# A mean-field analysis of two-player zero-sum games

arXiv: https://arxiv.org/abs/2002.06277 · HTML: https://mlsanity.com/paper/2002.06277/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Feb 2020.

## Built on by (most cited first, 28 of 28)

- [Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approach](https://mlsanity.com/paper/2506.03979/) (2025)
- [An Exponentially Converging Particle Method for the Mixed Nash Equilibrium of Continuous Games](https://mlsanity.com/paper/2211.01280/) (2022)
- [Two-Scale Gradient Descent Ascent Dynamics Finds Mixed Nash Equilibria of Continuous Games: A Mean-Field Perspective](https://mlsanity.com/paper/2212.08791/) (2022)
- [Symmetric Mean-field Langevin Dynamics for Distributional Minimax Problems](https://mlsanity.com/paper/2312.01127/) (2023)
- [A Policy Gradient Framework for Stochastic Optimal Control Problems with Global Convergence Guarantee](https://mlsanity.com/paper/2302.05816/) (2023)
- [Dual Training of Energy-Based Models with Overparametrized Shallow Neural Networks](https://mlsanity.com/paper/2107.05134/) (2021)
- [Nonconvex-Nonconcave Min-Max Optimization with a Small Maximization Domain](https://mlsanity.com/paper/2110.03950/) (2021)
- [Worst-case generation via minimax optimization in Wasserstein space](https://mlsanity.com/paper/2512.08176/) (2025)
- [Strategic Distribution Shift of Interacting Agents via Coupled Gradient Flows](https://mlsanity.com/paper/2307.01166/) (2023)
- [Expected Variational Inequalities](https://mlsanity.com/paper/2502.18605/) (2025)
- [On adversarial robustness and the use of Wasserstein ascent-descent dynamics to enforce it](https://mlsanity.com/paper/2301.03662/) (2023)
- [Analysis of Kernel Mirror Prox for Measure Optimization](https://mlsanity.com/paper/2403.00147/) (2024)
- [Inclusive KL Gradient Flows: Otto-Wasserstein, Fisher-Rao-Gaussian, and Local-Estimator Dynamics](https://mlsanity.com/paper/2411.00214/) (2024)
- [A Fisher-Rao gradient flow for entropic mean-field min-max games](https://mlsanity.com/paper/2405.15834/) (2024)
- [Generative Adversarial Network: Some Analytical Perspectives](https://mlsanity.com/paper/2104.12210/) (2021)
- [Convergence of GANs Training: A Game and Stochastic Control Methodology](https://mlsanity.com/paper/2112.00222/) (2021)
- [Simultaneous Transport Evolution for Minimax Equilibria on Measures](https://mlsanity.com/paper/2202.06460/) (2022)
- [PAPAL: A Provable PArticle-based Primal-Dual ALgorithm for Mixed Nash Equilibrium](https://mlsanity.com/paper/2303.00970/) (2023)
- [Convergence of two-timescale gradient descent ascent dynamics: finite-dimensional and mean-field perspectives](https://mlsanity.com/paper/2501.17122/) (2025)
- [Non-convex entropic mean-field optimization via Best Response flow](https://mlsanity.com/paper/2505.22760/) (2025)
- [WGAN with an Infinitely Wide Generator Has No Spurious Stationary Points](https://mlsanity.com/paper/2102.07541/) (2021)
- [Proximal Mean Field Learning in Shallow Neural Networks](https://mlsanity.com/paper/2210.13879/) (2022)
- [Large deviations for interacting particle dynamics for finding mixed equilibria in zero-sum games](https://mlsanity.com/paper/2206.15177/) (2022)
- [Towards Optimal Randomized Strategies in Adversarial Example Game](https://mlsanity.com/paper/2306.16738/) (2023)
- [Riemannian Bilevel Optimization](https://mlsanity.com/paper/2405.15816/) (2024)
- [Local exponential stability of mean-field Langevin descent-ascent and associated particle system](https://mlsanity.com/paper/2602.01564/) (2026)
- [On propagation of chaos for the Fisher-Rao gradient flow in entropic mean-field optimization](https://mlsanity.com/paper/2602.15094/) (2026)
- [The Off-Support Barrier: Why Semantic Safety Constraints Are Not Learning-Problem Invariants, and What Follows for Prior Design, Containment, and Verification](https://mlsanity.com/paper/2608.11243/) (2026)

## Abstract (arXiv)

Finding Nash equilibria in two-player zero-sum continuous games is a central problem in machine learning, e.g. for training both GANs and robust models. The existence of pure Nash equilibria requires strong conditions which are not typically met in practice. Mixed Nash equilibria exist in greater generality and may be found using mirror descent. Yet this approach does not scale to high dimensions. To address this limitation, we parametrize mixed strategies as mixtures of particles, whose positions and weights are updated using gradient descent-ascent. We study this dynamics as an interacting gradient flow over measure spaces endowed with the Wasserstein-Fisher-Rao metric. We establish global convergence to an approximate equilibrium for the related Langevin gradient-ascent dynamic. We prove a law of large numbers that relates particle dynamics to mean-field dynamics. Our method identifies mixed equilibria in high dimensions and is demonstrably effective for training mixtures of GANs.