[Paper Review] Lipschitz solutions to mean field games with a major player and applications

Charles Meynard|arXiv (Cornell University)|Mar 16, 2026

Stochastic processes and financial applications0 citations

TL;DR

The paper defines Lipschitz weak solutions for the master equations of mean field games with a major player, proves uniqueness under Lipschitz Coefficients, and establishes long-time existence under a joint monotonicity condition with a sufficiently large common-noise volatility; it also extends extragradient methods to this setting.

ABSTRACT

This paper introduces a notion of weak solution for the coupled system of master equations in mean field games with a major player. It extends the previously introduced notion of Lipschitz solutions in mean field games. By relying on a probabilistic representation of the system of master equations, we prove that there can exist at most one sufficiently smooth solution and that it is consistent with the associated Nash equilibrium. In this approach, coefficients are only required to be Lipschitz, in particular, no differentiability assumption with respect to probability measures is needed. In a second part, we apply this notion of solution to prove the existence and uniqueness of solutions to MFGs with a major player on intervals of arbitrary length. Our argument relies on assuming that the intensity of the Brownian common noise driving the state of the major player is sufficiently large, as well as a joint displacement monotonicity assumption between the coefficients of minor players and those of the major player. Most notably, this joint monotonicity allows us to prove that the threshold of volatility can be taken independently of the horizon of the game considered, without any long time decoupling assumption on the dependence between minor players and the major player. Finally, inspired by recent extragradient methods for mean field games, we present an algorithm that converges exponentially fast to the solution of the major-minor probabilistic system under this monotonicity assumption. Thanks to the generality of our approach, all results presented in this article hold for mean field games of controls with a major player.

Motivation & Objective

• Motivate and formalize a weak solution concept for the master equations in MFGs with a major player.
• Show that Lipschitz decoupling fields yield uniqueness and stability of the Nash equilibrium.
• Prove local and long-time existence results under Lipschitzcoefficient assumptions and a joint monotonicity condition.
• Demonstrate that sufficiently large major-player common noise enables horizon-independent volatility thresholds for existence on any time interval.
• Extend extragradient numerical schemes to solve the major-minor MFG system under monotonicity assumptions.

Proposed method

• Introduce a forward–backward probabilistic system (2.1) that links minor and major players through a decoupling field (U, ϕ).
• Define Lipschitz solutions via a Lipschitz decoupling field that satisfies (2.2) and (2.3) with representation properties.
• Prove Zϕ_t is almost surely bounded by the Lipschitz constant of ∇qϕ, establishing regularity without measure-differentiability assumptions.
• Establish uniqueness of strong solutions by a contraction argument on the FBSDE system under Hypothesis 2.5.
• Develop long-time existence results under a joint displacement monotonicity condition and a sufficiently large σ0, independent of horizon T (Theorem 3.6 and 3.8).
• Describe an extragradient-type iterative scheme that converges geometrically to the solution under the monotonicity assumptions.

Experimental results

Research questions

• RQ1 Can a weak (Lipschitz) solution concept for the master equations of MFGs with a major player be defined and shown to be unique when coefficients are Lipschitz?
• RQ2 Under what monotonicity and noise-regularity conditions can one guarantee existence (locally and long-time) of Lipschitz solutions for major-minor MFGs?
• RQ3 Does a sufficiently large common-noise volatility enable a horizon-independent existence threshold for any time interval?
• RQ4 Can extragradient-type algorithms be adapted to converge to the MFG with a major player under the proposed monotonicity framework?
• RQ5 How does the proposed framework extend to MFGs of controls with a major player?

Key findings

• A Lipschitz decoupling-field-based notion yields uniqueness and stability of solutions for the major-minor MFG system when coefficients are Lipschitz.
• Local-in-time existence of Lipschitz solutions is established without requiring differentiability with respect to probability measures.
• A volatility threshold σ0* for the major-player common noise is shown to give long-time existence on any horizon, under a joint displacement monotonicity assumption.
• A horizon-independent threshold for σ0 ensuring solvability is obtained under additional monotonicity assumptions.
• An extragradient numerical scheme is shown to converge geometrically to the solution under the stated monotonicity conditions.
• The results extend to extended MFGs and MFGs of controls with a major player, including additive common noise variants.