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Planning Problem for Continuous-Time Finite State Mean Field Game with Compact Action Space

Author

Listed:
  • Yurii Averboukh

    (Higher School of Economics
    Krasovskii Institute of Mathematics and Mechanics)

  • Aleksei Volkov

    (Krasovskii Institute of Mathematics and Mechanics)

Abstract

The planning problem for the mean field game implies that one tries to transfer the system of infinitely many identical rational agents from the given distribution to the final one using the choice of the terminal payoff. It can be formulated as the mean field game system with the boundary condition only on the distribution. In the paper, we consider the continuous-time finite state mean field game, assuming that the space of actions for each player is compact. It is shown that the planning problem in this case may not admit a solution even if the final distribution is reachable from the initial one. Further, we introduce the concept of generalized solution of the planning problem for the finite state mean field game based on the minimization of regret of a fictitious player. This minimal regret solution always exists. Additionally, the set of minimal regret solution is the closure of the set of classical solution of the planning problem, provided that the latter is nonempty. Finally, we examine the uniqueness of the solution to the planning problem using the Lasry–Lions monotonicity arguments.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:dyngam:v:14:y:2024:i:2:d:10.1007_s13235-023-00492-0
    DOI: 10.1007/s13235-023-00492-0
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    References listed on IDEAS

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    1. 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.
    2. Alessio Porretta, 2014. "On the Planning Problem for the Mean Field Games System," Dynamic Games and Applications, Springer, vol. 4(2), pages 231-256, June.
    3. V. N. Kolokoltsov & O. A. Malafeyev, 2017. "Mean-Field-Game Model of Corruption," Dynamic Games and Applications, Springer, vol. 7(1), pages 34-47, March.
    4. 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.
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