# Analysis of Multiscale Reinforcement Q-Learning Algorithms for Mean Field Control Games

arXiv: https://arxiv.org/abs/2405.17017 · HTML: https://mlsanity.com/paper/2405.17017/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in RL for games and combinatorial optimization, one of 15 directions in Deep reinforcement learning methods.
- Builds on 13 of its 38 references that are in the map.
- First posted on arXiv 27 May 2024.

## Placed in

- Deep reinforcement learning methods / [RL for games and combinatorial optimization](https://mlsanity.com/deeprl/rl-for-games-and-combinatorial-optimization/)

## Builds on (most cited first, 13 of 13)

- [Approximately Solving Mean Field Games via Entropy-Regularized Deep Reinforcement Learning](https://mlsanity.com/paper/2102.01585/) (2021)
- [Learning in Mean Field Games: A Survey](https://mlsanity.com/paper/2205.12944/) (2022)
- [Model-Free Mean-Field Reinforcement Learning: Mean-Field MDP and Mean-Field Q-Learning](https://mlsanity.com/paper/1910.12802/) (2019)
- [Mean-Field Controls with Q-learning for Cooperative MARL: Convergence and Complexity Analysis](https://mlsanity.com/paper/2002.04131/) (2020)
- [Unified Reinforcement Q-Learning for Mean Field Game and Control Problems](https://mlsanity.com/paper/2006.13912/) (2020)
- [On the Convergence of Model Free Learning in Mean Field Games](https://mlsanity.com/paper/1907.02633/) (2019)
- [Linear-Quadratic Mean-Field Reinforcement Learning: Convergence of Policy Gradient Methods](https://mlsanity.com/paper/1910.04295/) (2019)
- [Mean-field Markov decision processes with common noise and open-loop controls](https://mlsanity.com/paper/1912.07883/) (2019)
- [Actor-Critic learning for mean-field control in continuous time](https://mlsanity.com/paper/2303.06993/) (2023)
- [Continuous-time q-learning for mean-field control problems](https://mlsanity.com/paper/2306.16208/) (2023)
- [Convergence of Multi-Scale Reinforcement Q-Learning Algorithms for Mean Field Game and Control Problems](https://mlsanity.com/paper/2312.06659/) (2023)
- [Reinforcement Learning Algorithm for Mixed Mean Field Control Games](https://mlsanity.com/paper/2205.02330/) (2022)
- [Reinforcement Learning for Intra-and-Inter-Bank Borrowing and Lending Mean Field Control Game](https://mlsanity.com/paper/2207.03449/) (2022)

## Built on by (most cited first, 3 of 3)

- [Convergence of Actor-Critic Learning for Mean Field Games and Mean Field Control in Continuous Spaces](https://mlsanity.com/paper/2511.06812/) (2025)
- [Last Iterate Convergence in Monotone Mean Field Games](https://mlsanity.com/paper/2410.05127/) (2024)
- [Robust Q-learning for mean-field control under Wasserstein uncertainty in common noise](https://mlsanity.com/paper/2606.20356/) (2026)

## Abstract (arXiv)

Mean Field Control Games (MFCG), introduced in [Angiuli et al., 2022a], represent competitive games between a large number of large collaborative groups of agents in the infinite limit of number and size of groups. In this paper, we prove the convergence of a three-timescale Reinforcement Q-Learning (RL) algorithm to solve MFCG in a model-free approach from the point of view of representative agents. Our analysis uses a Q-table for finite state and action spaces updated at each discrete time-step over an infinite horizon. In [Angiuli et al., 2023], we proved convergence of two-timescale algorithms for MFG and MFC separately highlighting the need to follow multiple population distributions in the MFC case. Here, we integrate this feature for MFCG as well as three rates of update decreasing to zero in the proper ratios. Our technique of proof uses a generalization to three timescales of the two-timescale analysis in [Borkar, 1997]. We give a simple example satisfying the various hypothesis made in the proof of convergence and illustrating the performance of the algorithm.