The Master Equation and the Convergence Problem in Mean Field Games

AMS-201

Pierre Cardaliaguet, François Delarue, Jean-Michel Lasry , Pierre-Louis Lions

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English

Princeton University Press
22 October 2019

Differential calculus & equations; Game theory

Series: Annals of Mathematics Studies

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. This 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.

This groundbreaking 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.

By: Pierre Cardaliaguet, François Delarue, Jean-Michel Lasry, Pierre-Louis Lions
Imprint: Princeton University Press
Country of Publication: United States
Volume: 2
Dimensions: Height: 235mm, Width: 156mm,
ISBN: 9780691190716
ISBN 10: 0691190712
Series: Annals of Mathematics Studies
Pages: 224
Publication Date: 22 October 2019
Audience: College/higher education , Professional and scholarly , Primary , Undergraduate
Format: Paperback
Publisher's Status: Active

Pierre Cardaliaguet is professor of mathematics at Paris Dauphine University. François Delarue is professor of mathematics at the University of Nice Sophia Antipolis. Jean-Michel Lasry is associate researcher of mathematics at Paris Dauphine University. Pierre-Louis Lions is professor of partial differential equations and their applications at the Collège de France.

Reviews for The Master Equation and the Convergence Problem in Mean Field Games: (AMS-201)

"""This book . . . . is a major contribution to the state of the art in MFGs which is a must read for researchers in the field. . . . . The authors use the book format (and not a more compact paper format) to explain all their steps carefully. Because of its structured approach, it could be used as a textbook for an advanced course on the subject.""---Adhemar Bultheel, European Mathematical Society"