Two-Player Zero-Sum Games with Bandit Feedback

We study a two-player zero-sum game in which the row player aims to maximize their payoff against a competing column player, under an unknown payoff matrix estimated through bandit feedback. We propose three algorithms based on the Explore-Then-Commit (ETC) and action pair elimination frameworks. The first adapts it to zero-sum games, the second incorporates adaptive elimination that leverages the $\varepsilon$-Nash Equilibrium property to efficiently select the optimal action pair, and the third extends the elimination algorithm by employing non-uniform exploration. Our objective is to demonstrate the applicability of ETC and action pair elimination algorithms in a zero-sum game setting by focusing on learning pure strategy Nash Equilibria. A key contribution of our work is a derivation of instance-dependent upper bounds on the expected regret of our proposed algorithms, which has received limited attention in the literature on zero-sum games. Particularly, after $T$ rounds, we achieve an instance-dependent regret upper bounds of $O(Δ+ \sqrt{T})$ for ETC in zero-sum game setting and $O\left(\frac{\log (T Δ^2)}Δ\right)$ for the adaptive elimination algorithm and its variant with non-uniform exploration, where $Δ$ denotes the suboptimality gap. Therefore, our results indicate that the ETC and action pair elimination algorithms perform effectively in zero-sum game settings, achieving regret bounds comparable to existing methods while providing insight through instance-dependent analysis.

💡 Research Summary

This paper investigates learning in finite two‑player zero‑sum games (TPZSG) when the payoff matrix is unknown and only bandit feedback is available. In each round the row player selects an action i∈Sₓ, the column player selects j∈S_y, and both observe a noisy payoff rₜ with E