  This paper considers two-player zero-sum finite-horizon Markov games with
simultaneous moves. The study focuses on the challenging settings where the
value function or the model is parameterized by general function classes.
Provably efficient algorithms 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 algorithm by 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 both
players 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 can be bounded by a generalization of Witness rank to Markov games.
The model-free algorithm enjoys a $\sqrt{K}$-regret upper bound where $K$ is
the number of episodes.
