IDEAS home Printed from https://ideas.repec.org/a/spr/dyngam/v3y2013i4p523-536.html
   My bibliography  Save this article

The Derivation of Ergodic Mean Field Game Equations for Several Populations of Players

Author

Listed:
  • Ermal Feleqi

Abstract

This note contains a detailed derivation of the equations of the recent mean field games theory (abbr. MFG), developed by M. Huang, P.E. Caines, and R.P. Malhamé on one hand and by J.-M. Lasry and P.-L. Lions on the other, associated with a class of stochastic differential games, where the players belong to several populations, each of which consisting of a large number of similar and indistinguishable individuals, in the context of periodic diffusions and long-time-average (or ergodic) costs. After introducing a system of N Hamilton–Jacobi–Bellman (abbr. HJB) and N Kolmogorov–Fokker–Planck (abbr. KFP) equations for an N-player game belonging to such a class of games, the system of MFG equations (consisting of as many HJB equations, and of as many KFP equations as is the number of populations) is derived by letting the number of the members of each population go to infinity. For the sake of clarity and for reader’s convenience, the case of a single population of players, as formulated in the work of J.-M. Lasry and P.-L. Lions, is presented first. The note slightly improves the results in this case too, by dealing with more general dynamics and costs. Copyright Springer Science+Business Media New York 2013

