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Hierarchical Structures and Leadership Design in Mean-Field-Type Games with Polynomial Cost

Author

Listed:
  • Zahrate El Oula Frihi

    (Lab. of Probability and Statistics (LaPS), Department of Mathematics, Badji-Mokhtar University, B.P.12, Annaba 23000, Algeria)

  • Julian Barreiro-Gomez

    (Learning & Game Theory Laboratory, Engineering Division, New York University Abu Dhabi, Saadiyat Campus, PO Box 129188, Abu Dhabi 44966, UAE
    Research Center on Stability, Instability and Turbulence, New York University Abu Dhabi, Abu Dhabi 44966, UAE)

  • Salah Eddine Choutri

    (Learning & Game Theory Laboratory, Engineering Division, New York University Abu Dhabi, Saadiyat Campus, PO Box 129188, Abu Dhabi 44966, UAE
    Research Center on Stability, Instability and Turbulence, New York University Abu Dhabi, Abu Dhabi 44966, UAE)

  • Hamidou Tembine

    (Learning & Game Theory Laboratory, Engineering Division, New York University Abu Dhabi, Saadiyat Campus, PO Box 129188, Abu Dhabi 44966, UAE
    Research Center on Stability, Instability and Turbulence, New York University Abu Dhabi, Abu Dhabi 44966, UAE)

Abstract

This article presents a class of hierarchical mean-field-type games with multiple layers and non-quadratic polynomial costs. The decision-makers act in sequential order with informational differences. We first examine the single-layer case where each decision-maker does not have the information about the other control strategies. We derive the Nash mean-field-type equilibrium and cost in a linear state-and-mean-field feedback form by using a partial integro-differential system. Then, we examine the Stackelberg two-layer problem with multiple leaders and multiple followers. Numerical illustrations show that, in the symmetric case, having only one leader is not necessarily optimal for the total sum cost. Having too many leaders may also be suboptimal for the total sum cost. The methodology is extended to multi-level hierarchical systems. It is shown that the order of the play plays a key role in the total performance of the system. We also identify a specific range of parameters for which the Nash equilibrium coincides with the hierarchical solution independently of the number of layers and the order of play. In the heterogeneous case, it is shown that the total cost is significantly affected by the design of the hierarchical structure of the problem.

Suggested Citation

  • Zahrate El Oula Frihi & Julian Barreiro-Gomez & Salah Eddine Choutri & Hamidou Tembine, 2020. "Hierarchical Structures and Leadership Design in Mean-Field-Type Games with Polynomial Cost," Games, MDPI, vol. 11(3), pages 1-26, August.
    • Handle: RePEc:gam:jgames:v:11:y:2020:i:3:p:30-:d:395498
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    References listed on IDEAS

    as
    1. Panayiotis Theodossiou & Christos S. Savva, 2016. "Skewness and the Relation Between Risk and Return," Management Science, INFORMS, vol. 62(6), pages 1598-1609, June.
    2. Jingtao Shi, 2018. "Stochastic Leader-Follower Differential Game with Asymmetric Information," Chapters, in: Danijela Tuljak-Suban (ed.), Game Theory - Applications in Logistics and Economy, IntechOpen.
    3. 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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