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.

翻译：本文研究了复制动态（复制子动态）——乘性权重更新算法的连续极限——的流与博弈响应图之间的关系。我们解决了一个开放问题，证明在复制动态下，汇链分量（刻画动力系统长期行为的拓扑概念）始终存在，且可由博弈响应图的汇连通分量近似。具体而言，每个汇链分量包含响应图的一个汇连通分量，以及所有支撑集由同一连通分量中的纯策略组成的混合策略组合（称为该连通分量的内容）。作为推论，当响应图强连通时，所有策略组合都是链循环的。在与零和博弈共享响应图的任意两人博弈中，汇链分量唯一。在两人零和博弈与势博弈中，汇链分量与汇连通分量一一对应，我们猜想该性质对所有博弈成立。