Extended Mean Field Control Games with Moment Interactions: General Framework and Linear-Quadratic Model
Abstract
Mean field control games (MFCGs) provide a framework for studying non-cooperative games involving groups of cooperative players in the limit as both the number of players and the number of groups go to infinity.This class of games includes mean field games (MFGs) and mean field control (MFC) problems as special cases, corresponding to purely non-cooperative and purely cooperative settings, respectively. We incorporate interactions through the mean of actions, hence the terminology of extended MFCGs. We first show that the equilibrium control can be described using a coupled system of partial differential equations consisting of a Hamilton-Jacobi-Bellman equation and a Fokker-Planck equation. After introducing the general framework, we focus on a linear-quadratic (LQ) structure. We show that the equilibrium can be reduced to a system of ODEs, generalizing those obtained for MFGs and MFC problems. We then provide two numerical examples illustrating specific features of MFCGs.
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BibTeX
@misc{cao2026extended,
title = {{Extended Mean Field Control Games with Moment Interactions: General Framework and Linear-Quadratic Model}},
author = {Zhongyuan Cao and Mathieu Laurière and Andrew Shi and Jiefei Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12687},
eprint = {2609.12687},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12687},
}
RIS
TY - UNPB
TI - Extended Mean Field Control Games with Moment Interactions: General Framework and Linear-Quadratic Model
AU - Zhongyuan Cao
AU - Mathieu Laurière
AU - Andrew Shi
AU - Jiefei Yang
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12687
UR - https://arxiv.org/abs/2609.12687
AB - Mean field control games (MFCGs) provide a framework for studying non-cooperative games involving groups of cooperative players in the limit as both the number of players and the number of groups go to infinity.This class of games includes mean field games (MFGs) and mean field control (MFC) problems as special cases, corresponding to purely non-cooperative and purely cooperative settings, respectively. We incorporate interactions through the mean of actions, hence the terminology of extended MFCGs. We first show that the equilibrium control can be described using a coupled system of partial differential equations consisting of a Hamilton-Jacobi-Bellman equation and a Fokker-Planck equation. After introducing the general framework, we focus on a linear-quadratic (LQ) structure. We show that the equilibrium can be reduced to a system of ODEs, generalizing those obtained for MFGs and MFC problems. We then provide two numerical examples illustrating specific features of MFCGs.
ER -