Fictitious play (FP) is a history-based strategy to choose actions in normal-form games, where players best-respond to the empirical frequency of their opponents' past actions. While it is well-established that FP converges to the set of Nash equilibria (NE) in zero-sum games, the question of whether it converges to a single equilibrium point, especially when multiple equilibria exist, has remained an open challenge. In this paper, we establish that FP does not necessarily stabilize at a single equilibrium. Specifically, we identify a class of zero-sum games where pointwise convergence fails, regardless of the tie-breaking rules employed. We prove that two geometric conditions on the NE set (A1 and A2) and a technical assumption (A3) are sufficient to preclude pointwise convergence. Furthermore, we conjecture that the first two conditions alone may be sufficient to guarantee this non-convergence, suggesting a broader fundamental instability in FP dynamics.

翻译：虚构博弈（FP）是一种基于历史记录的策略，用于在正则形式博弈中选择行动，其中玩家对其对手过去行动的经验频率做出最佳响应。虽然已有充分证据表明，在零和博弈中，FP会收敛到纳什均衡（NE）的集合，但关于它是否会收敛到单一均衡点的问题，特别是在存在多个均衡时，仍是一个开放的挑战。在本文中，我们证明了FP不一定会在单一均衡处稳定。具体而言，我们确定了一类零和博弈，在这些博弈中，无论采用何种打破平局的规则，逐点收敛都会失败。我们证明了NE集合上的两个几何条件（A1和A2）以及一个技术性假设（A3）足以排除逐点收敛。此外，我们推测仅前两个条件可能就足以保证这种不收敛性，这表明FP动力学中存在更广泛的基本不稳定性。