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A Method for Multi-Leader–Multi-Follower Games by Smoothing the Followers’ Response Function

Author

Listed:
  • Atsushi Hori

    (Seikei University)

  • Daisuke Tsuyuguchi

    (Wakayama Prefectural Board of Education)

  • Ellen H. Fukuda

    (Kyoto University)

Abstract

The multi-leader–multi-follower game (MLMFG) involves two or more leaders and followers and serves as a generalization of the Stackelberg game and the single-leader–multi-follower game. Although MLMFG covers wide range of real-world applications, its research is still sparse. Notably, fundamental solution methods for this class of problems remain insufficiently established. A prevailing approach is to recast the MLMFG as an equilibrium problem with equilibrium constraints (EPEC) and solve it using a solver. Meanwhile, interpreting the solution to the EPEC in the context of MLMFG may be complex due to shared decision variables among all leaders, followers’ strategies that each leader can unilaterally change, but the variables are essentially controlled by followers. To address this issue, we introduce a response function of followers’ noncooperative game that is a function with leaders’ strategies as a variable. Employing this approach allows the MLMFG to be solved as a single-level differentiable variational inequality using a smoothing scheme for the followers’ response function. We also demonstrate that the sequence of solutions to the smoothed variational inequality converges to a stationary equilibrium of the MLMFG. Finally, we illustrate the behavior of the smoothing method by numerical experiments.

Suggested Citation

  • Atsushi Hori & Daisuke Tsuyuguchi & Ellen H. Fukuda, 2024. "A Method for Multi-Leader–Multi-Follower Games by Smoothing the Followers’ Response Function," Journal of Optimization Theory and Applications, Springer, vol. 203(1), pages 305-335, October.
  • Handle: RePEc:spr:joptap:v:203:y:2024:i:1:d:10.1007_s10957-024-02506-2
    DOI: 10.1007/s10957-024-02506-2
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    References listed on IDEAS

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    1. Hanif D. Sherali, 1984. "A Multiple Leader Stackelberg Model and Analysis," Operations Research, INFORMS, vol. 32(2), pages 390-404, April.
    2. Atsushi Hori & Masao Fukushima, 2019. "Gauss–Seidel Method for Multi-leader–follower Games," Journal of Optimization Theory and Applications, Springer, vol. 180(2), pages 651-670, February.
    3. Ming Hu & Masao Fukushima, 2012. "Smoothing approach to Nash equilibrium formulations for a class of equilibrium problems with shared complementarity constraints," Computational Optimization and Applications, Springer, vol. 52(2), pages 415-437, June.
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