A unifying framework for submodular mean field games

TL;DR

This paper introduces a comprehensive framework for submodular mean field games, establishing conditions for existence and approximation of equilibria across diverse models, including discrete, diffusive, and timing-based scenarios.

Contribution

It provides a unifying theoretical framework with verifiable conditions for analyzing various types of mean field games, expanding applicability to models with discontinuities and common noise.

Findings

Established existence of strong mean field equilibria under new conditions.

Unified approach applicable to discrete, diffusive, and timing-based models.

Utilized Tarski's fixed point theorem for analysis.

Abstract

We provide an abstract framework for submodular mean field games and identify verifiable sufficient conditions that allow to prove 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 sub-probability measures.