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July  2021, 8(3): 299-329. doi: 10.3934/jdg.2021017

A mean field game model for the evolution of cities

César Barilla 1, , Guillaume Carlier 2,, and  Jean-Michel Lasry 3,

1. 

Department of Economics, Columbia University, New York, USA
2. 

CEREMADE, Université Paris Dauphine, PSL Research University and INRIA-Paris, MOKAPLAN, Paris, France
3. 

CEREMADE, Université Paris Dauphine, PSL Research University, Paris, France

* Corresponding author: Guillaume Carlier

Received  December 2020 Published  May 2021

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 [4]. We present numerical results based on this approach, these simulations exhibit different behaviours with either agglomeration or segregation dominating depending on the initial conditions and parameters.

Keywords: Mean field games, convex duality, optimal transport, labour market equilibrium, Iterative Proportional Fitting procedure (IPFP).
Mathematics Subject Classification: Primary: 49L12, 91A14; Secondary: 65K10.
Citation: César Barilla, Guillaume Carlier, Jean-Michel Lasry. A mean field game model for the evolution of cities. Journal of Dynamics & Games, 2021, 8 (3) : 299-329. doi: 10.3934/jdg.2021017
References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

M. Cuturi, Sinkhorn distances: Lightspeed computation of optimal transport, in Advances in Neural Information Processing Systems, 2013, 2292–2300. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18] A. Galichon, Optimal Transport Methods in Economics, Princeton University Press, Princeton, NJ, 2016.  doi: 10.1515/9781400883592.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

G. Peyré, Entropic approximation of Wasserstein gradient flows, SIAM J. Imaging Sci., 8 (2015), 2323-2351.  doi: 10.1137/15M1010087.  Google Scholar

[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.  Google Scholar

[28]

R. T. Rockafellar, Integrals which are convex functionals. Ⅱ, Pacific J. Math., 39 (1971), 439-469.  doi: 10.2140/pjm.1971.39.439.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/gsm/058.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

M. Cuturi, Sinkhorn distances: Lightspeed computation of optimal transport, in Advances in Neural Information Processing Systems, 2013, 2292–2300. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18] A. Galichon, Optimal Transport Methods in Economics, Princeton University Press, Princeton, NJ, 2016.  doi: 10.1515/9781400883592.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

G. Peyré, Entropic approximation of Wasserstein gradient flows, SIAM J. Imaging Sci., 8 (2015), 2323-2351.  doi: 10.1137/15M1010087.  Google Scholar

[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.  Google Scholar

[28]

R. T. Rockafellar, Integrals which are convex functionals. Ⅱ, Pacific J. Math., 39 (1971), 439-469.  doi: 10.2140/pjm.1971.39.439.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, 58, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/gsm/058.  Google Scholar

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