Suggested Citation

  • Ermal Feleqi, 2013. "The Derivation of Ergodic Mean Field Game Equations for Several Populations of Players," Dynamic Games and Applications, Springer, vol. 3(4), pages 523-536, December.
    • Handle: RePEc:spr:dyngam:v:3:y:2013:i:4:p:523-536
    DOI: 10.1007/s13235-013-0088-5
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s13235-013-0088-5
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s13235-013-0088-5?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(4), pages 691-705, August.
    2. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(1), pages 225-228, February.
    3. Olivier Guéant & Pierre Louis Lions & Jean-Michel Lasry, 2011. "Mean Field Games and Applications," Post-Print hal-01393103, HAL.
    4. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(5), pages 879-883, October.
    5. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(2), pages 411-413, April.
    6. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(6), pages 1195-1198, December.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Piotr Więcek, 2020. "Discrete-Time Ergodic Mean-Field Games with Average Reward on Compact Spaces," Dynamic Games and Applications, Springer, vol. 10(1), pages 222-256, March.
    2. Masaaki Fujii & Akihiko Takahashi, 2021. "Equilibrium Price Formation with a Major Player and its Mean Field Limit," CARF F-Series CARF-F-509, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    3. Masaaki Fujii & Akihiko Takahashi, 2021. "``Equilibrium Price Formation with a Major Player and its Mean Field Limit''," CIRJE F-Series CIRJE-F-1162, CIRJE, Faculty of Economics, University of Tokyo.
    4. Aurell, Alexander & Djehiche, Boualem, 2019. "Modeling tagged pedestrian motion: A mean-field type game approach," Transportation Research Part B: Methodological, Elsevier, vol. 121(C), pages 168-183.
    5. Masaaki Fujii, 2019. "Probabilistic Approach to Mean Field Games and Mean Field Type Control Problems with Multiple Populations," Papers 1911.11501, arXiv.org, revised Nov 2020.
    6. Diogo Gomes & João Saúde, 2014. "Mean Field Games Models—A Brief Survey," Dynamic Games and Applications, Springer, vol. 4(2), pages 110-154, June.
    7. Masaaki Fujii, 2020. "Probabilistic Approach to Mean Field Games and Mean Field Type Control Problems with Multiple Populations," CARF F-Series CARF-F-497, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    8. Masaaki Fujii, 2019. "Probabilistic Approach to Mean Field Games and Mean Field Type Control Problems with Multiple Populations," CARF F-Series CARF-F-467, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    9. Masaaki Fujii & Akihiko Takahashi, 2021. "Equilibrium Price Formation with a Major Player and its Mean Field Limit," Papers 2102.10756, arXiv.org, revised Feb 2022.
    10. Eduardo Abi Jaber & Eyal Neuman & Moritz Vo{ss}, 2023. "Equilibrium in Functional Stochastic Games with Mean-Field Interaction," Papers 2306.05433, arXiv.org, revised Feb 2024.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. A. Bensoussan & K. Sung & S. Yam, 2013. "Linear–Quadratic Time-Inconsistent Mean Field Games," Dynamic Games and Applications, Springer, vol. 3(4), pages 537-552, December.
    2. A. Bensoussan & K. C. J. Sung & S. C. P. Yam & S. P. Yung, 2016. "Linear-Quadratic Mean Field Games," Journal of Optimization Theory and Applications, Springer, vol. 169(2), pages 496-529, May.
    3. Fabio Priuli, 2015. "Linear-Quadratic $$N$$ N -Person and Mean-Field Games: Infinite Horizon Games with Discounted Cost and Singular Limits," Dynamic Games and Applications, Springer, vol. 5(3), pages 397-419, September.
    4. Minyi Huang, 2013. "A Mean Field Capital Accumulation Game with HARA Utility," Dynamic Games and Applications, Springer, vol. 3(4), pages 446-472, December.
    5. Dario Bauso & Raffaele Pesenti & Marco Tolotti, 2016. "Opinion Dynamics and Stubbornness Via Multi-Population Mean-Field Games," Journal of Optimization Theory and Applications, Springer, vol. 170(1), pages 266-293, July.
    6. Paolo Dai Pra & Elena Sartori & Marco Tolotti, 2019. "Climb on the Bandwagon: Consensus and Periodicity in a Lifetime Utility Model with Strategic Interactions," Dynamic Games and Applications, Springer, vol. 9(4), pages 1061-1075, December.
    7. Yakut, Oguz, 2021. "Implementation of hydraulically driven barrel shooting control by utilizing artificial neural networks," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 1206-1223.
    8. X. Qin & G. Huang, 2009. "An Inexact Chance-constrained Quadratic Programming Model for Stream Water Quality Management," Water Resources Management: An International Journal, Published for the European Water Resources Association (EWRA), Springer;European Water Resources Association (EWRA), vol. 23(4), pages 661-695, March.
    9. Md. Yousuf Gazi & Khandakar Tahmida Tafhim, 2019. "Investigation of Heavy-mineral Deposits Using Multispectral Satellite Imagery in the Eastern Coastal Margin of Bangladesh," Earth Sciences Malaysia (ESMY), Zibeline International Publishing, vol. 3(2), pages 16-22, October.
    10. Billionnet, Alain, 2011. "Solving the probabilistic reserve selection problem," Ecological Modelling, Elsevier, vol. 222(3), pages 546-554.
    11. Minghe Sun, 2005. "Warm-Start Routines for Solving Augmented Weighted Tchebycheff Network Programs in Multiple-Objective Network Programming," INFORMS Journal on Computing, INFORMS, vol. 17(4), pages 422-437, November.
    12. François Clautiaux & Cláudio Alves & José Valério de Carvalho & Jürgen Rietz, 2011. "New Stabilization Procedures for the Cutting Stock Problem," INFORMS Journal on Computing, INFORMS, vol. 23(4), pages 530-545, November.
    13. Eichengreen, Barry & Kletzer, Kenneth & Mody, Ashoka, 2003. "Crisis Resolution: Next Steps," Santa Cruz Center for International Economics, Working Paper Series qt4cj974r4, Center for International Economics, UC Santa Cruz.
    14. Tansel, Aysit & Karao?lan, Deniz, 2016. "The Causal Effect of Education on Health Behaviors: Evidence from Turkey," IZA Discussion Papers 10020, Institute of Labor Economics (IZA).
    15. Di Feng & Bettina Klaus, 2022. "Preference revelation games and strict cores of multiple‐type housing market problems," International Journal of Economic Theory, The International Society for Economic Theory, vol. 18(1), pages 61-76, March.
    16. Anna Scherbina, 2021. "Assessing the Optimality of a COVID Lockdown in the United States," Economics of Disasters and Climate Change, Springer, vol. 5(2), pages 177-201, July.
    17. John McKay, 2005. "How Significant and Effective are North Korea's "Market Reforms"?," Global Economic Review, Taylor & Francis Journals, vol. 34(1), pages 83-97.
    18. Timothy K.M. Beatty & Erling Røed Larsen & Dag Einar Sommervoll, 2005. "Measuring the Price of Housing Consumption for Owners in the CPI," Discussion Papers 427, Statistics Norway, Research Department.
    19. Marco Bianchi & Carlos Tapia & Ikerne del Valle, 2020. "Monitoring domestic material consumption at lower territorial levels: A novel data downscaling method," Journal of Industrial Ecology, Yale University, vol. 24(5), pages 1074-1087, October.
    20. Sonmez, Tayfun & Utku Unver, M., 2005. "House allocation with existing tenants: an equivalence," Games and Economic Behavior, Elsevier, vol. 52(1), pages 153-185, July.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:dyngam:v:3:y:2013:i:4:p:523-536. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.