IDEAS home Printed from https://ideas.repec.org/p/bie/wpaper/743.html

Pasting of Equilibria and Donsker-type Results for Mean Field Games

Author

Listed:
  • Dianetti, Jodi

    (Center for Mathematical Economics, Bielefeld University)

  • Nendel, Max

    (Center for Mathematical Economics, Bielefeld University)

  • Tangpi, Ludovic

    (Center for Mathematical Economics, Bielefeld University)

  • Wang, Shichun

    (Center for Mathematical Economics, Bielefeld University)

Abstract

This paper studies the relation between equilibria in single-period, discrete-time and continuous-time mean field game models. First, for single-period mean field games, we establish the existence of equilibria and then prove the propagation of the Lasry-Lions monotonicity to the optimal equilibrium value, as a function of the realization of the initial condition and its distribution. Secondly, we prove a pasting property for equilibria; that is, we construct equilibria to multi-period discrete-time mean field games by recursively pasting the equilibria of suitably initialized single- period games. Then, we show that any sequence of equilibria of discrete-time mean field games with discretized noise converges (up to a subsequence) to some equilibrium of the continuous-time mean field game as the mesh size of the discretization tends to zero. When the cost functions of the game satisfy the Lasry-Lions monotonicity property, we strengthen this convergence result by providing a sharp convergence rate.

Suggested Citation

  • Dianetti, Jodi & Nendel, Max & Tangpi, Ludovic & Wang, Shichun, 2025. "Pasting of Equilibria and Donsker-type Results for Mean Field Games," Center for Mathematical Economics Working Papers 743, Center for Mathematical Economics, Bielefeld University.
    • Handle: RePEc:bie:wpaper:743
    as

    Download full text from publisher

    File URL: https://pub.uni-bielefeld.de/download/3006246/3006248
    File Function: First Version, 2024
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Chan, K. S. & Stramer, O., 1998. "Weak consistency of the Euler method for numerically solving stochastic differential equations with discontinuous coefficients," Stochastic Processes and their Applications, Elsevier, vol. 76(1), pages 33-44, August.
    2. Martin Hutzenthaler & Arnulf Jentzen & Thomas Kruse & Tuan Anh Nguyen, 2020. "A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations," Partial Differential Equations and Applications, Springer, vol. 1(2), pages 1-34, April.
    3. Jodi Dianetti & Giorgio Ferrari & Markus Fischer & Max Nendel, 2023. "A Unifying Framework for Submodular Mean Field Games," Mathematics of Operations Research, INFORMS, vol. 48(3), pages 1679-1710, August.
    4. Lacker, Daniel, 2015. "Mean field games via controlled martingale problems: Existence of Markovian equilibria," Stochastic Processes and their Applications, Elsevier, vol. 125(7), pages 2856-2894.
    5. Naci Saldi & Tamer Bas¸ ar & Maxim Raginsky, 2020. "Approximate Markov-Nash Equilibria for Discrete-Time Risk-Sensitive Mean-Field Games," Mathematics of Operations Research, INFORMS, vol. 45(4), pages 1596-1620, November.
    6. Gabriel Y. Weintraub & C. Lanier Benkard & Benjamin Van Roy, 2008. "Markov Perfect Industry Dynamics With Many Firms," Econometrica, Econometric Society, vol. 76(6), pages 1375-1411, November.
    7. Bouchard, Bruno & Touzi, Nizar, 2004. "Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 111(2), pages 175-206, June.
    8. Ludovic Tangpi & Shichun Wang, 2022. "Optimal Bubble Riding: A Mean Field Game with Varying Entry Times," Papers 2209.04001, arXiv.org, revised Jan 2024.
    9. Naci Saldi & Tamer Başar & Maxim Raginsky, 2023. "Partially Observed Discrete-Time Risk-Sensitive Mean Field Games," Dynamic Games and Applications, Springer, vol. 13(3), pages 929-960, September.
    10. René Carmona & Daniel B. Cooney & Christy V. Graves & Mathieu Laurière, 2022. "Stochastic Graphon Games: I. The Static Case," Mathematics of Operations Research, INFORMS, vol. 47(1), pages 750-778, February.
    11. Zhang, Xicheng, 2005. "Strong solutions of SDES with singular drift and Sobolev diffusion coefficients," Stochastic Processes and their Applications, Elsevier, vol. 115(11), pages 1805-1818, November.
    Full references (including those not matched with items on IDEAS)

