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.