We argue that the existing regret matchings for Nash equilibrium approximation conduct "jumpy" strategy updating when the probabilities of future plays are set to be proportional to positive regret measures. We propose a geometrical regret matching which features "smooth" strategy updating. Our approach is simple, intuitive and natural. The analytical and numerical results show that, continuously and "smoothly" suppressing "unprofitable" pure strategies is sufficient for the game to evolve towards Nash equilibrium, suggesting that in reality the tendency for equilibrium could be pervasive and irresistible. Technically, iterative regret matching gives rise to a sequence of adjusted mixed strategies for our study its approximation to the true equilibrium point. The sequence can be studied in metric space and visualized nicely as a clear path towards an equilibrium point. Our theory has limitations in optimizing the approximation accuracy.

中文翻译：

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几何遗憾匹配

我们认为，当未来游戏的概率设置为与正后悔措施成正比时，纳什均衡近似的现有后悔匹配会进行“跳跃”策略更新。我们提出了一种几何遗憾匹配，其特征是“平滑”策略更新。我们的方法简单、直观且自然。分析和数值结果表明，持续、“平稳”地抑制“无利可图”的纯策略足以使博弈向纳什均衡演化，这表明在现实中均衡的趋势可能是普遍且不可抗拒的。从技术上讲，迭代后悔匹配产生了一系列调整后的混合策略，用于我们研究其对真实平衡点的近似。可以在度量空间中研究该序列，并将其很好地可视化为通往平衡点的清晰路径。我们的理论在优化近似精度方面存在局限性。