A probabilistic weak formulation of mean field games and applications

TL;DR

This paper introduces a flexible weak formulation for mean field games that accommodates complex interactions, discontinuities, and history dependence, providing theoretical results and applications to economic and flocking models.

Contribution

It develops a general weak formulation for mean field games, proving existence and uniqueness, and constructs strategies for approximate Nash equilibria in finite-player games.

Findings

Existence and uniqueness of solutions are established.

A new class of multi-agent price impact models is analyzed.

Flocking models with proven equilibrium existence.

Abstract

Mean field games are studied by means of the weak formulation of stochastic optimal control. This approach allows the mean field interactions to enter through both state and control processes and take a form which is general enough to include rank and nearest-neighbor effects. Moreover, the data may depend discontinuously on the state variable, and more generally its entire history. Existence and uniqueness results are proven, along with a procedure for identifying and constructing distributed strategies which provide approximate Nash equlibria for finite-player games. Our results are applied to a new class of multi-agent price impact models and a class of flocking models for which we prove existence of equilibria.