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Existence and Uniqueness of Solutions for Bertrand and Cournot Mean Field Games

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Abstract

We study a system of partial differential equations used to describe Bertrand and Cournot competition among a continuum of producers of an exhaustible resource. By deriving new a priori estimates, we prove the existence of classical solutions under general assumptions on the data. Moreover, under an additional hypothesis we prove uniqueness.

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References

  1. Achdou, Y., Buera, J., Lasry, J.M., Lions, P.L., Moll, B.: Partial differential equation models in macroeconomics. Philos. Trans. R. Soc. Lond. A 372, 20130397 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Burger, M., Caffarelli, L., Markowich, P.A.: Partial differential equation models in the socio-economic sciences. Philos. Trans. R. Soc. Lond. A 372, 20130406 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cardaliaguet, P.: Weak solutions for first order mean field games with local coupling. arXiv:1305.7015 (2013)

  4. Cardaliaguet, P., Graber, P.J.: Mean field games systems of first order. ESAIM: COCV 21, 690–722 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cardaliaguet, P., Graber, P.J., Porretta, A., Tonon, D.: Second order mean field games with degenerate diffusion and local coupling. Nonlinear Differ. Equ. Appl. NoDEA 22, 1287–1317 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cardaliaguet, P., Lasry, J.-M., Lions, P.-L., Porretta, A.: Long time average of mean field games with a nonlocal coupling. SIAM J. Control Optim. 51, 3558–3591 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cardaliaguet, P., Lasry, J.-M., Lions, P.-L., Porretta, A., et al.: Long time average of mean field games. Netw. Heterog. Media 7, 279–301 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chan, P., Sircar, R.: Bertrand and Cournot mean field games. Appl. Math. Optim. 71, 1–37 (2014)

    MathSciNet  MATH  Google Scholar 

  9. Chan, P.,Sircar, R.: Fracking, renewables & mean field games. Available at SSRN 2632504 (2015)

  10. Evans, L.: Partial Differential Equations. Amer Mathematical Society, Graduate Studies in Mathematics Series (2010)

  11. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2015)

    MATH  Google Scholar 

  12. Gomes, D., Velho, R.M., Wolfram, M.-T.: Socio-economic applications of finite state mean field games. Philos. Trans. R. Soc. Lond. A 372, 20130405 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gomes, D.A., Mitake, H.: Existence for stationary mean-field games with congestion and quadratic hamiltonians. Nonlinear Diff. Equ. Appl. NoDEA 22, 1–14 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gomes, D.A., Pimentel, E., Sánchez-Morgado, H.: Time dependent mean-field games in the superquadratic case. arXiv preprint arXiv:1311.6684 (2013)

  15. Gomes, D.A., Pimentel, E., Sánchez-Morgado, H.: Time-dependent mean-field games in the subquadratic case. Commun. Partial Diff. Equ. 40, 40–76 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gomes, D.A., Voskanyan, V.K.: Short-time existence of solutions for mean field games with congestion, preprint

  17. Graber, P.J.: Weak solutions for mean field games with congestion. arXiv preprint arXiv:1503.04733 (2015)

  18. Guéant, O., Lasry, J.-M., Lions, P.-L.: Mean field games and applications. In: Paris-Princeton Lectures on Mathematical Finance 2010, pp. 205–266. Springer (2011)

  19. 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, 221–252 (2006)

    MathSciNet  MATH  Google Scholar 

  20. Ladyzhenskai-A, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasi-linear Equations of Parabolic Type. American Mathematical Society, Providence (1968)

    Google Scholar 

  21. Lasry, J.-M., Lions, P.-L.: Mean field games. Jpn. J. Math. 2, 229–260 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lucas Jr., R.E., Moll, B.: Knowledge growth and the allocation of time, tech. report, National Bureau of Economic Research (2011)

  23. Porretta, A.: Weak solutions to Fokker–Planck equations and mean field games. Arch. Ration. Mech. Anal. 216, 1–62 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. International Center for Decision and Risk Analysis, Naveen Jindal School of Management, The University of Texas at Dallas, 800 West Campbell Rd, SM30, Richardson, TX, 75080-3021, USA

    P. Jameson Graber & Alain Bensoussan

  2. Department of Systems Engineering and Engineering Management, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon Tong, Hong Kong

    Alain Bensoussan

Authors
  1. P. Jameson Graber
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  2. Alain Bensoussan
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Correspondence to P. Jameson Graber.

Additional information

Both authors are grateful to be supported in this work by the National Science Foundation under NSF Grant DMS-1303775. This research is also supported by the Research Grants Council of HKSAR (CityU 500113).

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Graber, P.J., Bensoussan, A. Existence and Uniqueness of Solutions for Bertrand and Cournot Mean Field Games. Appl Math Optim 77, 47–71 (2018). https://doi.org/10.1007/s00245-016-9366-0

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  • DOI: https://doi.org/10.1007/s00245-016-9366-0

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