Mean Field Games

When every player of a large crowd solves a control problem against all the others, two equations appear: a backward Hamilton–Jacobi–Bellman equation for one player’s value and a forward Fokker–Planck equation for the crowd. A mean field equilibrium is the fixed point of the pair, and everything else in the theory is a statement about that coupling.

This book follows the fixed point from a game of N players to the master equation. It proves when the equilibrium exists, when it is unique and what it costs against a planner, reads it again as a forward–backward stochastic equation, writes one equation for every initial law, and proves that the N-player game converges to it. It then computes equilibria, with four solvers and the conditions under which each converges, and carries the frame into four markets: systemic risk, traders executing at once, producers of an exhaustible resource and the wealth distribution.

One running model, the linear–quadratic crowd, is solved by hand in every chapter. Every main result is proved, and every algorithm is set out step by step.

Nikolaos Vardakis is a probabilist who works on optimal stopping and stochastic control, and solves the simplest case by hand before he trusts the general one.