Abstract: This paper considers two-player zero-sum finite-horizon Markov games with simultaneous moves. The study focuses on the challenging settings where the valuefunction or the model is parameterized by general function classes. Provably efficientalgorithms for both decoupled and coordinated settings are developed. In the decoupled setting where the agent controls a single player and plays against an arbitrary opponent, we propose a new model-free algorithm. The sample complexity is governed by the Minimax Eluder dimension—a new dimension of the function class in Markov games. As a special case, this method improves the state-of-the-art algorithmby a $\sqrt{d}$ factor in the regret when the reward function and transition kernel are parameterized with d-dimensional linear features. In the coordinated setting where bothplayers are controlled by the agent, we propose a model-based algorithm and a model-free algorithm. In the model-based algorithm, we prove that sample complexity canbe bounded by a generalization of Witness rank to Markov games. The model-freealgorithm enjoys a $\sqrt{K}$-regret upper bound where $K$ is the number of episodes. Ouralgorithms are based on new techniques of alternate optimism