  This paper makes progress towards learning Nash equilibria in two-player
zero-sum Markov games from offline data. Specifically, consider a
$\gamma$-discounted infinite-horizon Markov game with $S$ states, where the
max-player has $A$ actions and the min-player has $B$ actions. We propose a
pessimistic model-based algorithm with Bernstein-style lower confidence bounds
-- called VI-LCB-Game -- that provably finds an $\varepsilon$-approximate Nash
equilibrium with a sample complexity no larger than
$\frac{C_{\mathsf{clipped}}^{\star}S(A+B)}{(1-\gamma)^{3}\varepsilon^{2}}$ (up
to some log factor). Here, $C_{\mathsf{clipped}}^{\star}$ is some unilateral
clipped concentrability coefficient that reflects the coverage and distribution
shift of the available data (vis-\`a-vis the target data), and the target
accuracy $\varepsilon$ can be any value within
$\big(0,\frac{1}{1-\gamma}\big]$. Our sample complexity bound strengthens prior
art by a factor of $\min\{A,B\}$, achieving minimax optimality for the entire
$\varepsilon$-range. An appealing feature of our result lies in algorithmic
simplicity, which reveals the unnecessity of variance reduction and sample
splitting in achieving sample optimality.
