# Policy Optimization for Markov Games: Unified Framework and Faster Convergence

arXiv: https://arxiv.org/abs/2206.02640 · HTML: https://mlsanity.com/paper/2206.02640/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 45 of its 67 references that are in the map.
- First posted on arXiv 6 Jun 2022.

## Builds on (most cited first, 30 of 45)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Trust Region Policy Optimization](https://mlsanity.com/paper/1502.05477/) (2015)
- [The Surprising Effectiveness of PPO in Cooperative, Multi-Agent Games](https://mlsanity.com/paper/2103.01955/) (2021)
- [QMIX: Monotonic Value Function Factorisation for Deep Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/1803.11485/) (2018)
- [The StarCraft Multi-Agent Challenge](https://mlsanity.com/paper/1902.04043/) (2019)
- [Multi-Agent Reinforcement Learning: A Selective Overview of Theories and Algorithms](https://mlsanity.com/paper/1911.10635/) (2019)
- [Is Q-learning Provably Efficient?](https://mlsanity.com/paper/1807.03765/) (2018)
- [Is Independent Learning All You Need in the StarCraft Multi-Agent Challenge?](https://mlsanity.com/paper/2011.09533/) (2020)
- [On the Global Convergence Rates of Softmax Policy Gradient Methods](https://mlsanity.com/paper/2005.06392/) (2020)
- [Emergence of Grounded Compositional Language in Multi-Agent Populations](https://mlsanity.com/paper/1703.04908/) (2017)
- [The Hanabi Challenge: A New Frontier for AI Research](https://mlsanity.com/paper/1902.00506/) (2019)
- [Optimization, Learning, and Games with Predictable Sequences](https://mlsanity.com/paper/1311.1869/) (2013)
- [Fast Global Convergence of Natural Policy Gradient Methods with Entropy Regularization](https://mlsanity.com/paper/2007.06558/) (2020)
- [Fast Convergence of Regularized Learning in Games](https://mlsanity.com/paper/1507.00407/) (2015)
- [Global Optimality Guarantees For Policy Gradient Methods](https://mlsanity.com/paper/1906.01786/) (2019)
- [On the Convergence Rates of Policy Gradient Methods](https://mlsanity.com/paper/2201.07443/) (2022)
- [Adaptive Trust Region Policy Optimization: Global Convergence and Faster Rates for Regularized MDPs](https://mlsanity.com/paper/1909.02769/) (2019)
- [V-Learning -- A Simple, Efficient, Decentralized Algorithm for Multiagent RL](https://mlsanity.com/paper/2110.14555/) (2021)
- [Provable Self-Play Algorithms for Competitive Reinforcement Learning](https://mlsanity.com/paper/2002.04017/) (2020)
- [Near-Optimal Reinforcement Learning with Self-Play](https://mlsanity.com/paper/2006.12007/) (2020)
- [A Sharp Analysis of Model-based Reinforcement Learning with Self-Play](https://mlsanity.com/paper/2010.01604/) (2020)
- [Independent Policy Gradient Methods for Competitive Reinforcement Learning](https://mlsanity.com/paper/2101.04233/) (2021)
- [Learning Zero-Sum Simultaneous-Move Markov Games Using Function Approximation and Correlated Equilibrium](https://mlsanity.com/paper/2002.07066/) (2020)
- [Global Convergence of Multi-Agent Policy Gradient in Markov Potential Games](https://mlsanity.com/paper/2106.01969/) (2021)
- [Model-Based Multi-Agent RL in Zero-Sum Markov Games with Near-Optimal Sample Complexity](https://mlsanity.com/paper/2007.07461/) (2020)
- [Near-Optimal No-Regret Learning in General Games](https://mlsanity.com/paper/2108.06924/) (2021)
- [Last-iterate Convergence of Decentralized Optimistic Gradient Descent/Ascent in Infinite-horizon Competitive Markov Games](https://mlsanity.com/paper/2102.04540/) (2021)
- [Online Reinforcement Learning in Stochastic Games](https://mlsanity.com/paper/1712.00579/) (2017)
- [When Can We Learn General-Sum Markov Games with a Large Number of Players Sample-Efficiently?](https://mlsanity.com/paper/2110.04184/) (2021)
- [Fast Policy Extragradient Methods for Competitive Games with Entropy Regularization](https://mlsanity.com/paper/2105.15186/) (2021)

