Characterizing the limit behavior -- that is, the attractors -- of learning dynamics is one of the most fundamental open questions in game theory. In recent work in this front, it was conjectured that the attractors of the replicator dynamic are in one-to-one correspondence with the sink equilibria of the game -- the sink strongly connected components of a game's preference graph -- , and it was established that they do stand in at least one-to-many correspondence with them. We make threefold progress on the problem of characterizing attractors. First, we show through a topological construction that the one-to-one conjecture is false. Second, we make progress on the attractor characterization problem for two-player games by establishing that the one-to-one conjecture is true in the absence of a local pattern called a weak local source -- a pattern that is absent from zero-sum games. Finally, we look -- for the first time in this context -- at fictitious play, the longest-studied learning dynamic, and examine to what extent the conjecture generalizes there. We establish that under fictitious play, sink equilibria always contain attractors (sometimes strictly), and every attractor corresponds to a strongly connected set of nodes in the preference graph.

翻译：刻画学习动态的极限行为——即其吸引子——是博弈论中最基本的开放问题之一。在此前沿的最新研究中，曾有猜想认为复制动态的吸引子与博弈的汇点均衡——博弈偏好图的汇强连通分量——一一对应，并已证实它们之间至少存在一对多的对应关系。我们在刻画吸引子的问题上取得了三方面进展。首先，我们通过拓扑构造证明一一对应猜想不成立。其次，我们在双人博弈的吸引子刻画问题上取得进展，证明在不存在称为弱局部源（该模式在零和博弈中不存在）的局部模式时，一一对应猜想成立。最后，我们首次在此背景下研究虚构博弈这一历史最悠久的学习动态，并探讨该猜想在何种程度上可推广至此。我们证明在虚构博弈中，汇点均衡总是包含吸引子（有时是严格包含），且每个吸引子都对应偏好图中一个强连通的节点集。