  This paper considers the problem of designing optimal algorithms for
reinforcement learning in two-player zero-sum games. We focus on self-play
algorithms which learn the optimal policy by playing against itself without any
direct supervision. In a tabular episodic Markov game with $S$ states, $A$
max-player actions and $B$ min-player actions, the best existing algorithm for
finding an approximate Nash equilibrium requires $\tilde{\mathcal{O}}(S^2AB)$
steps of game playing, when only highlighting the dependency on $(S,A,B)$. In
contrast, the best existing lower bound scales as $\Omega(S(A+B))$ and has a
significant gap from the upper bound. This paper closes this gap for the first
time: we propose an optimistic variant of the \emph{Nash Q-learning} algorithm
with sample complexity $\tilde{\mathcal{O}}(SAB)$, and a new \emph{Nash
V-learning} algorithm with sample complexity $\tilde{\mathcal{O}}(S(A+B))$. The
latter result matches the information-theoretic lower bound in all
problem-dependent parameters except for a polynomial factor of the length of
each episode. In addition, we present a computational hardness result for
learning the best responses against a fixed opponent in Markov games---a
learning objective different from finding the Nash equilibrium.
