We propose a (toy) MFG model for the evolution of residents and firms densities, coupled both by labour market equilibrium conditions and competition for land use (congestion). This results in a system of two Hamilton-Jacobi-Bellman and two Fokker-Planck equations with a new form of coupling related to optimal transport. This MFG has a convex potential which enables us to find weak solutions by a variational approach. In the case of quadratic Hamiltonians, the problem can be reformulated in Lagrangian terms and solved numerically by an IPFP/Sinkhorn-like scheme as in [
Citation: |
[1] |
Y. Achdou, M. Bardi and M. Cirant, Mean field games models of segregation, Math. Models Methods Appl. Sci., 27 (2017), 75-113.
doi: 10.1142/S0218202517400036.![]() ![]() ![]() |
[2] |
A. Beck and L. Tetruashvili, On the convergence of block coordinate descent type methods, SIAM J. Optim., 23 (2013), 2037-2060.
doi: 10.1137/120887679.![]() ![]() ![]() |
[3] |
J.-D. Benamou, G. Carlier, M. Cuturi, L. Nenna and G. Peyré, Iterative Bregman projections for regularized transportation problems, SIAM J. Sci. Comput., 37 (2015), A1111–A1138.
doi: 10.1137/141000439.![]() ![]() ![]() |
[4] |
J.-D. Benamou, G. Carlier, S. Di Marino and L. Nenna, An entropy minimization approach to second-order variational mean-field games, Math. Models Methods Appl. Sci., 29 (2019), 1553-1583.
doi: 10.1142/S0218202519500283.![]() ![]() ![]() |
[5] |
Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math., 44 (1991), 375-417.
doi: 10.1002/cpa.3160440402.![]() ![]() ![]() |
[6] |
P. Cardaliaguet, Weak solutions for first order mean field games with local coupling, in Analysis and Geometry in Control Theory and Its Applications, Springer INdAM Ser., 11, Springer, Cham, 2015,111–158.
doi: 10.1007/978-3-319-06917-3_5.![]() ![]() ![]() |
[7] |
P. Cardaliaguet, G. Carlier and B. Nazaret, Geodesics for a class of distances in the space of probability measures, Calc. Var. Partial Differential Equations, 48 (2013), 395-420.
doi: 10.1007/s00526-012-0555-7.![]() ![]() ![]() |
[8] |
P. Cardaliaguet and P. J. Graber, Mean field games systems of first order, ESAIM Control Optim. Calc. Var., 21 (2015), 690-722.
doi: 10.1051/cocv/2014044.![]() ![]() ![]() |
[9] |
P. Cardaliaguet, P. J. Graber, A. Porretta and D. Tonon, Second order mean field games with degenerate diffusion and local coupling, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 1287-1317.
doi: 10.1007/s00030-015-0323-4.![]() ![]() ![]() |
[10] |
P. Cardaliaguet, J.-M. Lasry, P.-L. Lions and A. Porretta, Long time average of mean field games, Netw. Heterog. Media, 7 (2012), 279-301.
doi: 10.3934/nhm.2012.7.279.![]() ![]() ![]() |
[11] |
G. Carlier and I. Ekeland, Equilibrium structure of a bidimensional asymmetric city, Nonlinear Anal. Real World Appl., 8 (2007), 725-748.
doi: 10.1016/j.nonrwa.2006.02.008.![]() ![]() ![]() |
[12] |
L. Chizat, G. Peyré, B. Schmitzer and F.-X. Vialard, Scaling algorithms for unbalanced optimal transport problems, Math. Comp., 87 (2018), 2563-2609.
doi: 10.1090/mcom/3303.![]() ![]() ![]() |
[13] |
M. Cirant, Multi-population mean field games systems with Neumann boundary conditions, J. Math. Pures Appl. (9), 103 (2015), 1294-1315.
doi: 10.1016/j.matpur.2014.10.013.![]() ![]() ![]() |
[14] |
D. Cordero-Erausquin, Sur le transport de mesures périodiques, C. R. Acad. Sci. Paris Sér. I Math., 329 (1999), 199-202.
doi: 10.1016/S0764-4442(00)88593-6.![]() ![]() ![]() |
[15] |
M. Cuturi, Sinkhorn distances: Lightspeed computation of optimal transport, in Advances in Neural Information Processing Systems, 2013, 2292–2300.