    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. Uğur Aydin & Naci Saldi, 2026. "Robustness and Approximation of Discrete-Time Mean-Field Games Under Discounted Cost Criterion," Mathematics of Operations Research, INFORMS, vol. 51(1), pages 185-217, January.
    2. Daniel Lacker & Agathe Soret, 2023. "A Label-State Formulation of Stochastic Graphon Games and Approximate Equilibria on Large Networks," Mathematics of Operations Research, INFORMS, vol. 48(4), pages 1987-2018, November.
    3. Naci Saldi, 2026. "Linear Mean-Field Games with Discounted Cost," Mathematics of Operations Research, INFORMS, vol. 51(1), pages 358-397, January.
    4. Jodi Dianetti & Giorgio Ferrari & Markus Fischer & Max Nendel, 2023. "A Unifying Framework for Submodular Mean Field Games," Mathematics of Operations Research, INFORMS, vol. 48(3), pages 1679-1710, August.
    5. Han, Qiang & Liu, Yurong, 2026. "Tamed Euler interpolation multilevel Monte Carlo method for quadratic forward backward stochastic differential equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 240(C), pages 20-37.
    6. Grzybowski, Lukasz, 2005. "Essays on Economics of Network Industries: Mobile Telephony," Munich Dissertations in Economics 5561, University of Munich, Department of Economics.
    7. Fujii, Masaaki & Takahashi, Akihiko, 2019. "Solving backward stochastic differential equations with quadratic-growth drivers by connecting the short-term expansions," Stochastic Processes and their Applications, Elsevier, vol. 129(5), pages 1492-1532.
    8. Bouchard Bruno & Tan Xiaolu & Warin Xavier & Zou Yiyi, 2017. "Numerical approximation of BSDEs using local polynomial drivers and branching processes," Monte Carlo Methods and Applications, De Gruyter, vol. 23(4), pages 241-263, December.
    9. Samuel Daudin, 2022. "Optimal Control of Diffusion Processes with Terminal Constraint in Law," Journal of Optimization Theory and Applications, Springer, vol. 195(1), pages 1-41, October.
    10. Régis Chenavaz & Corina Paraschiv & Gabriel Turinici, 2017. "Dynamic Pricing of New Products in Competitive Markets: A Mean-Field Game Approach," Working Papers hal-01592958, HAL.
    11. Light, Bar & Weintraub, Gabriel, 2018. "Mean Field Equilibrium: Uniqueness, Existence, and Comparative Statics," Research Papers 3731, Stanford University, Graduate School of Business.
    12. Corbae, Dean & D’Erasmo, Pablo, 2020. "Rising bank concentration," Journal of Economic Dynamics and Control, Elsevier, vol. 115(C).
    13. C. Lanier Benkard & Przemyslaw Jeziorski & Gabriel Y. Weintraub, 2015. "Oblivious equilibrium for concentrated industries," RAND Journal of Economics, RAND Corporation, vol. 46(4), pages 671-708, October.
    14. Masaaki Fujii & Akihiko Takahashi, 2015. "Perturbative Expansion Technique for Non-linear FBSDEs with Interacting Particle Method," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 22(3), pages 283-304, September.
    15. Masaaki Fujii & Akihiko Takahshi, 2015. "Perturbative Expansion Technique for Non-linear FBSDEs with Interacting Particle Method," CIRJE F-Series CIRJE-F-954, CIRJE, Faculty of Economics, University of Tokyo.
    16. Mark Podolskij & Bezirgen Veliyev & Nakahiro Yoshida, 2018. "Edgeworth expansion for Euler approximation of continuous diffusion processes," CREATES Research Papers 2018-28, Department of Economics and Business Economics, Aarhus University.
    17. Grégory Jolivet & Bruno Jullien & Fabien Postel-Vinay, 2014. "Reputation and Pricing on the e-Market: Evidence from a Major French Platform," Sciences Po Economics Publications (main) hal-03460312, HAL.
    18. Bing-Chang Wang & Minyi Huang, 2024. "Mean Field Social Control for Production Output Adjustment with Noisy Sticky Prices," Dynamic Games and Applications, Springer, vol. 14(3), pages 716-732, July.
    19. Bouveret, Géraldine & Dumitrescu, Roxana & Tankov, Peter, 2022. "Technological change in water use: A mean-field game approach to optimal investment timing," Operations Research Perspectives, Elsevier, vol. 9(C).
    20. Jean-Franc{c}ois Chassagneux & Junchao Chen & Noufel Frikha, 2022. "Deep Runge-Kutta schemes for BSDEs," Papers 2212.14372, arXiv.org.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    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:bie:wpaper:743. 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: Bettina Weingarten (email available below). General contact details of provider: https://edirc.repec.org/data/imbiede.html .

    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.