Hierarchical structures and leadership design in mean-field-type games with polynomial cost

Zahrate El Oula Frihi, Julian Barreiro-Gomez, Salah Eddine Choutri, Hamidou Tembine

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Article number30
Pages (from-to)1-26
Number of pages26
Issue number3
StatePublished - Sep 2020


  • Design of hierarchical structure
  • Mean-field-type games
  • Mean-field-type hierarchical control

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty
  • Applied Mathematics


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