Geometrical Regret Matching

TL;DR

This paper introduces a geometrical regret matching method that ensures smooth strategy updates, leading to a more natural convergence towards Nash equilibrium compared to traditional jumpy methods.

Contribution

It proposes a new geometrical approach to regret matching that provides smooth strategy updates, improving the convergence process towards Nash equilibrium.

Findings

Smooth strategy updates facilitate convergence to Nash equilibrium.

The geometrical method offers a clear, visualizable path towards equilibrium.

Limitations exist in optimizing approximation accuracy.

Abstract

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.