# Mean Field Control and Mean Field Game Models with Several Populations

arXiv: https://arxiv.org/abs/1810.00783 · HTML: https://mlsanity.com/paper/1810.00783/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Oct 2018.

## Built on by (most cited first, 6 of 6)

- [Scaling up Mean Field Games with Online Mirror Descent](https://mlsanity.com/paper/2103.00623/) (2021)
- [On the Approximation of Cooperative Heterogeneous Multi-Agent Reinforcement Learning (MARL) using Mean Field Control (MFC)](https://mlsanity.com/paper/2109.04024/) (2021)
- [Oracle-free Reinforcement Learning in Mean-Field Games along a Single Sample Path](https://mlsanity.com/paper/2208.11639/) (2022)
- [Exploiting Approximate Symmetry for Efficient Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2408.15173/) (2024)
- [Reinforcement Learning for Finite Space Mean-Field Type Games](https://mlsanity.com/paper/2409.18152/) (2024)
- [Beyond Nominal Equilibria: Risk-Averse Multi-Population Mean-Field Games](https://mlsanity.com/paper/2610.09244/) (2026)

## Abstract (arXiv)

In this paper, we investigate the interaction of two populations with a large number of indistinguishable agents. The problem consists in two levels: the interaction between agents of a same population, and the interaction between the two populations. In the spirit of mean field type control (MFC) problems and mean field games (MFG), each population is approximated by a continuum of infinitesimal agents. We define four different problems in a general context and interpret them in the framework of MFC or MFG. By calculus of variations, we derive formally in each case the adjoint equations for the necessary conditions of optimality. Importantly, we find that in the case of a competition between two coalitions, one needs to rely on a system of Master equations in order to describe the equilibrium. Examples are provided, in particular linear-quadratic models for which we obtain systems of ODEs that can be related to Riccati equations.