![]() |
[16] |
D. A. Dawson and J. Gärtner, Large deviations from the McKean-Vlasov limit for weakly interacting diffusions, Stochastics, 20 (1987), 247-308.
doi: 10.1080/17442508708833446.![]() ![]() ![]() |
[17] |
H. Föllmer, Random fields and diffusion processes, in École d'Été de Probabilités de Saint-Flour XV–XVII, 1985–87, Lecture Notes in Math., 1362, Springer, Berlin, 1988,101–203.
doi: 10.1007/BFb0086180.![]() ![]() ![]() |
[18] |
A. Galichon, Optimal Transport Methods in Economics, Princeton University Press, Princeton, NJ, 2016.
doi: 10.1515/9781400883592.![]() ![]() ![]() |
[19] |
P. J. Graber, Optimal control of first-order Hamilton-Jacobi equations with linearly bounded Hamiltonian, Appl. Math. Optim., 70 (2014), 185-224.
doi: 10.1007/s00245-014-9239-3.![]() ![]() ![]() |
[20] |
P. J. Graber, A. R. Mészáros, F. J. Silva and D. Tonon, The planning problem in mean field games as regularized mass transport, Calc. Var. Partial Differential Equations, 58 (2019), 28pp.
doi: 10.1007/s00526-019-1561-9.![]() ![]() ![]() |
[21] |
J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. Ⅰ. Le cas stationnaire, C. R. Math. Acad. Sci. Paris, 343 (2006), 619-625.
doi: 10.1016/j.crma.2006.09.019.![]() ![]() ![]() |
[22] |
J.-M. Lasry and P.-L. Lions, Jeux à champ moyen. Ⅱ. Horizon fini et contrôle optimal, C. R. Math. Acad. Sci. Paris, 343 (2006), 679-684.
doi: 10.1016/j.crma.2006.09.018.![]() ![]() ![]() |
[23] |
J.-M. Lasry and P.-L. Lions, Mean field games, Jpn. J. Math., 2 (2007), 229-260.
doi: 10.1007/s11537-007-0657-8.![]() ![]() ![]() |
[24] |
R. E. Lucas Jr. and E. Rossi-Hansberg, On the internal structure of cities, Econometrica, 70 (2002), 1445-1476.
doi: 10.1111/1468-0262.00338.![]() ![]() ![]() |
[25] |
C. Orrieri, A. Porretta and G. Savaré, A variational approach to the mean field planning problem, J. Funct. Anal., 277 (2019), 1868-1957.
doi: 10.1016/j.jfa.2019.04.011.![]() ![]() ![]() |
[26] |
G. Peyré, Entropic approximation of Wasserstein gradient flows, SIAM J. Imaging Sci., 8 (2015), 2323-2351.
doi: 10.1137/15M1010087.![]() ![]() ![]() |
[27] |
G. Peyré and M. Cuturi, Computational optimal transport: With applications to data science, Foundations and Trends in Machine Learning, 11 (2019), 355-607.
doi: 10.1561/2200000073.![]() ![]() |
[28] |
R. T. Rockafellar, Integrals which are convex functionals. Ⅱ, Pacific J. Math., 39 (1971), 439-469.
doi: 10.2140/pjm.1971.39.439.![]() ![]() ![]() |
[29] |
F. Santambrogio, Optimal Transport for Applied Mathematicians. Calculus of Variations, PDEs, and Modeling, Progress in Nonlinear Differential Equations and their Applications, 87, Birkhäuser/Springer, Cham, 2015.
doi: 10.1007/978-3-319-20828-2.![]() ![]() ![]() |
[30] |
R. Sinkhorn, A relationship between arbitrary positive matrices and doubly stochastic matrices, Ann. Math. Statist., 35 (1964), 876-879.
doi: 10.1214/aoms/1177703591.![]() ![]() ![]() |
[31] |
C. Villani, Optimal Transport. Old and New, Grundlehren der Mathematischen Wissenschaften, 338, Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-540-71050-9.![]() ![]() ![]() |
[32] |
C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003.
doi: 10.1090/gsm/058.![]() ![]() ![]() |