On Discounted Infinite-Time Mean Field Games
Yongsheng Song, Zeyu Yang
TL;DR
This work studies discounted infinite-time mean field games by decoupling the system into a representative agent and a social equilibrium, and establishing Nash equilibria via the stochastic maximum principle, infinite-time FBSDEs, and an elliptic master equation. It provides solvability and regularity results for the resulting McKean–Vlasov FBSDEs, and shows how the master equation yields a feedback representation of the equilibrium through $\alpha^*(x,\mu)=\hat{\alpha}(x,\partial_x U(x,\mu))$. The paper further connects the FBSDE framework to distribution-dependent elliptic PDEs by proving that the value $\mathcal{V}(x,\mu)=Y_0^{x,\xi}$ is a viscosity solution and establishing a Yamada–Watanabe type weak uniqueness result for infinite-time FBSDEs, along with regularity results. As a concrete achievement, it presents a solvable linear-quadratic example of the elliptic master equation, illustrating the practical applicability of the proposed representation and providing a solid theoretical foundation for discounted infinite-horizon mean field games.
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 shares the same feedback form with the strategy of the social equilibrium, we say 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 the elliptic master equations, a class of distribution-dependent elliptic PDEs , we provide a representation for the Nash equilibrium. We prove the Yamada-Watanabe theorem and show the weak uniqueness for infinite-time FBSDEs. And 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.