# Policy Optimization for Linear-Quadratic Zero-Sum Mean-Field Type Games

arXiv: https://arxiv.org/abs/2009.02146 · HTML: https://mlsanity.com/paper/2009.02146/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Sep 2020.

## Built on by (most cited first, 4 of 4)

- [Towards a Theoretical Foundation of Policy Optimization for Learning Control Policies](https://mlsanity.com/paper/2210.04810/) (2022)
- [Independent RL for Cooperative-Competitive Agents: A Mean-Field Perspective](https://mlsanity.com/paper/2403.11345/) (2024)
- [Reinforcement Learning for Finite Space Mean-Field Type Games](https://mlsanity.com/paper/2409.18152/) (2024)
- [Robust Q-learning for mean-field control under Wasserstein uncertainty in common noise](https://mlsanity.com/paper/2606.20356/) (2026)

## Abstract (arXiv)

In this paper, zero-sum mean-field type games (ZSMFTG) with linear dynamics and quadratic utility are studied under infinite-horizon discounted utility function. ZSMFTG are a class of games in which two decision makers whose utilities sum to zero, compete to influence a large population of agents. In particular, the case in which the transition and utility functions depend on the state, the action of the controllers, and the mean of the state and the actions, is investigated. The game is analyzed and explicit expressions for the Nash equilibrium strategies are derived. Moreover, two policy optimization methods that rely on policy gradient are proposed for both model-based and sample-based frameworks. In the first case, the gradients are computed exactly using the model whereas they are estimated using Monte-Carlo simulations in the second case. Numerical experiments show the convergence of the two players' controls as well as the utility function when the two algorithms are used in different scenarios.