  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.
