Two-Player Zero-Sum Games with Bandit Feedback

We study a two-player zero-sum game (TPZSG) in which the row player aims to maximize their payoff against an adversarial column player, under an unknown payoff matrix estimated through bandit feedback. We propose and analyze two algorithms: ETC-TPZSG, which directly applies ETC to the TPZSG setting and ETC-TPZSG-AE, which improves upon it by incorporating an action pair elimination (AE) strategy that leverages the $\varepsilon$-Nash Equilibrium property to efficiently select the optimal action pair. Our objective is to demonstrate the applicability of ETC in a TPZSG setting by focusing on learning pure strategy Nash Equilibrium. A key contribution of our work is a derivation of instance-dependent upper bounds on the expected regret for both algorithms, has received limited attention in the literature on zero-sum games. Particularly, after $T$ rounds, we achieve an instance-dependent regret upper bounds of $O(\Delta + \sqrt{T})$ for ETC-TPZSG and $O(\frac{\log (T \Delta^2)}{\Delta})$ for ETC-TPZSG-AE, where $\Delta$ denotes the suboptimality gap. Therefore, our results indicate that ETC-based algorithms perform effectively in adversarial game settings, achieving regret bounds comparable to existing methods while providing insights through instance-dependent analysis.