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# The Replicator Dynamic, Chain Components and the Response Graph

*Proceedings of The 34th International Conference on Algorithmic Learning Theory*, PMLR 201:237-258, 2023.

#### Abstract

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 \emph{response graph}. We settle an open problem establishing that under the replicator, \emph{sink chain components}—a topological notion of long-run outcome of a dynamical system—always exist and are approximated by the \emph{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 \emph{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.