Mean-field games (MFGs) are models of large populations of rational agents who seek to optimize an objective function that takes into account their location and the distribution of the remaining agents. Here, we consider stationary MFGs with congestion and prove the existence of stationary solutions. Because moving in congested areas is difficult, agents prefer to move in non-congested areas. As a consequence, the model becomes singular near the zero density. The existence of stationary solutions was previously obtained for MFGs with quadratic Hamiltonians thanks to a very particular identity. Here, we develop robust estimates that give the existence of a solution for general subquadratic Hamiltonians.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
Achdou, Y., Laurière, M.: Mean field type control with congestion. Appl. Math. Optim. 73, 393–418 (2016)
Achdou, Y., Porretta, A.: Mean field games with congestion. Preprint (2016)
Cardaliaguet, P., Garber, P., Porretta, A., Tonon, D.: Second order mean field games with degenerate diffusion and local coupling. Preprint (2014)
Cirant, M.: Multi-population mean field games systems with Neumann boundary conditions. J. Math. Pures Appl. 103(5), 1294–1315 (2015). doi:10.1016/j.matpur.2014.10.013
Evangelista, D., Gomes, D., Nurbekyan, L.: Radially symmetric mean-field-games with congestion. ArXiv preprint (2017). arXiv:1703.07594v1 [math.AP]
Ferreira, R., Gomes, D.: Existence of weak solutions for stationary mean-field games through variational inequalities. Preprint (2016)
Gomes, D., Mitake, H.: Existence for stationary mean-field games with congestion and quadratic Hamiltonians. NoDEA Nonlinear Differ. Equ. Appl. 22(6), 1897–1910 (2015)
Gomes, D., Nurbekyan, L., Prazeres, M.: Explicit solutions of one-dimensional, first-order, stationary mean-field games with congestion. In: 2016 IEEE 55th Conference on Decision and Control, CDC 2016, pp. 4534–4539 (2016)
Gomes, D., Nurbekyan, L., Prazeres, M.: One-dimensional stationary mean-field games with local coupling. Dyn. Games Appl. (2017). doi:10.1007/s13235-017-0223-9
Gomes, D., Patrizi, S., Voskanyan, V.: On the existence of classical solutions for stationary extended mean field games. Nonlinear Anal. 99, 49–79 (2014)
Gomes, D., Pimentel, E.: Time dependent mean-field games with logarithmic nonlinearities. SIAM J. Math. Anal. 47(5), 3798–3812 (2015)
Gomes, D., Pimentel, E.: Local regularity for mean-field games in the whole space. Minimax Theory Appl. 01(1), 065–082 (2016)
Gomes, D., Pimentel, E., Sánchez-Morgado, H.: Time-dependent mean-field games in the subquadratic case. Commun. Partial Differ. Equ. 40(1), 40–76 (2015)
Gomes, D., Pires, G.E., Sánchez-Morgado, H.: A-priori estimates for stationary mean-field games. Netw. Heterog. Media 7(2), 303–314 (2012)
Gomes, D., Voskanyan, V.: Short-time existence of solutions for mean-field games with congestion. J. Lond. Math. Soc. (2) 92(3), 778–799 (2015)
Gomes, D., Pimentel, E., Sánchez-Morgado, H.: Time-dependent mean-field games in the superquadratic case. ESAIM Control Optim. Calc. Var. 22(2), 562–580 (2016)
Graber, J.: Weak solutions for mean field games with congestion. Preprint (2015)
Guéant, O.: An existence and uniqueness result for mean field games with congestion effect on graphs. Preprint (2011)
Huang, M., Caines, P.E., Malhamé, R.P.: Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized \(\epsilon \)-Nash equilibria. IEEE Trans. Autom. Control 52(9), 1560–1571 (2007)
Huang, M., Malhamé, R.P., Caines, P.E.: Large population stochastic dynamic games: closed-loop McKean–Vlasov systems and the Nash certainty equivalence principle. Commun. Inf. Syst. 6(3), 221–251 (2006)
Lasry, J.-M., Lions, P.-L.: Jeux à champ moyen. I. Le cas stationnaire. C. R. Math. Acad. Sci. Paris 343(9), 619–625 (2006)
Lasry, J.-M., Lions, P.-L.: Jeux à champ moyen. II. Horizon fini et contrôle optimal. C. R. Math. Acad. Sci. Paris 343(10), 679–684 (2006)
Lasry, J.-M., Lions, P.-L.: Mean field games. Jpn. J. Math. 2(1), 229–260 (2007)
Lions, P.-L.: Collége de France course on mean-field games (2007–2011). https://www.college-de-france.fr/site/pierre-louis-lions/_course.htm
Nurbekyan, L.: One-dimensional, non-local, first-order, stationary mean-field games with congestion: a Fourier approach. ArXiv preprint (2017). arXiv:1703.03954v1 [math.AP]
Pimentel, E., Voskanyan, V.: Regularity for second-order stationary mean-field games. Indiana Univ. Math. J. 66(1), 1–22 (2017)
Porretta, A.: On the planning problem for the mean field games system. Dyn. Games Appl. 4(2), 231–256 (2014)
Porretta, A.: Weak solutions to Fokker–Planck equations and mean field games. Arch. Ration. Mech. Anal. 216(1), 1–62 (2015)
D. Gomes and D. Evangelista were partially supported baseline and start-up funds from King Abdullah University of Science and Technology (KAUST).
About this article
Cite this article
Evangelista, D., Gomes, D.A. On the Existence of Solutions for Stationary Mean-Field Games with Congestion. J Dyn Diff Equat 30, 1365–1388 (2018). https://doi.org/10.1007/s10884-017-9615-1
- Mean-field games
- Congestion problems
- Stationary problems
Mathematics Subject Classification