An Ergodic Problem for Mean Field Games: Qualitative Properties and Numerical Simulations

This paper is devoted to some qualitative descriptions and some numerical results for ergodic Mean Field Games systems which arise, e.g., in the homogenization with a small noise limit. We shall consider either power type potentials or logarithmic type ones. In both cases, we shall establish some qualitative properties of the effective Hamiltonian $\bar H$ and of the effective drift $\bar b$. In particular we shall provide two cases where the effective system keeps/looses the Mean Field Games structure, namely where $\nabla_P \bar H(P,\alpha)$ coincides or not with $\bar b(P, \alpha)$. On the other hand, we shall provide some numerical tests validating the aforementioned qualitative properties of $\bar H$ and $\bar b$. In particular, we provide a numerical estimate of the discrepancy $\nabla_P \bar H(P,\alpha)-\bar b(P, \alpha)$.

[1]  Pierre-Louis Lions,et al.  Long time average of mean field games , 2012, Networks Heterog. Media.

[2]  Yves Achdou,et al.  Mean Field Games: Numerical Methods , 2010, SIAM J. Numer. Anal..

[3]  P. Lions,et al.  Jeux à champ moyen. I – Le cas stationnaire , 2006 .

[4]  P. Lions,et al.  Jeux à champ moyen. II – Horizon fini et contrôle optimal , 2006 .

[5]  Diogo A. Gomes,et al.  A stochastic Evans-Aronsson problem , 2013 .

[6]  Edgard A. Pimentel,et al.  Regularity for second order stationary mean-field games , 2015, 1503.06445.

[7]  Diogo A. Gomes,et al.  On the existence of classical solutions for stationary extended mean field games , 2013, 1305.2696.

[8]  Simone Cacace,et al.  A Generalized Newton Method for Homogenization of Hamilton-Jacobi Equations , 2016, SIAM J. Sci. Comput..

[9]  Diogo A. Gomes,et al.  Mean Field Games Models—A Brief Survey , 2013, Dynamic Games and Applications.

[10]  P. Lions,et al.  The Master Equation and the Convergence Problem in Mean Field Games , 2015, 1509.02505.

[11]  A. Bensoussan,et al.  Mean Field Games and Mean Field Type Control Theory , 2013 .

[12]  Diogo A. Gomes,et al.  Regularity Theory for Mean-Field Game Systems , 2016 .

[13]  Simone Cacace,et al.  A numerical method for Mean Field Games on networks , 2015 .

[14]  Pierre Cardaliaguet,et al.  Long time behavior of the master equation in mean field game theory , 2017, Analysis & PDE.

[15]  Marco Cirant,et al.  Stationary focusing mean-field games , 2016, 1602.04231.

[16]  P. Lions,et al.  Mean field games , 2007 .

[17]  Annalisa Cesaroni,et al.  Homogenization of a Mean Field Game System in the Small Noise Limit , 2016, SIAM J. Math. Anal..

[18]  Yves Achdou,et al.  Finite Difference Methods for Mean Field Games , 2013 .

[19]  Martino Bardi,et al.  Nonlinear elliptic systems and mean-field games , 2016 .

[20]  Pierre-Louis Lions,et al.  Long Time Average of Mean Field Games with a Nonlocal Coupling , 2013, SIAM J. Control. Optim..