  In this paper we examine the relationship between the flow of the replicator
dynamic, the continuum limit of Multiplicative Weights Update, and a game's
response graph. We settle an open problem establishing that under the
replicator, sink chain components -- a topological notion of long-run outcome
of a dynamical system -- always exist and are approximated by the sink
connected components of the game's response graph. More specifically, each sink
chain component contains a sink connected component of the response graph, as
well as all mixed strategy profiles whose support consists of pure profiles in
the same connected component, a set we call the content of the connected
component. As a corollary, all profiles are chain recurrent in games with
strongly connected response graphs. In any two-player game sharing a response
graph with a zero-sum game, the sink chain component is unique. In two-player
zero-sum and potential games the sink chain components and sink connected
components are in a one-to-one correspondence, and we conjecture that this
holds in all games.
