# MF-OMO: An Optimization Formulation of Mean-Field Games

arXiv: https://arxiv.org/abs/2206.09608 · HTML: https://mlsanity.com/paper/2206.09608/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Jun 2022.

## Built on by (most cited first, 9 of 9)

- [A Survey on Large-Population Systems and Scalable Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2209.03859/) (2022)
- [When is Mean-Field Reinforcement Learning Tractable and Relevant?](https://mlsanity.com/paper/2402.05757/) (2024)
- [MF-OML: Online Mean-Field Reinforcement Learning with Occupation Measures for Large Population Games](https://mlsanity.com/paper/2405.00282/) (2024)
- [Learning Regularized Graphon Mean-Field Games with Unknown Graphons](https://mlsanity.com/paper/2310.17531/) (2023)
- [Learning in Herding Mean Field Games: Single-Loop Algorithm with Finite-Time Convergence Analysis](https://mlsanity.com/paper/2408.04780/) (2024)
- [Scalable Task-Driven Robotic Swarm Control via Collision Avoidance and Learning Mean-Field Control](https://mlsanity.com/paper/2209.07420/) (2022)
- [MESOB: Balancing Equilibria & Social Optimality](https://mlsanity.com/paper/2307.07911/) (2023)
- [Last Iterate Convergence in Monotone Mean Field Games](https://mlsanity.com/paper/2410.05127/) (2024)
- [Bounded Rationality Equilibrium Learning in Mean Field Games](https://mlsanity.com/paper/2411.07099/) (2024)

## Abstract (arXiv)

This paper proposes a new mathematical paradigm to analyze discrete-time mean-field games. It is shown that finding Nash equilibrium solutions for a general class of discrete-time mean-field games is equivalent to solving an optimization problem with bounded variables and simple convex constraints, called MF-OMO. This equivalence framework enables finding multiple (and possibly all) Nash equilibrium solutions of mean-field games by standard algorithms. For instance, projected gradient descent is shown to be capable of retrieving all possible Nash equilibrium solutions when there are finitely many of them, with proper initializations. Moreover, analyzing mean-field games with linear rewards and mean-field independent dynamics is reduced to solving a finite number of linear programs, hence solvable in finite time. This framework does not rely on the contractive and the monotone assumptions and the uniqueness of the Nash equilibrium.