Equilibrium Numbers in Non-Square Bimatrix Games

2026-09-21

AI Summary

研究非方阵双矩阵博弈中纳什均衡的数量问题，通过多面体的组合性质及图论障碍方法给出新的界限。

Abstract

Bimatrix games may have an exponential number of mixed Nash equilibria if both dimensions of the game are allowed to grow. Bounds on their maximal number give structural insights that have been used to construct hard-to-solve games. We show new sharp or asymptotically sharp bounds on the (polynomial) number of equilibria for generic games where one dimension of the game is fixed and the number of strategies of the other player grows. These results go beyond the hitherto studied square games. Our methods employ combinatorial properties of polytopes, and recent obstructions that relate to the graph of those polytopes. For $n\ge5$, we construct $3\times n$ games that have all $2n+1$ vertices of the best-response polytope as equilibrium strategies, proved using a simple case of the 4-color theorem for planar graphs. Generic $4\times 5$ games are shown to have at most 17 equilibria, using computer calculations with existing datasets for all combinatorial types of the relevant polytopes. For $d\times n$ games, we construct games where all but a fraction of $O(1/n)$ of the maximum number of vertices are equilibrium strategies.

Problem

• Bimatrix Games
• Nash Equilibria
• Polytopes

Innovation

• non-square bimatrix games
• mixed Nash equilibria
• polytopes
• best-response polytope
• combinatorial types