This is a tutorial introduction to mean field games pioneered by Lasry and Lions. Beginning with motivations from physics, game theory and economics, the concept of mean field games is introduced. Specifically, this refers to the limiting case of a large population of (say) \(N\) players with identical controlled dynamics and objective function, each coupled to the rest in an average manner, i.e., via a term involving an average over all agents. In the limit as \(N\) tends to infinity, one gets a situation described by two interdependent partial differential equations. One is the Hamilton-Jacobi equation corresponding to the finite horizon control problem that runs backward in time. The other is the Liouville (in deterministic case) or Fokker-Planck-Kolmogorov (in stochastic case) equation that describes the evolution of the population profile in forward time. The issues addressed are existence and uniqueness of solutions and the concomitant “mean field” Nash equilibrium. By reversing the time in the Hamilton-Jacobi equation and then letting time tend to infinity in both equations, stationary solutions are obtained and their local stability, termed “eductive stability”, is analyzed by linearized dynamics around the stationary solution. The theory throughout is developed through a succession of concrete examples, first a toy example followed by an example concerning production of an exhaustible resource by a continuum of producers, then a model for Mexican wave, a model for population distribution, an investment model with a continuum of asset managers with an additional classification dimension, and finally a model of economic growth. Numerical issues are also discussed and illustrated.
For the entire collection see [Zbl 1198.91010].