We present here results obtained in the joint work with Delarue, Lasry and Lions [4] on the convergence, as tends to infinity, of a system of coupled Hamilton-Jacobi equations, the Nash system. This system arises in differential game theory. The limit problem can be expressed in terms of the “Mean Field Game” system (coupling a Hamilton-Jacobi equation with a Fokker-Planck equation), or, alternatively, in terms of the “master equation” (a kind of second order partial differential equation stated on the space of probability measures). We also discuss the behavior of the optimal trajectories, for which we show a propagation of chaos property.
@article{SLSEDP_2015-2016____A17_0,
author = {Cardaliaguet, Pierre},
title = {Mean field games: the master equation and the mean field limit},
journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
note = {talk:17},
pages = {1--10},
publisher = {Institut des hautes des scientifiques & Centre de mathtiques Laurent Schwartz, ole polytechnique},
year = {2015-2016},
doi = {10.5802/slsedp.99},
language = {en},
url = {http://www.numdam.org/articles/10.5802/slsedp.99/}
}
TY - JOUR AU - Cardaliaguet, Pierre TI - Mean field games: the master equation and the mean field limit JO - Séminaire Laurent Schwartz — EDP et applications N1 - talk:17 PY - 2015-2016 SP - 1 EP - 10 PB - Institut des hautes des scientifiques & Centre de mathtiques Laurent Schwartz, ole polytechnique UR - http://www.numdam.org/articles/10.5802/slsedp.99/ DO - 10.5802/slsedp.99 LA - en ID - SLSEDP_2015-2016____A17_0 ER -
%0 Journal Article %A Cardaliaguet, Pierre %T Mean field games: the master equation and the mean field limit %J Séminaire Laurent Schwartz — EDP et applications %Z talk:17 %D 2015-2016 %P 1-10 %I Institut des hautes des scientifiques & Centre de mathtiques Laurent Schwartz, ole polytechnique %U http://www.numdam.org/articles/10.5802/slsedp.99/ %R 10.5802/slsedp.99 %G en %F SLSEDP_2015-2016____A17_0
Cardaliaguet, Pierre. Mean field games: the master equation and the mean field limit. Séminaire Laurent Schwartz — EDP et applications (2015-2016), Exposé no. 17, 10 p. doi : 10.5802/slsedp.99. http://www.numdam.org/articles/10.5802/slsedp.99/
[1] Ambrosio, L., Gigli, N., Savaré, G. Gradient flows in metric spaces and in the space of probability measures. Second edition. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2008.
[2] Aumann R. Markets with a continuum of traders. Econometrica, 32(1/2), 1964.
[3] Bensoussan, A., Frehse, J. (2002). Smooth solutions of systems of quasilinear parabolic equations. ESAIM: Control, Optimisation and Calculus of Variations, 8, 169-193.
[4] Cardaliaguet P., Delarue F., Lasry J.-M., and Lions P.-L. (2015) The master equation and the convergence problem in mean field games. Preprint.
[5] Cardaliaguet P (2016) The convergence problem in mean field games with a local coupling. Preprint.
[6] Carmona, R., and Delarue, F. Probabilistic Theory of Mean Field Games with Applications. To appear.
[7] Friedman, A. (2013). Differential games. Courier Corporation.
[8] Huang, M., Malhamé, R.P. Caines, P.E. (2006). Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Communication in information and systems. Vol. 6, No. 3, pp. 221-252.
[9] Huang, M., Caines, P.E., Malhamé, R.P. (2007). Large-Population Cost-Coupled LQG Problems With Nonuniform Agents: Individual-Mass Behavior and Decentralized -Nash Equilibria. IEEE Transactions on Automatic Control, 52(9), p. 1560-1571.
[10] Huang, M., Caines, P.E., Malhamé, R.P. (2007). The Nash Certainty Equivalence Principle and McKean-Vlasov Systems: an Invariance Principle and Entry Adaptation. 46th IEEE Conference on Decision and Control, p. 121-123.
[11] Huang, M., Caines, P.E., Malhamé, R.P. (2007). An Invariance Principle in Large Population Stochastic Dynamic Games. Journal of Systems Science & Complexity, 20(2), p. 162-172.
[12] Lasry, J.-M., Lions, P.-L. Jeux à champ moyen. I. Le cas stationnaire. C. R. Math. Acad. Sci. Paris 343 (2006), no. 9, 619–625.
[13] Lasry, J.-M., Lions, P.-L. Jeux à champ moyen. II. Horizon fini et contrôle optimal. C. R. Math. Acad. Sci. Paris 343 (2006), no. 10, 679–684.
[14] Lasry, J.-M., Lions, P.-L. Mean field games. Jpn. J. Math. 2 (2007), no. 1, 229–260.
[15] Lions, P.-L. Cours au Collège de France. www.college-de-france.fr.
[16] Otto F., The geometry of dissipative evolution equations: the porous medium equation. Communications in Partial Differential Equations, 26 (1-2), 101-174, 2001.
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