Thompson Sampling Algorithm for Stochastic Games

TL;DR

This paper introduces a Thompson sampling algorithm for stochastic differential games with multiple players, providing regret bounds and convergence to Nash equilibrium in a complex dynamic setting.

Contribution

It develops a novel Thompson sampling approach with dynamic episodes for multi-player stochastic games, achieving regret bounds independent of the number of players.

Findings

Bayesian regret is bounded by O(√T log T) for each player.

Average regret per unit time approaches zero as T increases.

The algorithm converges to a Nash equilibrium in the game.

Abstract

We study a stochastic differential game with $N$ competitive players in a linear-quadratic framework with ergodic cost, where $d$-dimensional diffusion processes govern the state dynamics with an unknown common drift (matrix). Assuming a Gaussian prior on the drift, we use filtering techniques to update its posterior estimates. Based on these estimates, we propose a Thompson-sampling-based algorithm with dynamic episode lengths to approximate strategies. We show that the Bayesian regret for each player has an error bound of order $O(\sqrt{T\log(T)})$, where $T$ is the time-horizon, independent of the number of players. This implies that average regret per unit time goes to zero. Finally, we prove that the algorithm results in a Nash equilibrium.