The project is devoted to the study of Kinetic Mean Field Games (KMFGs), i.e. the limit of cooperative or non-cooperative games in very large populations, composed of interacting individuals, where each individual has a small influence on the global behaviour of the system and where the behaviour of the system depends on additional (internal) variables.
Though the observables of KMFGs do not depend on the additional variables themselves (as they are the moments of the distribution function with respect to these additional variables), such internal variables have to be taken into account in the models and they have an influence in its behaviour. Moreover, since different time scales can be involved in the phenomenon, the kinetic approach allows to treat systems that cannot be modelled by purely macroscopic equations.
The methods to be used come from the most recent developments in kinetic theory, as well as from the strategies developed in the last years for macroscopic mean field games. They include: existence theory of nontrivial remarkable states, linear and nonlinear asymptotic stability methods, entropy/entropy dissipation estimates, perturbative methods, renormalized solutions, etc. Our goal is, in particular, to extract information from systems which are far from known steady states.
Moreover, through the identification of a scale parameter, we will study the limit, as the parameter tends to zero, of the system.
Particular emphasis will be addressed to the numerical simulation of KMFG systems.
As the unknown function depends - in principle - on several variables and as the KMFG is in principle both forward and backward in time, the numerical study of kinetic mean field game systems requires new approaches and massive parallelization techniques. The easy access to teraflop-based manycore computing (i.e. either 1-card GPU-like office computing, or clusters of manycores HPC) will allow to handle KMFG equations at the numerical level and produce accurate numerical schemes.
Since the modelling is not completely stabilized in many of the problems under consideration, a collaboration with teams of economists or biologists is planned.
The research project proposed here foresees the hiring of excellent postdoctoral researchers, coming from national or international high level universities and aims to give a substantial contribution in developing KMFG theory.
Our development strategy consists in a fast dissemination of the scientific results that we will obtain. It will be done by producing publications in international reviewed journals and by means of communications in international conferences and workshops. Our works will provide contributions in applied mathematics journals but also, for example, in economics or biology journals.
The interaction with industries and agencies is planned.
Furthermore, we aim to widespread our results also in non-academic contexts, in order to show how applied mathematics can help the productive world.
Finally, we shall create a web site where the pieces of information on the ANR project will be available to the scientific community.
Monsieur Francesco Salvarani (ASSOCIATION LEONARD DE VINCI)
The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.
ALDV ASSOCIATION LEONARD DE VINCI
CEREMADE Centre De Recherche en Mathématiques de la Décision - Université Paris Dauphine
Help of the ANR 499,999 euros
Beginning and duration of the scientific project: February 2015 - 48 Months