Mittwoch, 25.07.2012 15:00 Uhr

Some aspects of Mean Field Games

Ermal Feleqi, Università degli Studi di Padova

Mean field games are a branch of game theory introduced by J-M. Lasry and P-L. Lions in order
to model the behavior of a very large number of rational agents with a limited information (or visibility)
of the game who optimize their decisions in view of the global (or macroscopic) informations
available to them and that result from the actions of all agents. The perspective of applications is
quite broad, e.g., in economics, finance, sociology, urban planning, engineering etc. Games with
e very large number of players are approximated by a “continuum limit“ (letting the number of the
players go to infinity) in analogy with certain “mean field“ approaches of statistical mechanics and
physics, and this justifies the name.

The focus on my talk will be on a class of ergodic stocastic differential games cou- pled only
through the costs with players belonging to N different populations. (Each population consists of
a large number of identical players, but the characteristics of the players vary from one population
to the other.) In this case the MFG model results in a diagonal system of 2N stationary PDEs:
N Hamilton-Jacobi-Bellman equations and N Kolmogorov-Fokker-Plank linear PDEs for the final
distribution of the players of each population. I will exhibit a wide range of sufficient conditions for
the solvability of these systems and their rigorous (or mathematical) derivation as a “continuum
limit“ of certain systems of PDEs associated with games with a finite number of players as the
cardinality of each population goes to infinity. In doing so, I do not only generalize previous work
of Lasry and Lions by considering more general dynamics, costs and several populations, but also
provide detailed proofs (which they do not in their articles).

I will end my talk by indicating some perspectives for future research, most in- triguing for me
being the possibility of formulating a very general “master equation“ in metric spaces of probability
measures. Its mathematical interpretation, study and degree of approximation of the related
games with a finite number of players seem very fascinating and promising topics to me.

Zeit: 15:00 Uhr

Ort: SG 11, Seminargebäude, RWTH Aachen, Wüllnerstr. 5b, 52062 Aachen