In this paper we characterise the long-run behaviour of the replicator dynamic in two-player zero-sum games (symmetric or otherwise). Specifically, we prove that every zero-sum game possesses a unique global attractor, which we then characterise. Most surprisingly, this attractor depends only on each player's preference order over their own strategies and not on the cardinal payoff values. Consequently, it is structurally stable. The attractor is defined by a finite directed graph we call the game's fundamental graph. If the game is symmetric, this graph is a tournament whose nodes are strategies; if the game is not symmetric, this graph is the game's response graph. In both cases the attractor can be computed in time quasilinear in the size of the game. We discuss the consequences of our results on chain recurrence and equilibria in games.

翻译：本文刻画了双人零和博弈（对称或非对称情形）中复制动态的长期行为。具体而言，我们证明每个零和博弈存在唯一的全局吸引子，并对其进行了刻画。最令人惊讶的是，该吸引子仅取决于每位玩家对其自身策略的偏好顺序，而与基数支付值无关。因此，它具有结构稳定性。该吸引子由一个有限有向图定义，我们称其为博弈的基础图。若博弈是对称的，则该图是一个以策略为节点的竞赛图；若博弈非对称，则该图为博弈的响应图。在两种情形下，该吸引子均可在博弈规模拟线性时间内计算。我们讨论了这些结果对博弈中链循环与均衡的启示。