We study the typical structure of games in terms of their connectivity properties. A game is said to be `connected' if it has a pure Nash equilibrium and the property that there is a best-response path from every action profile which is not a pure Nash equilibrium to every pure Nash equilibrium, and it is generic if it has no indifferences. In previous work we showed that, among all $n$-player $k$-action generic games that admit a pure Nash equilibrium, the fraction that are connected tends to $1$ as $n$ gets sufficiently large relative to $k$. The present paper considers the large-$k$ regime, which behaves differently: we show that the connected fraction tends to $1-ζ_n$ as $k$ gets large, where $ζ_n>0$. In other words, a constant fraction of many-action games are not connected. However, $ζ_n$ is small and tends to $0$ rapidly with $n$, so as $n$ increases all but a vanishingly small fraction of many-player-many-action games are connected. Since connectedness is conducive to equilibrium convergence we obtain, by implication, that there is a simple adaptive dynamic that is guaranteed to lead to a pure Nash equilibrium in all but a vanishingly small fraction of generic games that have one. Our results are based on new probabilistic and combinatorial arguments which allow us to address the large-$k$ regime that the approach used in our previous work could not tackle. We thus complement our previous work to provide a more complete picture of game connectivity across different regimes.

翻译：本研究从连通性角度探讨博弈的典型结构。若博弈存在纯纳什均衡，且满足从任意非均衡行动组合出发均存在最佳响应路径可达每个纯纳什均衡，则称该博弈具有"连通性"；若博弈不存在无差异情形，则称其具有"一般性"。前期研究表明：在全体存在纯纳什均衡的$n$位参与者$k$种行动的一般性博弈中，当$n$相对于$k$充分大时，连通博弈的比例趋近于$1$。本文聚焦于大$k$情形（其表现出不同特性）：证明当$k$增大时，连通比例趋近于$1-ζ_n$，其中$ζ_n>0$。这意味着在多行动博弈中存在恒定比例的非连通博弈。然而$ζ_n$数值较小且随$n$增大而急速趋近于$0$，因此随着参与者数量增加，几乎所有多参与者-多行动博弈都具有连通性。由于连通性有助于均衡收敛，本研究进一步推论：在几乎所有存在纯纳什均衡的一般性博弈中，存在一种简单自适应动态能确保收敛至纯纳什均衡。研究结论基于新的概率论与组合数学论证方法，这些方法能处理前期研究无法解决的大$k$情形，从而与前期工作共同构建了不同参数区间下博弈连通性的完整理论图景。