The Replicator Dynamic, Chain Components and the Response Graph

TL;DR

This paper links the long-term behavior of the replicator dynamic to the structure of a game's response graph, establishing existence and approximation of sink chain components and their relation to game properties.

Contribution

It proves that sink chain components always exist under the replicator dynamic and are approximated by sink connected components of the response graph, clarifying long-term outcomes.

Findings

Sink chain components always exist under the replicator dynamic.

Sink chain components are approximated by sink connected components of the response graph.

In certain classes of games, these components are in one-to-one correspondence.

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 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.