Geometrical regret matching: A new dynamics to Nash equilibrium

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 that features “smooth” strategy updating. Our approach is simple, intuitive, and natural. The analytical and numerical results show that “smoothly” suppressing “unprofitable” pure strategies is sufficient for the game to evolve toward 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 us to examine its approximation to the true equilibrium point. The sequence can be studied in the metric space and visualized nicely as a clear path toward an equilibrium point. Our theory has limitations in optimizing the approximation accuracy.