# A case study on stochastic games on large graphs in mean field and sparse regimes

arXiv: https://arxiv.org/abs/2005.14102 · HTML: https://mlsanity.com/paper/2005.14102/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 May 2020.

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

- [Learning Graphon Mean Field Games and Approximate Nash Equilibria](https://mlsanity.com/paper/2112.01280/) (2021)
- [A Survey on Large-Population Systems and Scalable Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2209.03859/) (2022)
- [Learning Sparse Graphon Mean Field Games](https://mlsanity.com/paper/2209.03880/) (2022)
- [Neural McKean-Vlasov Processes: Distributional Dependence in Diffusion Processes](https://mlsanity.com/paper/2404.09402/) (2024)
- [Learning Mean Field Control on Sparse Graphs](https://mlsanity.com/paper/2501.17079/) (2025)
- [Learning Distributed Equilibria in Linear-Quadratic Stochastic Differential Games: An α-Potential Approach](https://mlsanity.com/paper/2602.16555/) (2026)
- [Finite-Agent Stochastic Differential Games on Large Graphs: II. Graph-Based Architectures](https://mlsanity.com/paper/2509.12484/) (2025)
- [Directed Chain Generative Adversarial Networks](https://mlsanity.com/paper/2304.13131/) (2023)
- [Stationary Robust Mean-Field Games under Model Mismatches](https://mlsanity.com/paper/2606.22579/) (2026)

## Abstract (arXiv)

We study a class of linear-quadratic stochastic differential games in which each player interacts directly only with its nearest neighbors in a given graph. We find a semi-explicit Markovian equilibrium for any transitive graph, in terms of the empirical eigenvalue distribution of the graph's normalized Laplacian matrix. This facilitates large-population asymptotics for various graph sequences, with several sparse and dense examples discussed in detail. In particular, the mean field game is the correct limit only in the dense graph case, i.e., when the degrees diverge in a suitable sense. Even though equilibrium strategies are nonlocal, depending on the behavior of all players, we use a correlation decay estimate to prove a propagation of chaos result in both the dense and sparse regimes, with the sparse case owing to the large distances between typical vertices. Without assuming the graphs are transitive, we show also that the mean field game solution can be used to construct decentralized approximate equilibria on any sufficiently dense graph sequence.