We analyse the typical structure of games in terms of the connectivity properties of their best-response graphs. Our central result shows that, among games that are `generic' (without indifferences) and that have a pure Nash equilibrium, all but a small fraction are \emph{connected}, meaning that every action profile that is not a pure Nash equilibrium can reach every pure Nash equilibrium via best-response paths. This has important implications for dynamics in games. In particular, we show that there are simple, uncoupled, adaptive dynamics for which period-by-period play converges almost surely to a pure Nash equilibrium in all but a small fraction of generic games that have one (which contrasts with the known fact that there is no such dynamic that leads almost surely to a pure Nash equilibrium in \emph{every} generic game that has one). We build on recent results in probabilistic combinatorics for our characterisation of game connectivity.

翻译：我们通过研究博弈的最佳响应图的连通性特征，分析了博弈的典型结构。核心结果表明，在无差异且存在纯纳什均衡的“一般性”博弈中，除极小比例外，其余博弈均为连通性博弈——即每个非纯纳什均衡的行动轨迹均能通过最佳响应路径抵达所有纯纳什均衡。这一结论对博弈动态具有重要启示：特别地，我们证明存在一种简单的非耦合自适应动态机制，使得在除极小比例外的所有存在纯纳什均衡的一般性博弈中，周期迭代博弈几乎必然收敛于纯纳什均衡（这与已知结论形成对比——已知不存在能保证在每一个存在纯纳什均衡的一般性博弈中几乎必然收敛至纯纳什均衡的此类动态）。我们基于概率组合学的最新成果，完成了对博弈连通性的刻画。