4 Presentations

• 
  02:00 PM - 02:25 PM

  On the convergence problem for first order mean field games

  • Francisco José Silva Alvarez, presenter, XLIM, Université de Limoges
  • Markus Fischer, University of Padua

  In this talk we provide a simple justification of the first order MFG system, first introduced by Lasry and Lions in 2007, as a PDE characterization of Nash equilibria for symmetric deterministic differential games with a continuum of players. Our main result shows that such equilibria can be found as the limit of Nash equilibria of suitable differential games with a finite number of players.

• 
  02:25 PM - 02:50 PM

  Finite mean field games: fictitious play and convergence analysis

  • Saeed Hadikhanloo, presenter, Ecole Polytechnique
  • Francisco José Silva Alvarez, XLIM, Université de Limoges

  In this talk, based on an ongoing work with F. Silva (U. Limoges and TSE), we consider a class of finite state and discrete time Mean Field Games (MFGs) introduced by Gomes, Mohr and Rigao Souza in 2009. In this framework we first study an adaptation of the fictitious play procedure for continuous MFGs, introduced recently by Cardaliaguet and Hadikhanloo, and we prove the convergence to the solution of the finite MFG. In the second part of the talk, we consider a first order continuous MFG and an associated family of finite MFGs, parameterized by a finite time and space grid. We prove that, as the time and space steps tend to 0, the solutions of the finite MFGs converge to a solution of the continuous one.

• 
  02:50 PM - 03:15 PM

  Proximal methods for stationary Mean Field Games with local couplings: Theory and algorithms

  • Luis Briceño-Arias, presenter, Universidad Técnica Federico Santa María
  • Francisco José Silva Alvarez, XLIM, Université de Limoges
  • Dante Kalise, Mathematics Department, La Sapienza - University of Rome 1

  We address the numerical approximation of Mean Field Games with local couplings. For power-like Hamiltonians, we consider both unconstrained and constrained stationary systems with density constraints in order to model hard congestion effects. For finite difference discretizations of the Mean Field Game system, we follow a variational approach. We prove that the aforementioned schemes can be obtained as the optimality system of suitably defined optimization problems. Next, assuming next that the coupling term is monotone, we study and compare several proximal type globally convergent first-order methods for solving the convex optimization problems. Each step of the proposed algorithms is easy computable, which leads to efficient implementations.

• 
  03:15 PM - 03:40 PM

  On Mean Field Games with Common Noise based on Stable-Like Processes

  • Vassili Kolokoltsov, University of Warwick, Department of Statistics
  • Marianna Troeva, presenter, North-Eastern Federal University, Research Institute of Mathematics

  We study Mean Field games with common noise based on nonlinear stable-like processes. The MFG limit is specified by a single quasi-linear deterministic infinite-dimensional partial differential second order backward equation. The main result is that any it’s solution provides an 1/N-Nash equilibrium for the initial game of N agents. Our basic approach is based on interpreting the common noise as a kind of binary interaction of agents and our previous results on regularity and sensitivity with respect to the initial conditions of the solution to the nonlinear McKean-Vlasov SPDE arising in games of this type.

  Keywords: Mean Field games, common noise, stable-like processes, McKean-Vlasov SPDE, regularity, sensitivity, Nash equilibrium