Fictitious play is a history-based learning process in which players best respond to the empirical distribution of their opponents' past play. Although classical results show that fictitious play converges to the set of Nash equilibria in zero-sum games, this set convergence does not imply convergence to a single equilibrium when the equilibrium set is non-singleton. This paper shows that pointwise convergence fails in a strong sense. Suppose the equilibrium set of a player has positive measure and consists only of fully mixed strategies. Then, under any tie-breaking rule, the empirical strategy of that player cannot converge to any equilibrium point, provided it is initialized outside the equilibrium set -- in particular, when the player starts with no prior beliefs. The proof identifies two mechanisms behind this instability. In the interior of the equilibrium set, the dynamics retain inertia that prevents settling. At the boundary, the opponent's deviations from its unique equilibrium action steadily unbalance the cumulative payoffs that drive best responses, so that not all actions can remain competitive, as a fully mixed limit would require. The results clarify the gap between convergence to an equilibrium set and convergence to an equilibrium point in fictitious play.

翻译：虚构博弈是一种基于历史的学习过程，其中玩家对其对手过去博弈的经验分布做出最优反应。尽管经典结果表明虚构博弈在零和博弈中收敛于纳什均衡集，但当均衡集非单点时，该集合收敛并不意味着收敛于单一均衡。本文证明，点态收敛在强意义上失败。假设某玩家的均衡集具有正测度且仅由完全混合策略构成。那么，在任何破平规则下，只要该玩家的经验策略初始化为均衡集之外——特别是当玩家从无先验信念开始时——它就无法收敛于任何均衡点。证明识别了这种不稳定性背后的两种机制：在均衡集内部，动态保持惯性，阻止了定点收敛；在边界处，对手偏离其唯一均衡行动的行为会持续打破驱动最优反应的累积收益的平衡，使得并非所有行动都能保持竞争力——而完全混合极限要求所有行动均能保持竞争力。这些结果阐明了虚构博弈中收敛于均衡集与收敛于均衡点之间的差距。