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A Probabilistic Approach to Extended Finite State Mean Field Games

Author

Listed:
  • René Carmona

    (Department of Operations Research and Financial Engineering, Princeton University, Princeton, New Jersey 08544)

  • Peiqi Wang

    (Bank of America, New York, New York 10281)

Abstract

We develop a probabilistic approach to continuous-time finite state mean field games. Based on an alternative description of continuous-time Markov chains by means of semimartingales and the weak formulation of stochastic optimal control, our approach not only allows us to tackle the mean field of states and the mean field of control at the same time, but also extends the strategy set of players from Markov strategies to closed-loop strategies. We show the existence and uniqueness of Nash equilibrium for the mean field game as well as how the equilibrium of a mean field game consists of an approximative Nash equilibrium for the game with a finite number of players under different assumptions of structure and regularity on the cost functions and transition rate between states.

Suggested Citation

  • René Carmona & Peiqi Wang, 2021. "A Probabilistic Approach to Extended Finite State Mean Field Games," Mathematics of Operations Research, INFORMS, vol. 46(2), pages 471-502, May.
    • Handle: RePEc:inm:ormoor:v:46:y:2021:i:2:p:471-502
    DOI: 10.1287/moor.2020.1071
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    References listed on IDEAS

    as
    1. Pierre Cardaliaguet & Charles-Albert Lehalle, 2016. "Mean Field Game of Controls and An Application To Trade Crowding," Papers 1610.09904, arXiv.org, revised Sep 2017.
    2. Romuald Elie & Thibaut Mastrolia & Dylan Possamaï, 2019. "A Tale of a Principal and Many, Many Agents," Mathematics of Operations Research, INFORMS, vol. 44(2), pages 440-467, May.
    3. Cecchin, Alekos & Pelino, Guglielmo, 2019. "Convergence, fluctuations and large deviations for finite state mean field games via the Master Equation," Stochastic Processes and their Applications, Elsevier, vol. 129(11), pages 4510-4555.
    4. Samuel N. Cohen & Robert J. Elliott, 2008. "Comparisons for backward stochastic differential equations on Markov chains and related no-arbitrage conditions," Papers 0810.0055, arXiv.org, revised Jan 2010.
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    Citations

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    Cited by:

    1. Alexander Aurell & René Carmona & Gökçe Dayanıklı & Mathieu Laurière, 2022. "Finite State Graphon Games with Applications to Epidemics," Dynamic Games and Applications, Springer, vol. 12(1), pages 49-81, March.
    2. Berenice Anne Neumann, 2024. "A myopic adjustment process for mean field games with finite state and action space," International Journal of Game Theory, Springer;Game Theory Society, vol. 53(1), pages 159-195, March.
    3. Yurii Averboukh & Aleksei Volkov, 2024. "Planning Problem for Continuous-Time Finite State Mean Field Game with Compact Action Space," Dynamic Games and Applications, Springer, vol. 14(2), pages 285-303, May.
    4. Lu-ping Liu & Wen-sheng Jia, 2024. "Well-Posedness for Mean Field Games with Finite State and Action Space," Journal of Optimization Theory and Applications, Springer, vol. 201(1), pages 36-53, April.
    5. René Carmona, 2022. "The influence of economic research on financial mathematics: Evidence from the last 25 years," Finance and Stochastics, Springer, vol. 26(1), pages 85-101, January.

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