非方形双矩阵博弈中的均衡数量
Equilibrium Numbers in Non-Square Bimatrix Games
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中文总结 AI 辅助
该研究针对非方形双矩阵博弈，利用多胞形组合性质与四色定理，给出固定维度下均衡数量的精确或渐近界限，并构造达到或接近上限的博弈实例。
中文摘要 AI 辅助
双矩阵博弈在博弈的两个维度都允许增长时，可能具有指数数量的混合纳什均衡。对其最大数量的界限提供了结构性的洞见，这些洞见已被用于构造难以求解的博弈。我们针对一类博弈（其中一个维度固定，另一玩家的策略数量增长）的（多项式）均衡数量，展示了新的精确或渐近精确的界限。这些结果超越了迄今研究的方形博弈。我们的方法利用了多胞形的组合性质，以及最近与这些多胞形图相关的障碍。对于$n\ge5$，我们构造了$3\times n$博弈，其最佳响应多胞形的所有$2n+1$个顶点均为均衡策略，这通过平面图四色定理的一个简单情形得以证明。对于一般的$4\times 5$博弈，利用现有数据集对相关多胞形的所有组合类型进行计算机计算，我们证明其最多有17个均衡。对于$d\times n$博弈，我们构造了博弈，其中除$O(1/n)$比例外的最大顶点数均为均衡策略。
英文摘要
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.
发表机构
- Goethe-Universität（法兰克福歌德大学）
- London School of Economics（伦敦政治经济学院）
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