  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.
