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Probabilistic Approach to Mean Field Games and Mean Field Type Control Problems with Multiple Populations

Author

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  • Masaaki Fujii

    (Quantitative Finance Course, Graduate School of Economics, The University of Tokyo)

Abstract

In this work, we systematically investigate mean field games and mean field type control problems with multiple populations. We study the mean field limits of the three different situations; (i) every agent is non-cooperative; (ii) the agents within each population are cooperative; and (iii) the agents in some populations are cooperative. We provide several sets of sufficient conditions for the existence of a mean field equilibrium for each case. We also show that, under appropriate conditions, each mean field solution actually provides an approximate Nash equilibrium for the corresponding game with a large but finite number of agents.

Suggested Citation

  • 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.
    • Handle: RePEc:cfi:fseres:cf497
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    References listed on IDEAS

    as
    1. Masaaki Fujii & Akihiko Takahashi & Masayuki Takahashi, 2017. "Asymptotic Expansion as Prior Knowledge in Deep Learning Method for high dimensional BSDEs," Papers 1710.07030, arXiv.org, revised Mar 2019.
    2. Masaaki Fujii & Akihiko Takahashi, 2020. "A Finite Agent Equilibrium in an Incomplete Market and its Strong Convergence to the Mean-Field Limit," CARF F-Series CARF-F-495, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    3. Fujii, Masaaki & Takahashi, Akihiko, 2018. "Quadratic–exponential growth BSDEs with jumps and their Malliavin’s differentiability," Stochastic Processes and their Applications, Elsevier, vol. 128(6), pages 2083-2130.
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    7. Masaaki Fujii & Akihiko Takahashi, 2020. "A Mean Field Game Approach to Equilibrium Pricing with Market Clearing Condition," CARF F-Series CARF-F-473, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    8. Clémence Alasseur & Imen Ben Taher & Anis Matoussi, 2020. "An Extended Mean Field Game for Storage in Smart Grids," Journal of Optimization Theory and Applications, Springer, vol. 184(2), pages 644-670, February.
    9. Charles-Albert Lehalle & Charafeddine Mouzouni, 2019. "A mean field game of portfolio trading and its consequences on perceived correlations," Working Papers hal-02003143, HAL.
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    11. Guanxing Fu & Ulrich Horst, 2018. "Mean-Field Leader-Follower Games with Terminal State Constraint," Papers 1809.04401, arXiv.org.
    12. 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.
    13. Olivier F'eron & Peter Tankov & Laura Tinsi, 2020. "Price formation and optimal trading in intraday electricity markets," Papers 2009.04786, arXiv.org, revised Jun 2021.
    14. Masaaki Fujii & Akihiko Takahashi, 2020. "A Finite Agent Equilibrium in an Incomplete Market and its Strong Convergence to the Mean-Field Limit," CIRJE F-Series CIRJE-F-1156, CIRJE, Faculty of Economics, University of Tokyo.
    15. Masaaki Fujii & Akihiko Takahashi, 2020. "Strong Convergence to the Mean-Field Limit of A Finite Agent Equilibrium," Papers 2010.09186, arXiv.org, revised Dec 2021.
    16. Masaaki Fujii & Akihiko Takahashi & Masayuki Takahashi, 2019. "Asymptotic Expansion as Prior Knowledge in Deep Learning Method for High dimensional BSDEs," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 26(3), pages 391-408, September.
    17. Li-Hsien Sun, 2019. "Systemic Risk and Heterogeneous Mean Field Type Interbank Network," Papers 1907.03082, arXiv.org, revised Sep 2019.
    18. Morlais, Marie-Amelie, 2010. "A new existence result for quadratic BSDEs with jumps with application to the utility maximization problem," Stochastic Processes and their Applications, Elsevier, vol. 120(10), pages 1966-1995, September.
    19. Masaaki Fujii & Akihiko Takahashi, 2020. "A Mean Field Game Approach to Equilibrium Pricing with Market Clearing Condition," CIRJE F-Series CIRJE-F-1144, CIRJE, Faculty of Economics, University of Tokyo.
    20. David Evangelista & Yuri Thamsten, 2020. "On finite population games of optimal trading," Papers 2004.00790, arXiv.org, revised Feb 2021.
    21. Olivier F'eron & Peter Tankov & Laura Tinsi, 2020. "Price formation and optimal trading in intraday electricity markets with a major player," Papers 2011.07655, arXiv.org.
    22. Masaaki Fujii & Akihiko Takahashi & Masayuki Takahashi, 2019. "Asymptotic Expansion as Prior Knowledge in Deep Learning Method for high dimensional BSDEs (Forthcoming in Asia-Pacific Financial Markets)," CARF F-Series CARF-F-456, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    23. Masaaki Fujii & Akihiko Takahashi, 2020. "A Mean Field Game Approach to Equilibrium Pricing with Market Clearing Condition," Papers 2003.03035, arXiv.org, revised Sep 2021.
    24. 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.
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