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

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  • 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
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    References listed on IDEAS

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    1. Olivier Guéant & Pierre Louis Lions & Jean-Michel Lasry, 2011. "Mean Field Games and Applications," Post-Print hal-01393103, HAL.
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    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. 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.
    4. Marco Cirant & Davide Francesco Redaelli, 2025. "Some Remarks on Linear-Quadratic Closed-Loop Games with Many Players," Dynamic Games and Applications, Springer, vol. 15(2), pages 558-591, May.
    5. 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.
    6. Daria Ghilli & Cristiano Ricci & Giovanni Zanco, 2024. "A mean field game model for COVID-19 with human capital accumulation," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 77(1), pages 533-560, February.
    7. 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.
    8. Piotr Więcek, 2024. "Multiple-Population Discrete-Time Mean Field Games with Discounted and Total Payoffs: The Existence of Equilibria," Dynamic Games and Applications, Springer, vol. 14(4), pages 997-1026, September.
    9. 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.
    10. 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.
    11. 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.
    12. Diogo Gomes & João Saúde, 2014. "Mean Field Games Models—A Brief Survey," Dynamic Games and Applications, Springer, vol. 4(2), pages 110-154, June.
    13. 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.

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