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Convergence, fluctuations and large deviations for finite state mean field games via the Master Equation

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  • Cecchin, Alekos
  • Pelino, Guglielmo

Abstract

We show the convergence of finite state symmetric N-player differential games, where players control their transition rates from state to state, to a limiting dynamics given by a finite state Mean Field Game system made of two coupled forward–backward ODEs. We exploit the so-called Master Equation, which in this finite-dimensional framework is a first order PDE in the simplex of probability measures, obtaining the convergence of the feedback Nash equilibria, the value functions and the optimal trajectories. The convergence argument requires only the regularity of a solution to the Master Equation. Moreover, we employ the convergence results to prove a Central Limit Theorem and a Large Deviation Principle for the evolution of the N-player empirical measures. The well-posedness and regularity of solution to the Master Equation are also studied, under monotonicity assumptions.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:11:p:4510-4555
    DOI: 10.1016/j.spa.2018.12.002
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    References listed on IDEAS

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    1. A. Bensoussan & K. C. J. Sung & S. C. P. Yam & S. P. Yung, 2016. "Linear-Quadratic Mean Field Games," Journal of Optimization Theory and Applications, Springer, vol. 169(2), pages 496-529, May.
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    4. Adam Shwartz & Alan Weiss, 2005. "Large Deviations with Diminishing Rates," Mathematics of Operations Research, INFORMS, vol. 30(2), pages 281-310, May.
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    Cited by:

    1. Asaf Cohen & Ethan Zell, 2026. "Asymptotic Nash Equilibria of Finite-State Ergodic Markovian Mean Field Games," Mathematics of Operations Research, INFORMS, vol. 51(2), pages 1139-1173, May.
    2. Talbi, Mehdi, 2024. "A finite-dimensional approximation for partial differential equations on Wasserstein space," Stochastic Processes and their Applications, Elsevier, vol. 177(C).
    3. Jialiang Luo & Harry Zheng, 2023. "Deep Neural Network Solution for Finite State Mean Field Game with Error Estimation," Dynamic Games and Applications, Springer, vol. 13(3), pages 859-896, September.
    4. Christoph Belak & Daniel Hoffmann & Frank T. Seifried, 2020. "Continuous-Time Mean Field Games with Finite StateSpace and Common Noise," Working Paper Series 2020-05, University of Trier, Research Group Quantitative Finance and Risk Analysis.
    5. 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.
    6. Erhan Bayraktar & Alekos Cecchin & Asaf Cohen & François Delarue, 2022. "Finite State Mean Field Games with Wright–Fisher Common Noise as Limits of N -Player Weighted Games," Mathematics of Operations Research, INFORMS, vol. 47(4), pages 2840-2890, November.

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