  Model-based algorithms -- algorithms that explore the environment through
building and utilizing an estimated model -- are widely used in reinforcement
learning practice and theoretically shown to achieve optimal sample efficiency
for single-agent reinforcement learning in Markov Decision Processes (MDPs).
However, for multi-agent reinforcement learning in Markov games, the current
best known sample complexity for model-based algorithms is rather suboptimal
and compares unfavorably against recent model-free approaches. In this paper,
we present a sharp analysis of model-based self-play algorithms for multi-agent
Markov games. We design an algorithm -- Optimistic Nash Value Iteration
(Nash-VI) for two-player zero-sum Markov games that is able to output an
$\epsilon$-approximate Nash policy in $\tilde{\mathcal{O}}(H^3SAB/\epsilon^2)$
episodes of game playing, where $S$ is the number of states, $A,B$ are the
number of actions for the two players respectively, and $H$ is the horizon
length. This significantly improves over the best known model-based guarantee
of $\tilde{\mathcal{O}}(H^4S^2AB/\epsilon^2)$, and is the first that matches
the information-theoretic lower bound $\Omega(H^3S(A+B)/\epsilon^2)$ except for
a $\min\{A,B\}$ factor. In addition, our guarantee compares favorably against
the best known model-free algorithm if $\min \{A,B\}=o(H^3)$, and outputs a
single Markov policy while existing sample-efficient model-free algorithms
output a nested mixture of Markov policies that is in general non-Markov and
rather inconvenient to store and execute. We further adapt our analysis to
designing a provably efficient task-agnostic algorithm for zero-sum Markov
games, and designing the first line of provably sample-efficient algorithms for
multi-player general-sum Markov games.
