In this paper we characterise the long-run behaviour of the replicator dynamic in zero-sum games (symmetric or non-symmetric). Specifically, we prove that every zero-sum game possesses a unique global replicator 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, defined by a finite directed graph we call the game's preference graph. When the game is symmetric, this graph is a tournament whose nodes are strategies; when the game is not symmetric, this graph is the game's response graph. We discuss the consequences of our results on chain recurrence and Nash equilibria.

翻译：本文刻画了零和博弈（对称或非对称）中复制动力学的长期行为。具体而言，我们证明每一个零和博弈均存在唯一的全局复制吸引子，并对其进行了刻画。最令人惊讶的是，该吸引子仅取决于每个玩家对其自身策略的偏好顺序，而非基数收益值，并由一个称为博弈偏好图的有限有向图所定义。当博弈对称时，该图是一个以策略为节点的竞赛图；当博弈非对称时，该图是博弈的响应图。我们讨论了所得结果对链递归性和纳什均衡的影响。