The Master Equation and the Convergence Problem in Mean Field Games

By Pierre Cardaliaguet, Franccois Delarue, Jean-Michel Lasry, Pierre-Louis Lions. 2015

590 citations recorded by the source corpus.

This book describes the latest advances in the theory of mean field games, which are optimal control problems with a continuum of players, each of them interacting with the whole statistical distribution of a population. While it originated in economics, this theory now has applications in areas as diverse as mathematical finance, crowd phenomena, epidemiology, and cybersecurity. Because mean field games concern the interactions of infinitely many players in an optimal control framework, one expects them to appear as the limit for Nash equilibria of differential games with finitely many players as the number of players tends to infinity. The book rigorously establishes this convergence, which has been an open problem until now. The limit of the system associated with differential games with finitely many players is described by the so-called master equation, a nonlocal transport equation in the space of measures. After defining a suitable notion of differentiability in the space of measures, the authors provide a complete self-contained analysis of the master equation. Their analysis includes the case of common noise problems in which all the players are affected by a common Brownian motion. They then go on to explain how to use the master equation to prove the mean field limit. The book presents two important new results in mean field games that contribute to a unified theoretical framework for this exciting and fast-developing area of mathematics.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4c39761aa3822299f1b66786ff5b933c9545c729

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Citations in the corpus

When Inequality Matters for Macro and Macro Matters for Inequality (2018, unknown): “The fully recursive version is what Cardaliaguet et al. (2015) term the “Master equation for Mean Field Games with common noise” whereas the hybrid-notation system (2) to (7) corresponds to their “MFG system with common noise.”
On non-uniqueness and uniqueness of solutions in finite-horizon Mean Field Games (2017, unknown): “The rigorous justiﬁcation of the PDE (1.1) as limit of the systems of Bellman equations for such games as N → ∞ was proved recently by Cardaliaguet et al. [11] using in a crucial way the convexity and monotonicity conditions leading to uniqueness for (1.1).”
Mean field games master equations with nonseparable Hamiltonians and displacement monotonicity (2021, background): “The arguments are more or less standard, see [22, 26], and we leave the details to interested readers.”
From the master equation to mean field game limit theory: a central limit theorem (2018, methodology): “This is indeed the strategy of [6], where it is shown that”
Wellposedness of Second Order Master Equations for Mean Field Games with Nonsmooth Data (2019, methodology): “The groundbreaking paper CardaliaguetDelarue-Lasry-Lions [16], by using a PDE approach, obtained global wellposedness of the master equation with general Hamiltonian and common noise.”
Viscosity solutions to parabolic master equations and McKean–Vlasov SDEs with closed-loop controls (2018, background): “Initiated independently by Caines, Huang, & Malhame [6] and Lasry & Lions [25], the mean field game and related topics have received very strong attention in the past decade.”
Finite state mean field games with Wright–Fisher common noise (2019, methodology): “We achieve this by analyzing the master equation of this game, which is a second-order partial differential equation whose stochastic characteristics are the stochastic forward-backward system that describes the fixed-point condition of the mean field game; see [8].”
Some remarks on mean field games (2018, methodology): “Even though [4, 20] provide a detailed study of the so-called master equation, several problems remain open.”
Weak quantitative propagation of chaos via differential calculus on the space of measures (2019, methodology): “The important object in our study, similarly to [6], is the PDE written on the space [0, T ]×P2(Rd), which corresponds to the lifted semigroup and comes from the Itô’s formula of functionals of measures established in [4] and [9].”
Superposition and mimicking theorems for conditional McKean–Vlasov equations (2020, background): “Nonlinear analogues of these backward PDEs appear in the guise of the master equation from the theory of mean field games [20, 15] and in mean field control problems [60, 17].”
Global Well‐Posedness of Master Equations for Deterministic Displacement Convex Potential Mean Field Games (2020, methodology): “In particular, as we can see this in [9], it provides an important tool to prove the convergence of Nash equilibria of N–player differential games to the Mean Field Games system, as N → +∞.”