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开放问题之一。在这一前沿的近期研究中，曾有猜想认为复制动态的吸引子与博弈的汇点均衡——即博弈偏好图的汇点强连通分量——存在一一对应关系，并且已证实它们之间至少存在一对多的对应关系。本文证明这一一对应猜想是错误的。我们通过三个定理证伪该猜想：第一个定理证伪了该猜想的一种更强形式，而较弱形式则在两人博弈和$N$人博弈（$N>2$）情形中分别被证伪。通过揭示猜想失败的具体机制，我们阐明了刻画复制动态吸引子所面临的核心障碍，并提出了应对这些障碍的新思路。所有三个反例都源于称为局部源点的对象——即位于汇点均衡内部却具有“局部排斥性”的点；我们证明局部源点的缺失是一一对应性质成立的必要条件，但并非充分条件。我们进一步补充了充分条件：引入称为伪凸性的汇点均衡局部性质，并证明当两人博弈的汇点均衡具有伪凸性时，它们能精确定义吸引子。伪凸性推广了此前已知该猜想成立的各类情形——如零和博弈与势博弈，并将这些情形重新表述为简单的图性质。