Quadratic mean field games

Publication date: Available online 21 January 2019Source: Physics ReportsAuthor(s): Denis Ullmo, Igor Swiecicki, Thierry GobronAbstractMean field games were introduced independently by J-M. Lasry and P-L. Lions, and by M. Huang, R.P. Malhamé and P.E. Caines, in order to bring a new approach to optimization problems with a large number of interacting agents. The description of such models split in two parts, one describing the evolution of the density of players in some parameter space, the other the value of a cost functional each player tries to minimize for himself, anticipating on the rational behavior of the others.Quadratic Mean Field Games form a particular class among these systems, in which the dynamics of each player is governed by a controlled Langevin equation with an associated cost functional quadratic in the control parameter. In such cases, there exists a deep relationship with the non-linear Schrödinger equation in imaginary time, connection which lead to effective approximation schemes as well as a better understanding of the behavior of Mean Field Games.The aim of this paper is to serve as an introduction to Quadratic Mean Field Games and their connection with the non-linear Schrödinger equation, providing to physicists a good entry point into this new and exciting field.

Source: Physics Reports - January 22, 2019 Category: Physics Source Type: research