On Discounted Infinite-Time Mean Field Games

TL;DR

This paper develops a theoretical framework for infinite-time discounted mean field games, using stochastic maximum principles, FBSDEs, and elliptic PDEs to characterize Nash equilibria and prove uniqueness and existence results.

Contribution

It introduces a novel approach combining FBSDEs and elliptic PDEs to analyze infinite-time mean field games with discounting, establishing existence and uniqueness of equilibria.

Findings

Constructed Nash equilibrium via stochastic maximum principle.

Proved weak uniqueness for infinite-time FBSDEs.

Linked solutions of FBSDEs to viscosity solutions of elliptic PDEs.

Abstract

In this paper, we study the infinite-time mean field games with discounting, establishing an equilibrium where individual optimal strategies collectively regenerate the mean-field distribution. To solve this problem, we partition all agents into a representative player and the social equilibrium. When the optimal strategy of the representative player has the same feedback form as the strategy in the social equilibrium, we say that the system achieves a Nash equilibrium. We construct a Nash equilibrium using the stochastic maximum principle and infinite-time forward-backward stochastic differential equations (FBSDEs). By employing elliptic master equations, a class of distribution-dependent elliptic partial differential equations (PDEs), we provide a representation for the Nash equilibrium strategies. We prove the Yamada-Watanabe type theorem and show weak uniqueness for infinite-time FBSDEs. Furthermore, we prove that the solutions to a system of infinite-time FBSDEs can be employed to construct viscosity solutions for a class of distribution-dependent elliptic PDEs.