Geometrical Regret Matching of Mixed Strategies
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We argue that the existing regret matchings for equilibrium approximation lead to "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 has a "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 equilibrium, suggesting that in reality the tendency 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 analyzed in metric space and visualized nicely as a clear path towards an equilibrium point. Our theory has limitations in optimizing the approximation accuracy.
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