A Probabilistic Approach to Discounted Infinite Horizon and Invariant Mean Field Games

TL;DR

This paper extends the probabilistic weak formulation of discounted infinite horizon mean field games, proving existence, uniqueness, and convergence of solutions, and also addresses invariant mean field games under relaxed conditions.

Contribution

It introduces a probabilistic framework for infinite horizon mean field games, establishes solution existence and uniqueness, and quantifies finite to infinite horizon approximation convergence.

Findings

Proved existence and uniqueness of solutions for infinite horizon games.

Quantified convergence rates of finite horizon solutions to the infinite horizon case.

Extended the framework to solve invariant mean field games.

Abstract

This paper considers discounted infinite horizon mean field games by extending the probabilistic weak formulation of the game as introduced by Carmona and Lacker (2015). Under similar assumptions as in the finite horizon game, we prove existence and uniqueness of solutions for the extended infinite horizon game. The key idea is to construct local versions of the previously considered stable topologies. Further, we analyze how sequences of finite horizon games approximate the infinite horizon one. Under a weakened Lasry-Lions monotonicity condition, we can quantify the convergence rate of solutions for the finite horizon games to the one for the infinite horizon game using a novel stability result for mean field games. Lastly, applying our results allows to solve the invariant mean field game as well.