IDEAS home Printed from https://ideas.repec.org/a/gam/jrisks/v6y2018i3p90-d167248.html
   My bibliography  Save this article

Mean Field Game with Delay: A Toy Model

Author

Listed:
  • Jean-Pierre Fouque

    (Department of Statistics & Applied Probability, University of California, Santa Barbara, CA 93106-3110, USA
    Work supported by NSF grants DMS-1409434 and DMS-1814091.)

  • Zhaoyu Zhang

    (Department of Statistics & Applied Probability, University of California, Santa Barbara, CA 93106-3110, USA)

Abstract

We study a toy model of linear-quadratic mean field game with delay. We “lift” the delayed dynamic into an infinite dimensional space, and recast the mean field game system which is made of a forward Kolmogorov equation and a backward Hamilton-Jacobi-Bellman equation. We identify the corresponding master equation. A solution to this master equation is computed, and we show that it provides an approximation to a Nash equilibrium of the finite player game.

Suggested Citation

  • Jean-Pierre Fouque & Zhaoyu Zhang, 2018. "Mean Field Game with Delay: A Toy Model," Risks, MDPI, vol. 6(3), pages 1-17, September.
    • Handle: RePEc:gam:jrisks:v:6:y:2018:i:3:p:90-:d:167248
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-9091/6/3/90/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-9091/6/3/90/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Vassili Kolokoltsov & Marianna Troeva & Wei Yang, 2014. "On the Rate of Convergence for the Mean-Field Approximation of Controlled Diffusions with Large Number of Players," Dynamic Games and Applications, Springer, vol. 4(2), pages 208-230, June.
    2. René Carmona & Jean-Pierre Fouque & Seyyed Mostafa Mousavi & Li-Hsien Sun, 2018. "Systemic Risk and Stochastic Games with Delay," Journal of Optimization Theory and Applications, Springer, vol. 179(2), pages 366-399, November.
    3. Giorgio Fabbri & Fausto Gozzi & Andrzej Swiech, 2017. "Stochastic Optimal Control in Infinite Dimensions - Dynamic Programming and HJB Equations," Post-Print hal-01505767, HAL.
    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. Zachary Feinstein & Andreas Sojmark, 2019. "A Dynamic Default Contagion Model: From Eisenberg-Noe to the Mean Field," Papers 1912.08695, arXiv.org.
    2. Hanchao Liu & Dena Firoozi, 2024. "Hilbert Space-Valued LQ Mean Field Games: An Infinite-Dimensional Analysis," Papers 2403.01012, arXiv.org, revised Jul 2024.
    3. Djehiche, Boualem & Gozzi, Fausto & Zanco, Giovanni & Zanella, Margherita, 2022. "Optimal portfolio choice with path dependent benchmarked labor income: A mean field model," Stochastic Processes and their Applications, Elsevier, vol. 145(C), pages 48-85.
    4. 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.
    5. Li-Hsien Sun, 2022. "Mean Field Games with Heterogeneous Groups: Application to Banking Systems," Journal of Optimization Theory and Applications, Springer, vol. 192(1), pages 130-167, January.
    6. Li-Hsien Sun, 2019. "Systemic Risk and Heterogeneous Mean Field Type Interbank Network," Papers 1907.03082, arXiv.org, revised Sep 2019.

