple equilibria on learning algorithms and show that at least four dimensions are required to visualize higher-dimensional dynamics.</p>

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On the uniqueness of Nash equilibria in polymatrix games

  • 期刊论文

  • 发表日期：

    2026年5月21日

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  • James P. Bailey

摘要

We prove necessary conditions for unique, fully-mixed equilibria in polymatrix zero-sum games; specifically, a unique, fully-mixed equilibrium cannot occur when the dimension of the combined strategy space is odd, e.g., there is no 3-agent 2-strategy zero-sum game where all agents have a unique fully mixed Nash equilibrium. We also provide necessary and sufficient conditions for unique equilibria in unconstrained polymatrix games, an important setting for motivating learning algorithms. Specifically, we show that (i) there is almost always a unique Nash equilibrium when no agent controls more than half of the strategies and the dimension of the combined strategy space is even, and (ii) there is never a unique Nash equilibrium if (i) is violated. As a result, multiple equilibria frequently occur in the mutliagent setting. Finally, we study the impact of multiple equilibria on learning algorithms and show that at least four dimensions are required to visualize higher-dimensional dynamics.