The Attractor of the Replicator Dynamic in Zero-Sum Games

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.