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Seminario di Equazioni Deifferenziali e Applicazioni: “On the convergence problem in mean field games: a two state model without uniqueness”

Lunedì 9 Aprile 2018, ore 12:30 - Aula 2BC30 - Alekos Cecchin

ARGOMENTI: Seminari

Lunedì 9 Aprile 2018 alle ore 12:30 in Aula 2BC30, Alekos Cecchin (Università di Padova) terrà un seminario dal titolo “On the convergence problem in mean field games: a two state model without uniqueness”.

Abstract
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 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 solution. We prove that the N-player game always admits a limit, which depends on the initial distribution. 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.

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