Abstract: In this work, we develop provably efficient reinforcement learning algorithms for two-player zero-sum Markov games with simultaneous moves. \n We consider a family of Markov games where the reward function and\n transition kernel possess a linear structure. Two settings are studied: In the offline setting, we control both players and the goal is to find the Nash Equilibrium efficiently by minimizing the worst-case duality gap. In the online setting, we control a single player and play against an arbitrary opponent; the goal is to minimize the regret. For both settings, we propose an optimistic variant of the least-squares minimax value iteration algorithm.\n We show that our algorithm is computationally efficient and provably achieves an $\tilde O(\sqrt{d^3 H^3 T} )$ upper bound on the duality gap and regret, without requiring additional assumptions on the sampling model. \n We highlight that our setting requires overcoming several new challenges that are absent in MDPs or turn-based Markov games. In particular, to achieve optimism under the simultaneous-move games, we construct both upper and lower confidence bounds of the value function, and then derive the optimistic policy by solving a general-sum matrix game with these bounds as the payoff matrices.\n As finding the Nash Equilibrium of this general-sum game is computationally hard, our algorithm instead solves for a Coarse Correlated Equilibrium (CCE), which can be obtained efficiently via linear programming. To our best knowledge, such a CCE-based mechanism for implementing optimism has not appeared in the literature and might be of interest in its own right.