×

Mean field games models of segregation. (English) Zbl 1355.91003

Summary: This paper introduces and analyzes some models in the framework of mean field games (MFGs) describing interactions between two populations motivated by the studies on urban settlements and residential choice by Thomas Schelling. For static games, a large population limit is proved. For differential games with noise, the existence of solutions is established for the systems of partial differential equations of MFG theory, in the stationary and in the evolutive case. Numerical methods are proposed with several simulations. In the examples and in the numerical results, particular emphasis is put on the phenomenon of segregation between the populations.

MSC:

91A07 Games with infinitely many players
49N70 Differential games and control
35K55 Nonlinear parabolic equations
91A40 Other game-theoretic models
91D10 Models of societies, social and urban evolution
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Achdou, Y., Buera, F. J., Lasry, J.-M., Lions, P.-L. and Moll, B., Partial differential equation models in macroeconomics, Phil. Trans. R. Soc. A372 (2014) 20130397. · Zbl 1353.91027
[2] Achdou, Y., Camilli, F. and Capuzzo-Dolcetta, I., Mean field games: Numerical methods for the planning problem, SIAM J. Control Optim.50 (2012) 77-109. · Zbl 1242.91014
[3] Achdou, Y., Camilli, F. and Capuzzo-Dolcetta, I., Mean field games: Convergence of a finite difference method, SIAM J. Numer. Anal.51 (2013) 2585-2612. · Zbl 1286.91022
[4] Achdou, Y. and Capuzzo-Dolcetta, I., Mean field games: Numerical methods, SIAM J. Numer. Anal.48 (2010) 1136-1162. · Zbl 1217.91019
[5] Achdou, Y. and Porretta, A., Convergence of a finite difference scheme to weak solutions of the system of partial differential equation arising in mean field games, SIAM J. Numer. Anal.54 (2016) 161-186. · Zbl 1382.65273
[6] Anderson, R. F. and Orey, S., Small random perturbation of dynamical systems with reflecting boundary, Nagoya Math. J.60 (1976) 189-216. · Zbl 0324.60063
[7] Bardi, M. and Feleqi, E., Nonlinear elliptic systems and mean field games, Nonlinear Differential Equations Appl.23 (2016) 23-44. · Zbl 1358.35198
[8] Bardi, M. and Priuli, F. S., Linear-quadratic \(N\)-person and mean-field games with ergodic cost, SIAM J. Control Optim.50 (2014) 3022-3052. · Zbl 1308.91008
[9] Barr, J. and Tassier, T., Segregation and strategic neighborhood interaction, Eastern Econ. J.34 (2008) 480-503.
[10] Bellomo, N. and Gibelli, L., Toward a mathematical theory of behavioral-social dynamics for pedestrian crowds.Math. Models Methods Appl. Sci.25 (2015) 2417-2437. · Zbl 1325.91042
[11] Benamou, J.-D. and Carlier, G., Augmented Lagrangian methods for transport optimization, mean field games and degenerate elliptic equations, J. Optim. Theor. Appl.167 (2015) 1-26. · Zbl 1326.49074
[12] Bensoussan, A., Frehse, J. and Yam, Ph., Mean Field Games and Mean Field Type Control Theory (Springer, 2013). · Zbl 1287.93002
[13] Bruch, E. E. and Mare, R. D., Neighborhood choice and neighborhood change, Amer. J. Soc.112 (2006) 667-709.
[14] P. Cardaliaguet, Notes on Mean Field Games, unpublished notes (2013), https://www.ceremade.dauphine.fr/cardalia/index.html.
[15] 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:1509.02505. · Zbl 1430.91002
[16] Cardaliaguet, P., Lasry, J.-M., Lions, P.-L. and Porretta, A., Long time average of mean field games, Netw. Heterog. Media7 (2012) 279-301. · Zbl 1270.35098
[17] Carlini, E. and Silva, F. J., A fully discrete semi-Lagrangian scheme for a first order mean field game problem, SIAM J. Numer. Anal.52 (2014) 45-67. · Zbl 1300.65064
[18] Carlini, E. and Silva, F. J., A semi-Lagrangian scheme for a degenerate second order mean field game system, Discrete Contin. Dynam. Syst.35 (2015) 4269-4292. · Zbl 1332.65138
[19] M. Cirant, Nonlinear PDEs in Ergodic Control, Mean Field Games and Prescribed Curvature Problems, Ph.D. thesis, Università di Padova (2013).
[20] Cirant, M., Multi-population mean field games systems with Neumann boundary conditions, J. Math. Pures Appl.103 (2015) 1294-1315. · Zbl 1320.35347