## Built on by (most cited first, 16 of 16)

- [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)
- [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)
- [Faster Last-iterate Convergence of Policy Optimization in Zero-Sum Markov Games](https://mlsanity.com/paper/2210.01050/) (2022)
- [Regret Minimization and Convergence to Equilibria in General-sum Markov Games](https://mlsanity.com/paper/2207.14211/) (2022)
- [Minimax-Optimal Multi-Agent RL in Markov Games With a Generative Model](https://mlsanity.com/paper/2208.10458/) (2022)
- [Near-Optimal Policy Optimization for Correlated Equilibrium in General-Sum Markov Games](https://mlsanity.com/paper/2401.15240/) (2024)
- [O(T-1) Convergence of Optimistic-Follow-the-Regularized-Leader in Two-Player Zero-Sum Markov Games](https://mlsanity.com/paper/2209.12430/) (2022)
- [Symmetric (Optimistic) Natural Policy Gradient for Multi-agent Learning with Parameter Convergence](https://mlsanity.com/paper/2210.12812/) (2022)
- [Near-Optimal Phi-Regret Learning in Extensive-Form Games](https://mlsanity.com/paper/2208.09747/) (2022)
- [Optimistic Policy Gradient in Multi-Player Markov Games with a Single Controller: Convergence Beyond the Minty Property](https://mlsanity.com/paper/2312.12067/) (2023)
- [Unified Policy Optimization for Continuous-action Reinforcement Learning in Non-stationary Tasks and Games](https://mlsanity.com/paper/2208.09452/) (2022)
- [Convergence to Nash Equilibrium and No-regret Guarantee in (Markov) Potential Games](https://mlsanity.com/paper/2404.06516/) (2024)
- [Near Optimal Convergence to Coarse Correlated Equilibrium in General-Sum Markov Games](https://mlsanity.com/paper/2511.02157/) (2025)
- [Convex Markov Games and Beyond: New Proof of Existence, Characterization and Learning Algorithms for Nash Equilibria](https://mlsanity.com/paper/2602.12181/) (2026)
- [widetildeO(T-1) Convergence to (Coarse) Correlated Equilibria in Full-Information General-Sum Markov Games](https://mlsanity.com/paper/2403.07890/) (2024)

## Abstract (arXiv)

This paper studies policy optimization algorithms for multi-agent reinforcement learning. We begin by proposing an algorithm framework for two-player zero-sum Markov Games in the full-information setting, where each iteration consists of a policy update step at each state using a certain matrix game algorithm, and a value update step with a certain learning rate. This framework unifies many existing and new policy optimization algorithms. We show that the state-wise average policy of this algorithm converges to an approximate Nash equilibrium (NE) of the game, as long as the matrix game algorithms achieve low weighted regret at each state, with respect to weights determined by the speed of the value updates. Next, we show that this framework instantiated with the Optimistic Follow-The-Regularized-Leader (OFTRL) algorithm at each state (and smooth value updates) can find an mathcalwidetildeO(T-5/6) approximate NE in T iterations, and a similar algorithm with slightly modified value update rule achieves a faster mathcalwidetildeO(T-1) convergence rate. These improve over the current best mathcalwidetildeO(T-1/2) rate of symmetric policy optimization type algorithms. We also extend this algorithm to multi-player general-sum Markov Games and show an mathcalwidetildeO(T-3/4) convergence rate to Coarse Correlated Equilibria (CCE). Finally, we provide a numerical example to verify our theory and investigate the importance of smooth value updates, and find that using "eager" value updates instead (equivalent to the independent natural policy gradient algorithm) may significantly slow down the convergence, even on a simple game with H=2 layers.