Skip to main content
SpringerLink
Log in

One-Dimensional Forward–Forward Mean-Field Games

  • Published:
Applied Mathematics & Optimization Submit manuscript

Abstract

While the general theory for the terminal-initial value problem for mean-field games (MFGs) has achieved a substantial progress, the corresponding forward–forward problem is still poorly understood—even in the one-dimensional setting. Here, we consider one-dimensional forward–forward MFGs, study the existence of solutions and their long-time convergence. First, we discuss the relation between these models and systems of conservation laws. In particular, we identify new conserved quantities and study some qualitative properties of these systems. Next, we introduce a class of wave-like equations that are equivalent to forward–forward MFGs, and we derive a novel formulation as a system of conservation laws. For first-order logarithmic forward–forward MFG, we establish the existence of a global solution. Then, we consider a class of explicit solutions and show the existence of shocks. Finally, we examine parabolic forward–forward MFGs and establish the long-time convergence of the solutions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Achdou, Y., Capuzzo-Dolcetta, I.: Mean field games: numerical methods. SIAM J. Numer. Anal. 48(3), 1136–1162 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Achdou, Y., Cirant, M., Bardi, M.: Mean-field games models of segregation. (2016, preprint). arXiv:1607.04453

  3. Al-Mulla, N., Ferreira, R., Gomes, D.: Two numerical approaches to stationary mean-field games. Dyn Games Appl (2016). doi:10.1007/s13235-016-0203-5

  4. Barron, E.N., Evans, L.C., Jensen, R.: The infinity Laplacian, Aronsson’s equation and their generalizations. Trans. Am. Math. Soc. 360(1), 77–101 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cacace, S., Camilli, F.: Ergodic problems for Hamilton–Jacobi equations: yet another but efficient numerical method. (2016, preprint). arXiv:1601.07107

  6. Cagnetti, F., Gomes, D., Mitake, H., Tran, H.V.: A new method for large time behavior of degenerate viscous Hamilton–Jacobi equations with convex Hamiltonians. Ann. Inst. H. Poincaré Anal. Non Linéaire 32(1), 183–200 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Camilli, F., Festa, A., Schieborn, D.: An approximation scheme for a Hamilton–Jacobi equation defined on a network. Appl. Numer. Math. 73, 33–47 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Camilli, F., Carlini, E., Marchi, C.: A model problem for mean field games on networks. Discret. Contin. Dyn. Syst. 35(9), 4173–4192 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cardaliaguet, P.: Weak solutions for first order mean field games with local coupling. In: Analysis and Geometry in Control Theory and its Applications, Springer INdAM Series, vol. 11, pp 111–158. Springer, Cham (2015)

  10. Cardaliaguet, P., Graber, P.J.: Mean field games systems of first order. ESAIM Control Optim. Calc. Var. 21(3), 690–722 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chueh, K.N., Conley, C.C., Smoller, J.A.: Positively invariant regions for systems of nonlinear diffusion equations. Indiana Univ. Math. J. 26(2), 373–392 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cirant, M.: Nonlinear PDEs in ergodic control, mean-field games and prescribed curvature problems. Thesis (2013)

  13. Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics, Volume 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 3rd edn. Springer, Berlin (2010)

    Google Scholar 

  14. Davini, A., Siconolfi, A.: A generalized dynamical approach to the large time behavior of solutions of Hamilton–Jacobi equations. SIAM J. Math. Anal. 38(2), 478–502 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Demoulini, S., Stuart, D.M.A., Tzavaras, A.E.: Construction of entropy solutions for one-dimensional elastodynamics via time discretisation. Ann. Inst. H. Poincaré Anal. Non Linéaire 17(6), 711–731 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. DiPerna, R.J.: Convergence of approximate solutions to conservation laws. Arch. Ration. Mech. Anal. 82(1), 27–70 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fathi, A.: Sur la convergence du semi-groupe de Lax-Oleinik. C. R. Acad. Sci. Paris Sér. I Math. 327, 267–270 (1998)

    Article  MathSciNet  Google Scholar 

  18. Ferreira, R., Gomes, D.: Existence of weak solutions for stationary mean-field games through variational inequalities. (2015, preprint). arXiv:1512.05828

  19. 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)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gomes, D., Patrizi, S., Voskanyan, V.: On the existence of classical solutions for stationary extended mean field games. Nonlinear Anal. 99, 49–79 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gomes, D., Patrizi, S.: Obstacle mean-field game problem. Interfaces Free Bound. 17(1), 55–68 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Gomes, D., Pimentel, E.: Time dependent mean-field games with logarithmic nonlinearities. SIAM J. Math. Anal. 47(5), 3798–3798 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  23. Gomes, D., Pimentel, E.: Local regularity for mean-field games in the whole space. Minimax Theory Appl. 1(1), 065–082 (2016)

  24. Gomes, D., Pimentel, E.: Regularity for mean-field games systems with initial–initial boundary conditions: subquadratic case. In: Bourguignon, J.P., Jeltsch, R., Pinto, A., Viana, M. (eds.) Dynamics, Games and Science, CIM Series in Mathematical Sciences, vol. 1, Chap. 15. Springer-Verlag (2015)

  25. Gomes, D., Nurbekyan, L., Prazeres, M.: Explicit solutions of one-dimensional first-order stationary mean-field games with a generic nonlinearity. Accepted for publication in 55th IEEE Conference on Decision and Control (2016)

  26. Gomes, D., Nurbekyan, L., Prazeres, M.: Explicit solutions of one-dimensional first-order stationary mean-field games with congestion. Preprint (2016)

  27. Ishii, H.: Asymptotic solutions for large time of Hamilton–Jacobi equations in Euclidean \(n\) space. Ann. Inst. H. Poincaré Anal. Non Linéaire 25(2), 231–266 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. 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)

    Article  MathSciNet  Google Scholar 

  29. 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)

    Article  MathSciNet  Google Scholar 

  30. Lasry, J.-M., Lions, P.-L., Guéant, O.: Mean Field Games and Applications. Paris-Princeton Lectures on Mathematical Finance (2010)

  31. Namah, G., Roquejoffre, J.M.: Comportement asymptotique des solutions d’une classe d’équations paraboliques et de Hamilton–Jacobi. C. R. Acad. Sci. Paris Sér. I Math. 324(12), 1367–1370 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  32. Porretta, A.: Weak solutions to Fokker–Planck equations and mean field games. Arch. Ration. Mech. Anal. 216(1), 1–62 (2015)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors were supported by KAUST baseline and start-up funds.

Author information

Authors and Affiliations

  1. CEMSE Division, King Abdullah University of Science and Technology (KAUST), Thuwal, 23955-6900, Saudi Arabia

    Diogo A. Gomes, Levon Nurbekyan & Marc Sedjro

Authors
  1. Diogo A. Gomes
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Levon Nurbekyan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Marc Sedjro
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Diogo A. Gomes.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gomes, D.A., Nurbekyan, L. & Sedjro, M. One-Dimensional Forward–Forward Mean-Field Games. Appl Math Optim 74, 619–642 (2016). https://doi.org/10.1007/s00245-016-9384-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00245-016-9384-y

Keywords

Search

Navigation