Unconditional well-posedness of the master equation for monotone mean field games of controls

TL;DR

This work proves unconditional well-posedness for the master equation of monotone mean field games of controls (MFGC) by a bottom-up compactness approach that avoids the generalized method of characteristics. It establishes uniform, high-order derivative bounds for the $N$-player Nash systems and shows these bounds yield tight compactness, allowing convergence to a classical solution of the master equation under displacement semi-monotonicity or Lasry–Lions monotonicity. The analysis handles both idiosyncratic and common noise and relies on a careful fixed-point structure via the map $\Phi$ and its finite-dimensional approximations $\bm{a}^N$. In the displacement semi-monotone setting, one obtains a global-in-time solution; in the LL-monotone setting, global well-posedness follows from propagation of LL-monotonicity. Overall, the paper advances the understanding of MFGC master equations by showing regularity and solvability emerge from monotonicity and data regularity rather than extra structure on fixed-point mappings.

Abstract

We establish the first unconditional well-posedness result for the master equation associated with a general class of mean field games of controls. Our analysis covers games with displacement monotone or Lasry--Lions monotone data, as well as those with a small time horizon. By unconditional, we mean that all assumptions are imposed solely at the level of the Lagrangian and the terminal cost. In particular, we do not require any a priori regularity or structural assumptions on the additional fixed-point mappings arising from the control interactions; instead we show that these fixed-point mappings are well-behaved as a consequence of the regularity and the monotonicity of the data. Our approach is bottom-up in nature, unlike most previous results which rely on a generalized method of characteristics. In particular, we build a classical solution of the master equation by showing that the solutions of the corresponding $N$-player Nash systems are compact, in an appropriate sense, and that their subsequential limit points must be solutions to the master equation. Compactness is obtained via uniform-in-$N$ decay estimates for derivatives of the $N$-player value functions. The underlying games are driven by non-degenerate idiosyncratic Brownian noise, and our results allow for the presence of common noise with constant intensity.

Theorems & Definitions (72)

• Theorem 1.1: Informal summary of main results
• Remark 2.5
• Remark 2.7
• Lemma 2.8
• Theorem 2.9: Classical solutions under displacement semi-monotonicity
• Theorem 2.10: Classical solutions under Lasry--Lions monotonicity
• Remark 2.11
• Proposition 3.1
• Proposition 3.2
• Lemma 3.3