# Finite mean field games: fictitious play and convergence to a first order continuous mean field game

arXiv: https://arxiv.org/abs/1805.05940 · HTML: https://mlsanity.com/paper/1805.05940/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 May 2018.

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

- [Scaling up Mean Field Games with Online Mirror Descent](https://mlsanity.com/paper/2103.00623/) (2021)
- [Reinforcement Learning for Mean Field Games, with Applications to Economics](https://mlsanity.com/paper/2106.13755/) (2021)
- [Learning Sparse Graphon Mean Field Games](https://mlsanity.com/paper/2209.03880/) (2022)
- [Population-aware Online Mirror Descent for Mean-Field Games by Deep Reinforcement Learning](https://mlsanity.com/paper/2403.03552/) (2024)
- [Primal and Dual Analysis of Entropic Fictitious Play for Finite-sum Problems](https://mlsanity.com/paper/2303.02957/) (2023)
- [Population-aware Online Mirror Descent for Mean-Field Games with Common Noise by Deep Reinforcement Learning](https://mlsanity.com/paper/2509.03030/) (2025)
- [Exploration noise for learning linear-quadratic mean field games](https://mlsanity.com/paper/2107.00839/) (2021)
- [Last Iterate Convergence in Monotone Mean Field Games](https://mlsanity.com/paper/2410.05127/) (2024)
- [Stationary Robust Mean-Field Games under Model Mismatches](https://mlsanity.com/paper/2606.22579/) (2026)

## Abstract (arXiv)

In this article we consider finite Mean Field Games (MFGs), i.e. with finite time and finite states. We adopt the framework introduced in Gomes Mohr and Souza in 2010, and study two seemly unexplored subjects. In the first one, we analyze the convergence of the fictitious play learning procedure, inspired by the results in continuous MFGs. In the second one, we consider the relation of some finite MFGs and continuous first order MFGs. Namely, given a continuous first order MFG problem and a sequence of refined space/time grids, we construct a sequence finite MFGs whose solutions admit limits points and every such limit point solves the continuous first order MFG problem.