2 Presentations

• 
  04:10 PM - 04:35 PM

  On the convergence problem in mean field games: a two state model without uniqueness

  • Alekos Cecchin, presenter, University of Padua
  • Paolo Dai Pra, University of Padua
  • Markus Fischer, University of Padua
  • Guglielmo Pelino, University of Padua

  Mean field games represent limit models for symmetric non-zero sum dynamic games when the number N of players tends to infinity. We consider games in continuous and finite horizon time where the position of each agent belongs to {-1,1}. A rigorous study of the convergence of the feedback Nash equilibria to the limit is made through the so-called master equation, which in this case can be written as a scalar conservation law in one space dimension. If there is uniqueness of mean field game solutions, i.e. under monotonicity assumpions, then the master equation possesses a smooth solution which can be used to prove the convergence of the value functions of the N players and a propagation of chaos property for the associated optimal trajectories. We consider here an example with anti-monotonous cost, and show that the mean fielg game has exactly three solutions. We prove that the N-player game always admits a limit: it selects one mean field game solution, so there is propagation of chaos. The value functions also converge and the limit is the entropy solution to the master equation. Moreover, viewing the mean field game system as the necessary conditions for optimality of a deterministic control problem, we show that the N-player game selects the optimum of this problem.

• 
  04:35 PM - 05:00 PM

  Mean Field Games of Pure Jump Type with Common Noise

  • Astrid Hilbert, presenter, Linnaeus University
  • Christine Grün, University Toulouse Capitole

  In this paper we study a mean field game with pure jump dynamics, where all players are subject to the same additional Brownian noise. Moreover, we study well-posedness and regularity of the solution of the stochastic partial differential equation with jumps that replaces the McKean-Vlasov equation. Finally, we show that the solution of the master equation, which in this case solves a second order partial differential equation in the space of probability measures, provides an approximate Nash-equilibrium.