The Attractor of the Replicator Dynamic in Zero-Sum Games

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.