[21] M. Cirant and G. Verzini, Bifurcation and segregation in quadratic two-populations mean field games systems, to appear in ESAIM Control Optim. Calc. Var. (2016). · Zbl 1371.35110
[22] Cristiani, E., Piccoli, B. and Tosin, A., Multiscale Modeling of Pedestrian Dynamics (Springer, 2014). · Zbl 1314.00081
[23] Feleqi, E., The derivation of ergodic mean field game equations for several populations of players, Dynam. Games Appl.3 (2013) 523-536. · Zbl 1314.91019
[24] M. Fischer, On the connection between symmetric \(N\)-player games and mean field games, preprint (2014), arXiv:1405.1345, to appear in Ann. Appl. Probab.
[25] S. Grauwin, F. Goffette-Nagot and P. Jensen, Dynamic models of residential segregation: Brief review, analytical resolution and study of the introduction of coordination. Working paper 09-14, Groupe d’Analyse et de Théorie Économique UMR 5824 du CNRS.
[26] Grauwin, S., Goffette-Nagot, F. and Jensen, P., Dynamic models of residential segregation: An analytical solution, J. Public Econ.96 (2012) 124-141.
[27] Gomes, D. A., Mohr, J. and Souza, R. R., Discrete time, finite state space mean field games, J. Math. Pures Appl.93 (2010) 308-328. · Zbl 1192.91028
[28] Gomes, D., Nurbekyan, L. and Pimentel, E., Economic Models and Mean-Field Games Theory, (Instituto Nacional de Matemática Pura e Aplicada, 2015). · Zbl 1335.91003
[29] D. Gomes, L. Nurbekyan and M. Sedjro, One-dimensional forward-forward mean field games, preprint (2016), arXiv:1606.09064. · Zbl 1371.49031
[30] Gomes, D., Pimentel, E. and Sánchez-Morgado, H., Time-dependent mean field games in the subquadratic case, Comm. Partial Differential Equations40 (2015) 40-76. · Zbl 1322.35053
[31] Gomes, D., Pimentel, E. and Sánchez-Morgado, H., Time-dependent mean-field games in the superquadratic case, ESAIM Control Optim. Calc. Var.22 (2016) 562-580. · Zbl 1339.35090
[32] Gomes, D., Pimentel, E. and Voskanyan, V., Regularity Theory for Mean-Field Game Systems (Springer, 2016). · Zbl 1391.91003
[33] Guéant, O., Mean field games equations with quadratic Hamiltonian: A specific approach, Math. Models Methods Appl. Sci.22 (2012) 1250022. · Zbl 1252.91013
[34] Guéant, O., New numerical methods for mean field games with quadratic costs, Netw. Heterog. Media7 (2012) 315-336. · Zbl 1270.35020
[35] Guéant, O., Lasry, J.-M. and Lions, P.-L., Mean field games and applications, in Paris-Princeton Lectures on Mathematical Finance 2010, , (Springer, 2011).
[36] Huang, M., Caines, P. E. and Malhamé, R. P., Large-population cost-coupled LQG problems with nonuniform agents: Individual-mass behavior and decentralized \(\epsilon \)-Nash equilibria, IEEE Trans. Automat. Control52 (2007) 1560-1571. · Zbl 1366.91016
[37] Huang, M., Malhamé, R. P. and Caines, P. E., Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Commun. Inf. Syst.6 (2006) 221-251. · Zbl 1136.91349
[38] A. Lachapelle, Human crowds and groups interactions: A mean field games approach, preprint (2010), hal-00484097.
[39] Lachapelle, A. and Wolfram, M. T., On a mean field game approach modeling congestion and aversion in pedestrian crowds, Trans. Res.: Part B: Methods45 (2011) 1572-1589.
[40] Ladyzhenskaya, O. A., Solonnikov, V. A. and Ural’seva, N. N., Linear and Quasilinear Equations of Parabolic Type (Amer. Math. Soc., 1967). · Zbl 0164.12302
[41] Lasry, J.-M. and Lions, P.-L., Mean field games, Jpn. J. Math.2 (2007) 229-260. · Zbl 1156.91321
[42] Nourian, M. and Caines, P. E., \( \epsilon \)-Nash mean field game theory for nonlinear stochastic dynamical systems with major and minor agents, SIAM J. Control Optim.51 (2013) 3302-3331. · Zbl 1275.93067
[43] J. Rauch, Seeing around corners, The Atlantic (2012), www.theatlantic.com/rauch.
[44] Schelling, T. C., Dynamic models of segregation, J. Math. Soc.1 (1971) 143-186. · Zbl 1355.91061
[45] Schelling, T. C., Micromotives and Macrobehavior (Norton, 1978).
[46] Stroock, D. W. and Varadhan, S. R. S., Diffusion processes with boundary conditions, Commun. Pure Appl. Math.24 (1971) 147-225. · Zbl 0227.76131
[47] Zhang, J., A dynamic model of residential segregation, J. Math. Soc.28 (2004) 147-170. · Zbl 1101.91311
[48] Zhang, J., Tipping and residential segregation: A unified Schelling model, J. Region. Sci.51 (2011) 167-193.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.