## Provably Efficient Fictitious Play Policy Optimization for Zero-Sum Markov Games with Structured Transitions

### Shuang Qiu · Xiaohan Wei · Jieping Ye · Zhaoran Wang · Zhuoran Yang

While single-agent policy optimization in a fixed environment has attracted a lot of research attention recently in the reinforcement learning community, much less is known theoretically when there are multiple agents playing in a potentially competitive environment. We take steps forward by proposing and analyzing new fictitious play policy optimization algorithms for two-player zero-sum Markov games with structured but unknown transitions. We consider two classes of transition structures: factored independent transition and single-controller transition. For both scenarios, we prove tight $\widetilde{\mathcal{O}}(\sqrt{T})$ regret bounds after $T$ steps in a two-agent competitive game scenario. The regret of each player is measured against a potentially adversarial opponent who can choose a single best policy in hindsight after observing the full policy sequence. Our algorithms feature a combination of Upper Confidence Bound (UCB)-type optimism and fictitious play under the scope of simultaneous policy optimization in a non-stationary environment. When both players adopt the proposed algorithms, their overall optimality gap is $\widetilde{\mathcal{O}}(\sqrt{T})$.