    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. Yan, Tingjin & Chiu, Mei Choi & Wong, Hoi Ying, 2023. "Portfolio liquidation with delayed information," Economic Modelling, Elsevier, vol. 126(C).
    2. Robert Balkin & Hector D. Ceniceros & Ruimeng Hu, 2023. "Stochastic Delay Differential Games: Financial Modeling and Machine Learning Algorithms," Papers 2307.06450, arXiv.org.
    3. Hanchao Liu & Dena Firoozi, 2024. "Hilbert Space-Valued LQ Mean Field Games: An Infinite-Dimensional Analysis," Papers 2403.01012, arXiv.org, revised Jul 2024.
    4. Emmanuelle Augeraud-Véron & Raouf Boucekkine & Vladimir Veliov, 2019. "Distributed Optimal Control Models in Environmental Economics: A Review," AMSE Working Papers 1902, Aix-Marseille School of Economics, France.
    5. Tiago Roux Oliveira & Victor Hugo Pereira Rodrigues & Miroslav Krstić & Tamer Başar, 2021. "Nash Equilibrium Seeking in Quadratic Noncooperative Games Under Two Delayed Information-Sharing Schemes," Journal of Optimization Theory and Applications, Springer, vol. 191(2), pages 700-735, December.
    6. William Lefebvre & Enzo Miller, 2021. "Linear-quadratic stochastic delayed control and deep learning resolution," Working Papers hal-03145949, HAL.
    7. Arvind Shrivats & Dena Firoozi & Sebastian Jaimungal, 2020. "A Mean-Field Game Approach to Equilibrium Pricing in Solar Renewable Energy Certificate Markets," Papers 2003.04938, arXiv.org, revised Aug 2021.
    8. William Lefebvre & Enzo Miller, 2021. "Linear-quadratic stochastic delayed control and deep learning resolution," Papers 2102.09851, arXiv.org, revised Feb 2021.
    9. Li-Hsien Sun, 2022. "Mean Field Games with Heterogeneous Groups: Application to Banking Systems," Journal of Optimization Theory and Applications, Springer, vol. 192(1), pages 130-167, January.
    10. Boucekkine, Raouf & Fabbri, Giorgio & Federico, Salvatore & Gozzi, Fausto, 2022. "A dynamic theory of spatial externalities," Games and Economic Behavior, Elsevier, vol. 132(C), pages 133-165.
    11. Raouf Boucekkine & Giorgio Fabbri & Salvatore Federico & Fausto Gozzi, 2019. "Growth and agglomeration in the heterogeneous space: a generalized AK approach," Journal of Economic Geography, Oxford University Press, vol. 19(6), pages 1287-1318.
    12. Arvind V. Shrivats & Dena Firoozi & Sebastian Jaimungal, 2022. "A mean‐field game approach to equilibrium pricing in solar renewable energy certificate markets," Mathematical Finance, Wiley Blackwell, vol. 32(3), pages 779-824, July.
    13. Bruno Bouchard & Boualem Djehiche & Idris Kharroubi, 2020. "Quenched Mass Transport of Particles Toward a Target," Journal of Optimization Theory and Applications, Springer, vol. 186(2), pages 345-374, August.
    14. Masiero, Federica & Orrieri, Carlo & Tessitore, Gianmario & Zanco, Giovanni, 2021. "Semilinear Kolmogorov equations on the space of continuous functions via BSDEs," Stochastic Processes and their Applications, Elsevier, vol. 136(C), pages 1-56.
    15. Raouf Boucekkine & Giorgio Fabbri & Salvatore Federico & Fausto Gozzi, 2020. "A dynamic theory of spatial externalities," Working Papers halshs-02613177, HAL.
    16. William Lefebvre & Enzo Miller, 2021. "Linear-quadratic stochastic delayed control and deep learning resolution," Post-Print hal-03145949, HAL.
    17. Alain Bensoussan & Boualem Djehiche & Hamidou Tembine & Sheung Chi Phillip Yam, 2020. "Mean-Field-Type Games with Jump and Regime Switching," Dynamic Games and Applications, Springer, vol. 10(1), pages 19-57, March.
    18. Dena Firoozi & Arvind V Shrivats & Sebastian Jaimungal, 2021. "Principal agent mean field games in REC markets," Papers 2112.11963, arXiv.org, revised Jun 2022.
    19. Alexander M. G. Cox & Sigrid Kallblad & Martin Larsson & Sara Svaluto-Ferro, 2021. "Controlled Measure-Valued Martingales: a Viscosity Solution Approach," Papers 2109.00064, arXiv.org, revised Aug 2023.
    20. Faggian, Silvia & Gozzi, Fausto & Kort, Peter M., 2021. "Optimal investment with vintage capital: Equilibrium distributions," Journal of Mathematical Economics, Elsevier, vol. 96(C).

    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:gam:jrisks:v:6:y:2018:i:3:p:90-:d:167248. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.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.