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 on 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. Here, we show that the one-to-one conjecture is false. We disprove this conjecture over the course of three theorems: the first disproves a stronger form of the conjecture, while the weaker form is disproved separately in the two-player and $N$-player ($N>2$) cases. By showing how the conjecture fails, we lay out the obstacles that lie ahead for characterizing attractors of the replicator, and introduce new ideas with which to tackle them. All three counterexamples derive from an object called a local source -- a point lying within the sink equilibrium, and yet which is `locally repelling'; we prove that the absence of local sources is necessary, but not sufficient, for the one-to-one property to be true. We complement this with a sufficient condition: we introduce a local property of a sink equilibrium called pseudoconvexity, and establish that when the sink equilibria of a two-player game are pseudoconvex then they precisely define the attractors. Pseudoconvexity generalizes the previous cases -- such as zero-sum games and potential games -- where this conjecture was known to hold, and reformulates these cases in terms of a simple graph property.

翻译：刻画学习动态的极限行为——即吸引子——是博弈论中最基本的未解问题之一。在近期相关研究中，有猜想认为复制动态的吸引子与博弈的汇点均衡（即博弈偏好图的汇强连通分量）存在一一对应关系，并已证实二者至少存在一对多对应关系。本文证明该一一对应猜想不成立。我们通过三个定理证伪该猜想：第一个定理证伪了猜想的强化形式，而弱化形式分别在双人博弈与$N$人博弈（$N>2$）情形下被证伪。通过揭示猜想失败的原因，我们阐明了刻画复制动态吸引子所面临的核心障碍，并提出了解决这些问题的新思路。所有三个反例均源于称为局部源点的对象——该点虽位于汇点均衡内部，却具有局部排斥性；我们证明局部源点的缺失是一一对应性质成立的必要非充分条件。作为补充，我们提出一个充分条件：引入称为伪凸性的汇点均衡局部性质，并证明当双人博弈的汇点均衡具有伪凸性时，它们能精确定义吸引子。伪凸性推广了先前已知猜想成立的案例（如零和博弈与势博弈），并将这些案例重构为简单的图性质。