Mean Field Game systems describe equilibrium configurations in differential games with infinitely many infinitesimal interacting agents. We introduce a learning procedure (similar to the Fictitious Play) for these games and show its convergence when the Mean Field Game is potential.
Accepté le :
DOI : 10.1051/cocv/2016004
Mots-clés : Mean field games, learning
@article{COCV_2017__23_2_569_0,
author = {Cardaliaguet, Pierre and Hadikhanloo, Saeed},
title = {Learning in mean field games: {The} fictitious play},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {569--591},
publisher = {EDP-Sciences},
volume = {23},
number = {2},
year = {2017},
doi = {10.1051/cocv/2016004},
mrnumber = {3608094},
zbl = {1365.35183},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv/2016004/}
}
TY - JOUR AU - Cardaliaguet, Pierre AU - Hadikhanloo, Saeed TI - Learning in mean field games: The fictitious play JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 569 EP - 591 VL - 23 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2016004/ DO - 10.1051/cocv/2016004 LA - en ID - COCV_2017__23_2_569_0 ER -
%0 Journal Article %A Cardaliaguet, Pierre %A Hadikhanloo, Saeed %T Learning in mean field games: The fictitious play %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 569-591 %V 23 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2016004/ %R 10.1051/cocv/2016004 %G en %F COCV_2017__23_2_569_0
Cardaliaguet, Pierre; Hadikhanloo, Saeed. Learning in mean field games: The fictitious play. ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 2, pp. 569-591. doi : 10.1051/cocv/2016004. http://archive.numdam.org/articles/10.1051/cocv/2016004/
, , , and , Partial differential equation models in macroeconomics. Philos. Trans. R. Soc. A: Math. Phys. Eng. Sci. 372 (2014) 20130397. | DOI | MR | Zbl
, Uninsured Idiosyncratic Risk and Aggregate Saving. The Quarterly Journal of Economics 109 (1994) 659–84. | DOI
, Transport equation and Cauchy problem for BV vector fields. Inv. Math. 158 (2004) 227–260. | DOI | MR | Zbl
L. Ambrosio, N. Gigli and G. Savarè, Gradient flows in metric spaces and in the space of probability measures. Lectures in Mathematics ETH Zürich. Birkhäuser-Verlag, Basel (2008). | MR | Zbl
, Iterative solution of games by Fictitious Play. Activity Anal. Prod. Alloc. 13 (1951) 374–376. | MR | Zbl
P. Cannarsa and C. Sinestrari, Semiconcave functions, Hamilton–Jacobi equations and optimal control. Birkhäuser, Boston (2004). | MR | Zbl
P. Cardaliaguet, Weak solutions for first order mean field games with local coupling. Preprint . | HAL | MR
, Long time average of first order mean field games and weak KAM theory. Dyn. Games Appl. 3 (2013) 473–488. | DOI | MR | Zbl
, and , Geodesics for a class of distances in the space of probability measures. Calc. Var. Partial Differ. Eq. 48 (2013) 395–420. | DOI | MR | Zbl
P. Cardaliaguet, F. Delarue, J.-M. Lasry and P.-L. Lions, The master equation and the convergence problem in mean field games. Preprint (2015). | arXiv
, , and , Second order mean field games with degenerate diffusion and local coupling. Nonlin. Differ. Equ. Appl. 22 (2015) 1287–1317. | DOI | MR | Zbl
and , Probabilist analysis of Mean-Field Games. SIAM J. Control Optim. 51 (2013) 2705–2734. | DOI | MR | Zbl
and , Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (1989) 511–547. | DOI | MR | Zbl
, Mean field games equations with quadratic hamiltonian: a specific approach. Math. Models Methods Appl. Sci. 22 (2012) 1250022. | DOI | MR | Zbl
O. Guéant, P.-L. Lions and J.-M. Lasry, Mean Field Games and Applications. Paris-Princeton Lectures on Mathematical Finance 2010, edited by P. Tankov, P.-L. Lions, J.-P. Laurent, J.-M. Lasry, M. Jeanblanc, D. Hobson, O. Guéant, S. Crépey, A. Cousin. Springer, Berlin (2011) 205–266. | MR | Zbl
D. Fudenberg and D.K. Levine, The theory of learning in games. MIT Press, Cambridge, MA (1998). | MR | Zbl
M. Huang, P.E. Caines and R.P. Malhamé, Individual and mass behaviour in large population stochastic wireless power control problems: centralized and Nash equilibrium solutions. Proc. of 42nd IEEE Conf. Decision Contr., Maui, Hawaii (2003) 98–103.
, and , Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle. Commun. Inform. Syst. 6 (2006) 221–252. | DOI | MR | Zbl
O.A. Ladyženskaja, V.A. Solonnikov and N.N. Ural’ceva, Linear and quasilinear equations of parabolic type. In vol. 23 of Translations of Mathematical Monographs. American Mathematical Society, Providence, R.I. (1967). | MR
and , Jeux à champ moyen. I. Le cas stationnaire. C. R. Math. Acad. Sci. Paris 343 (2006) 619–625. | DOI | MR | Zbl
and , Jeux à champ moyen. II. Horizon fini et contrôle optimal. C. R. Math. Acad. Sci. Paris 343 (2006) 679–684. | DOI | MR | Zbl
and , Mean field games. Jpn J. Math. 2 (2007) 229–260. | DOI | MR | Zbl
P.L. Lions, Cours au Collège de France. Available at www.college-de-france.fr.
K. Miyasawa, On the convergence of the learning process in a non-zero-sum two-person game. Princeton University, NJ (1961).
and , Potential games. Games Econ. Behav. 14 (1996) 124–143. | DOI | MR | Zbl
and , Fictitious play property for games with identical interests. J. Econ. Theory 68 (1996) 258–265. | DOI | MR | Zbl
J. Robinson, An iterative method of solving a game. Ann. Math. (1951) 296–301. | MR | Zbl
, Some topics in two-person games. Ann. Math. Stud. 5 (1964) 1–28. | MR | Zbl
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