Many well-studied learning dynamics, such as fictitious play and the replicator, are known to not converge in general $N$-player games. The simplest mode of non-convergence is cyclical or periodic behavior. Such cycles are fundamental objects, and have inspired a number of significant insights in the field, beginning with the pioneering work of Shapley (1964). However a central question remains unanswered: which cycles are stable under game dynamics? In this paper we give a complete and computational answer to this question for the two best-studied dynamics, fictitious play/best-response dynamics and the replicator dynamic. We show (1) that a periodic sequence of profiles is stable under one of these dynamics if and only it is stable under the other, and (2) we provide a polynomial-time spectral stability test to determine whether a given periodic sequence is stable under either dynamic. Finally, we give an entirely `structural' sufficient condition for stability: every cycle that is a sink equilibrium of the preference graph of the game is stable, and moreover it is an attractor of the replicator dynamic. This result generalizes the famous theorems of Shapley (1964) and Jordan (1993), and extends the frontier of recent work relating the preference graph to the replicator attractors.

翻译：许多被深入研究的博弈学习动态，如虚拟博弈和复制者动态，已知在一般$N$人博弈中并不收敛。最简单的非收敛模式是循环或周期行为。此类循环是基本对象，并已启发该领域的许多重要见解，始于Shapley（1964）的开创性工作。然而，一个核心问题仍未得到解答：哪些循环在博弈动态下是稳定的？本文针对两种研究最深入的动态——虚拟博弈/最优反应动态与复制者动态，给出了一个完整且可计算的答案。我们证明：（1）一个策略剖面序列在一种动态下稳定当且仅当它在另一种动态下也稳定；（2）我们提供了一个多项式时间的谱稳定性检验方法，用于判定给定周期序列在任一动态下是否稳定。最后，我们给出了一个完全“结构性”的稳定性充分条件：博弈偏好图的每个汇平衡循环都是稳定的，并且它是复制者动态的一个吸引子。这一结果推广了Shapley（1964）和Jordan（1993）的著名定理，并拓展了近期将偏好图与复制者动态吸引子相关联的研究前沿。