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Lipschitz solutions to mean field games with a major player and applications

Charles Meynard

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.

Lipschitz solutions to mean field games with a major player and applications

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.
Paper Structure (20 sections, 18 theorems, 291 equations)

This paper contains 20 sections, 18 theorems, 291 equations.

Table of Contents

  1. Introduction
  2. General introduction
  3. Setting of the study
  4. Main contributions
  5. Organization of the paper
  6. Notation
  7. Preliminary results
  8. Lipschitz solutions
  9. Notion of solution
  10. Nash equilibrium
  11. Formulation of the equilibrium
  12. Link with Lipschitz solutions
  13. The case of additive common noise
  14. Global wellposedness of Lipschitz solutions to the master system
  15. General study under Hilbertian monotonicity
  16. ...and 5 more sections

Key Result

Lemma 2.4

Let $T_c\leq T$ and $(U,\varphi):[0,T_c)\times \mathbb{R}^d\times \mathbb{R}^{d^0}\times\mathcal{P}_2(\mathbb{R}^d)\to \mathbb{R}^d\times \mathbb{R}$ be a couple of Lipschitz functions uniformly on $[0,t]$ for any $t<T_c$. Let $\tau\in (0,T_c)$, then $(U,\varphi)$ is a Lipschitz solution to MFGMP$(T is a Lipschitz solution to MFGMP$(T-\tau,\varphi(\tau,\cdot),U(\tau,\cdot),H_z,L,F,G)$ on $[0,T_c-\

Theorems & Definitions (64)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.6
  • ...and 54 more