# Learning Zero-Sum Simultaneous-Move Markov Games Using Function Approximation and Correlated Equilibrium

arXiv: https://arxiv.org/abs/2002.07066 · HTML: https://mlsanity.com/paper/2002.07066/

## Key facts

- Cited by 100 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Feb 2020.

## Built on by (most cited first, 30 of 100)

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- [Independent Policy Gradient for Large-Scale Markov Potential Games: Sharper Rates, Function Approximation, and Game-Agnostic Convergence](https://mlsanity.com/paper/2202.04129/) (2022)
- [Nearly Minimax Optimal Offline Reinforcement Learning with Linear Function Approximation: Single-Agent MDP and Markov Game](https://mlsanity.com/paper/2205.15512/) (2022)
- [The Complexity of Markov Equilibrium in Stochastic Games](https://mlsanity.com/paper/2204.03991/) (2022)
- [The Power of Exploiter: Provable Multi-Agent RL in Large State Spaces](https://mlsanity.com/paper/2106.03352/) (2021)
- [Efficient and Optimal Algorithms for Contextual Dueling Bandits under Realizability](https://mlsanity.com/paper/2111.12306/) (2021)
- [Pessimistic Minimax Value Iteration: Provably Efficient Equilibrium Learning from Offline Datasets](https://mlsanity.com/paper/2202.07511/) (2022)
- [Towards General Function Approximation in Zero-Sum Markov Games](https://mlsanity.com/paper/2107.14702/) (2021)
- [Provably Efficient Reinforcement Learning in Decentralized General-Sum Markov Games](https://mlsanity.com/paper/2110.05682/) (2021)
- [Sample-Efficient Learning of Stackelberg Equilibria in General-Sum Games](https://mlsanity.com/paper/2102.11494/) (2021)
- [On Improving Model-Free Algorithms for Decentralized Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2110.05707/) (2021)
- [Provably Efficient Model-Free Constrained RL with Linear Function Approximation](https://mlsanity.com/paper/2206.11889/) (2022)
- [Provably Efficient Offline Multi-agent Reinforcement Learning via Strategy-wise Bonus](https://mlsanity.com/paper/2206.00159/) (2022)
- [Breaking the Curse of Multiagency: Provably Efficient Decentralized Multi-Agent RL with Function Approximation](https://mlsanity.com/paper/2302.06606/) (2023)
- [Uncoupled and Convergent Learning in Two-Player Zero-Sum Markov Games with Bandit Feedback](https://mlsanity.com/paper/2303.02738/) (2023)
- [When is Offline Two-Player Zero-Sum Markov Game Solvable?](https://mlsanity.com/paper/2201.03522/) (2022)
- [Sample-Efficient Reinforcement Learning of Partially Observable Markov Games](https://mlsanity.com/paper/2206.01315/) (2022)
- [Breaking the Curse of Multiagents in a Large State Space: RL in Markov Games with Independent Linear Function Approximation](https://mlsanity.com/paper/2302.03673/) (2023)
- [Near-Optimal Learning of Extensive-Form Games with Imperfect Information](https://mlsanity.com/paper/2202.01752/) (2022)
- [Contrastive UCB: Provably Efficient Contrastive Self-Supervised Learning in Online Reinforcement Learning](https://mlsanity.com/paper/2207.14800/) (2022)
- [Faster Last-iterate Convergence of Policy Optimization in Zero-Sum Markov Games](https://mlsanity.com/paper/2210.01050/) (2022)
- [Almost Optimal Algorithms for Two-player Zero-Sum Linear Mixture Markov Games](https://mlsanity.com/paper/2102.07404/) (2021)
- [Can Reinforcement Learning Find Stackelberg-Nash Equilibria in General-Sum Markov Games with Myopic Followers?](https://mlsanity.com/paper/2112.13521/) (2021)
- [ESCHER: Eschewing Importance Sampling in Games by Computing a History Value Function to Estimate Regret](https://mlsanity.com/paper/2206.04122/) (2022)
- [Regret Minimization and Convergence to Equilibria in General-sum Markov Games](https://mlsanity.com/paper/2207.14211/) (2022)

## Abstract (arXiv)

We develop provably efficient reinforcement learning algorithms for two-player zero-sum finite-horizon Markov games with simultaneous moves. To incorporate function approximation, we consider a family of Markov games where the reward function and transition kernel possess a linear structure. Both the offline and online settings of the problems are considered. In the offline setting, we control both players and aim to find the Nash Equilibrium by minimizing the duality gap. In the online setting, we control a single player playing against an arbitrary opponent and aim to minimize the regret. For both settings, we propose an optimistic variant of the least-squares minimax value iteration algorithm. We show that our algorithm is computationally efficient and provably achieves an tilde O(√d3 H3 T ) upper bound on the duality gap and regret, where d is the linear dimension, H the horizon and T the total number of timesteps. Our results do not require additional assumptions on the sampling model. Our setting requires overcoming several new challenges that are absent in Markov decision processes or turn-based Markov games. In particular, to achieve optimism with simultaneous moves, we construct both upper and lower confidence bounds of the value function, and then compute the optimistic policy by solving a general-sum matrix game with these bounds as the payoff matrices. As finding the Nash Equilibrium of a general-sum game is computationally hard, our algorithm instead solves for a Coarse Correlated Equilibrium (CCE), which can be obtained efficiently. To our best knowledge, such a CCE-based scheme for optimism has not appeared in the literature and might be of interest in its own right.