A Unifying Framework for Submodular Mean Field Games

We provide an abstract framework for submodular mean field games and identify verifiable sufficient conditions that allow us to prove the existence and approximation of strong mean field equilibria in models where data may not be continuous with respect to the measure parameter and common noise is allowed. The setting is general enough to encompass qualitatively different problems, such as mean field games for discrete time finite space Markov chains, singularly controlled and reflected diffusions, and mean field games of optimal timing. Our analysis hinges on Tarski's fixed point theorem, along with technical results on lattices of flows of probability and subprobability measures.

Automated summary

This study provides an abstract framework for investigating submodular mean field games. It establishes verifiable sufficient conditions for the existence and approximation of strong mean field equilibria in models with discontinuous data and common noise. The framework is general enough to encompass a variety of problems, such as mean field games for finite space Markov chains in discrete time, singularly controlled and reflected diffusions, and mean field games of optimal timing. The analysis relies on Tarski's fixed point theorem and technical results on lattices of flows of probability and subprobability measures.