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Mean field games master equations: from discrete to continuous state space

Charles Bertucci, Alekos Cecchin

TL;DR

This work establishes a rigorous link between discrete finite-state mean field games and their continuous-state master equation by analyzing space-discretizations of diffusive dynamics with time kept continuous. It develops two complementary routes: a rate-based convergence via the MFG system and classical master equations when no common noise, and a monotone-solution/compactness framework for the common-noise setting, including a mollification-based alternative. When the limit master equation is smooth, the authors obtain explicit convergence rates for the master value and optimal trajectories; in the absence of smoothness, they rely on monotone-solution stability to guarantee convergence, with a rate available in certain regimes. The results significantly connect numerical discretizations to the analytical master equation, offering practical convergence guarantees and insights into how common noise affects the convergence mechanism and rate.

Abstract

This paper studies the convergence of mean field games with finite state space to mean field games with a continuous state space. We examine a space discretization of a diffusive dynamics, which is reminiscent of the Markov chain approximation method in stochasctic control, but also of finite difference numerical schemes; time remains continuous in the discretization, and the time horizon is arbitrarily long. We are mainly interested in the convergence of the solution of the associated master equations as the number of states tends to infinity. We present two approaches, to treat the case without or with common noise, both under monotonicity assumptions. The first one uses the system of characteristics of the master equation, which is the MFG system, to establish a convergence rate for the master equations without common noise and the associated optimal trajectories, both in case there is a smooth solution to the limit master equation and in case there is not. The second approach relies on the notion of monotone solutions introduced by Bertucci. In the presence of common noise, we show convergence of the master equations, with a convergence rate if the limit master equation is smooth, otherwise by compactness arguments.

Mean field games master equations: from discrete to continuous state space

TL;DR

This work establishes a rigorous link between discrete finite-state mean field games and their continuous-state master equation by analyzing space-discretizations of diffusive dynamics with time kept continuous. It develops two complementary routes: a rate-based convergence via the MFG system and classical master equations when no common noise, and a monotone-solution/compactness framework for the common-noise setting, including a mollification-based alternative. When the limit master equation is smooth, the authors obtain explicit convergence rates for the master value and optimal trajectories; in the absence of smoothness, they rely on monotone-solution stability to guarantee convergence, with a rate available in certain regimes. The results significantly connect numerical discretizations to the analytical master equation, offering practical convergence guarantees and insights into how common noise affects the convergence mechanism and rate.

Abstract

This paper studies the convergence of mean field games with finite state space to mean field games with a continuous state space. We examine a space discretization of a diffusive dynamics, which is reminiscent of the Markov chain approximation method in stochasctic control, but also of finite difference numerical schemes; time remains continuous in the discretization, and the time horizon is arbitrarily long. We are mainly interested in the convergence of the solution of the associated master equations as the number of states tends to infinity. We present two approaches, to treat the case without or with common noise, both under monotonicity assumptions. The first one uses the system of characteristics of the master equation, which is the MFG system, to establish a convergence rate for the master equations without common noise and the associated optimal trajectories, both in case there is a smooth solution to the limit master equation and in case there is not. The second approach relies on the notion of monotone solutions introduced by Bertucci. In the presence of common noise, we show convergence of the master equations, with a convergence rate if the limit master equation is smooth, otherwise by compactness arguments.
Paper Structure (25 sections, 16 theorems, 180 equations)

This paper contains 25 sections, 16 theorems, 180 equations.

Table of Contents

  1. Introduction
  2. General introduction
  3. Bibliographical comments
  4. Main results of the paper
  5. Organization of the paper
  6. Notation
  7. The continuous and discrete models
  8. The limit MFG model
  9. Assumptions
  10. The discrete MFG model
  11. Heuristic derivation of the limit master equation
  12. Convergence results in the absence of a common noise
  13. Classical solutions
  14. Another discretization
  15. Convergence through the MFG system
  16. ...and 10 more sections

Key Result

Lemma 3.1

Let $(u^n,\mu^n)$ be a solution of mfgsystem:n for given initial time $t_0 \in [0,T]$. This solution satisfies and then there exists $M > 0$ such that for any $s \in [0,T], \widetilde{\mu^n} \in \mathcal{P}(S^n), n \geq 1$ and $x \in S^n$

Theorems & Definitions (38)

  • Remark 2.1
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4: Convergence of trajectories
  • proof
  • Theorem 3.5
  • ...and 28 more