IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v180y2019i2d10.1007_s10957-018-1391-5.html
   My bibliography  Save this article

Gauss–Seidel Method for Multi-leader–follower Games

Author

Listed:
  • Atsushi Hori

    (Nanzan University)

  • Masao Fukushima

    (Nanzan University)

Abstract

The multi-leader–follower game has many applications such as the bilevel structured market in which two or more enterprises, called leaders, have initiatives, and the other firms, called followers, observe the leaders’ decisions and then decide their own strategies. A special case of the game is the Stackelberg model, or the single-leader–follower game, which has been studied for many years. The Stackelberg game may be reformulated as a mathematical program with equilibrium constraints, which has also been studied extensively in recent years. On the other hand, the multi-leader–follower game may be formulated as an equilibrium problem with equilibrium constraints, in which each leader’s problem is an mathematical program with equilibrium constraints. However, finding an equilibrium point of an equilibrium problem with equilibrium constraints is much more difficult than solving a single mathematical program with equilibrium constraints, because each leader’s problem contains those variables which are common to other players’ problems. Moreover, the constraints of each leader’s problem depend on the other rival leaders’ strategies. In this paper, we propose a Gauss–Seidel type algorithm with a penalty technique for solving an equilibrium problem with equilibrium constraints associated with the multi-leader–follower game, and then suggest a refinement procedure to obtain more accurate solutions. We discuss convergence of the algorithm and report some numerical results to illustrate the behavior of the algorithm.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:180:y:2019:i:2:d:10.1007_s10957-018-1391-5
    DOI: 10.1007/s10957-018-1391-5
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-018-1391-5
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-018-1391-5?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Ming Hu & Masao Fukushima, 2011. "Variational Inequality Formulation of a Class of Multi-Leader-Follower Games," Journal of Optimization Theory and Applications, Springer, vol. 151(3), pages 455-473, December.
    2. Holger Scheel & Stefan Scholtes, 2000. "Mathematical Programs with Complementarity Constraints: Stationarity, Optimality, and Sensitivity," Mathematics of Operations Research, INFORMS, vol. 25(1), pages 1-22, 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.
    4. Jong-Shi Pang & Masao Fukushima, 2005. "Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games," Computational Management Science, Springer, vol. 2(1), pages 21-56, January.
    5. Yihsu Chen & Benjamin Hobbs & Sven Leyffer & Todd Munson, 2006. "Leader-Follower Equilibria for Electric Power and NO x Allowances Markets," Computational Management Science, Springer, vol. 3(4), pages 307-330, September.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. J. S. Pang, 2007. "Partially B-Regular Optimization and Equilibrium Problems," Mathematics of Operations Research, INFORMS, vol. 32(3), pages 687-699, August.
    2. Zhe Yang & Yan Ju, 2016. "Existence and generic stability of cooperative equilibria for multi-leader-multi-follower games," Journal of Global Optimization, Springer, vol. 65(3), pages 563-573, July.
    3. Addis Belete Zewde & Semu Mitiku Kassa, 2021. "Multilevel multi-leader multiple-follower games with nonseparable objectives and shared constraints," Computational Management Science, Springer, vol. 18(4), pages 455-475, October.
    4. