In 1964 Shapley devised a family of games for which fictitious play fails to converge to Nash equilibrium. The games are two-player non-zero-sum with 3 pure strategies per player. Shapley assumed that each player played a specific pure strategy in the first round. We show that if we use random (mixed) strategy profile initializations we are able to converge to Nash equilibrium approximately 1/3 of the time for a representative game in this class.

翻译：1964年，沙普利设计了一类博弈，其中虚拟博弈无法收敛至纳什均衡。该类博弈为双人非零和博弈，每位玩家各有3种纯策略。沙普利假设每位玩家在首轮博弈中采用特定的纯策略。我们证明，若采用随机（混合）策略剖面初始化，对于该类博弈中的代表性博弈，约有三分之一的概率能够收敛至纳什均衡。