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Quadratic Mean Field Games
Mean field games were introduced independently by J-M. Lasry and P-L. Lions,
and by M. Huang, R.P. Malham\'e 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 Schr\"odinger
equation in imaginary time, connexion 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 connexion with the non-linear Schr\"odinger equation, providing
to physicists a good entry point into this new and exciting field.Comment: 62 pages, 4 figure
Two-scale homogenization of a stationary mean-field game
In this paper, we characterize the asymptotic behavior of a first-order
stationary mean-field game (MFG) with a logarithm coupling, a quadratic
Hamiltonian, and a periodically oscillating potential. This study falls into
the realm of the homogenization theory, and our main tool is the two-scale
convergence. Using this convergence, we rigorously derive the two-scale
homogenized and the homogenized MFG problems, which encode the so-called
macroscopic or effective behavior of the original oscillating MFG. Moreover, we
prove existence and uniqueness of the solution to these limit problems.Comment: 36 page
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