We analyse the typical structure of games in terms of the connectivity properties of their best-response graphs. In particular, we show that almost every 'large' generic game that has a pure Nash equilibrium is connected, meaning that every non-equilibrium action profile can reach every pure Nash equilibrium via best-response paths. This has implications for dynamics in games: many adaptive dynamics, such as the best-response dynamic with inertia, lead to equilibrium in connected games. It follows that there are simple, uncoupled, adaptive dynamics for which period-by-period play converges almost surely to a pure Nash equilibrium in almost every 'large' generic game that has one. We build on recent results in probabilistic combinatorics for our characterisation of game connectivity.

翻译：我们通过博弈的最佳响应图的连通性特征来分析博弈的典型结构。特别地，我们证明几乎所有存在纯纳什均衡的“大型”一般博弈都是连通的，这意味着每个非均衡行动组合都能通过最佳响应路径到达每个纯纳什均衡。这对博弈中的动力学具有重要意义：许多自适应动力学，例如带惯性的最佳响应动力学，在连通博弈中会导致均衡。由此可得，在几乎所有存在纯纳什均衡的“大型”一般博弈中，存在简单、非耦合的自适应动力学，使得逐期博弈几乎必然收敛到纯纳什均衡。我们基于概率组合学的最新成果来刻画博弈连通性。