Uniqueness of solutions to MFG systems with large discount

TL;DR

This work addresses the question of uniqueness for a class of Mean Field Game systems under a large discount factor $\lambda$, identifying an asymptotic uniqueness regime beyond classical monotonicity assumptions. The authors develop uniform a priori estimates for $(u_\lambda,m_\lambda)$, prove that $\|\lambda Du_\lambda-DF(\cdot,m_\lambda)\|_\infty$ decays like $O(1/\sqrt{\lambda})$, and use a nonlocal-in-time Grönwall framework together with Duhamel-type representations to deduce uniqueness for sufficiently large $\lambda$. They establish the convergence of $\lambda Du$ to $DF$ and show that a unique solution persists in both infinite and finite horizon settings under explicit thresholds, with robustness to viscosity. The results illuminate a non-monotone, asymptotic pathway to uniqueness and lay groundwork for studying long-time behavior and selection phenomena in MFGs with discounting.

Abstract

We prove that solutions to a class of Mean Field Game systems with discount are unique provided that the discount factor is large enough, and the Lagrangian term is (proportionally) small enough. This identifies an asymptotic uniqueness regime that falls outside the usual ones involving monotonicity.

Theorems & Definitions (20)

• Theorem
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3: Representation formula for $u$
• proof
• Proposition 2.4: Representation formula for $\lambda Du$
• proof
• Lemma 2.5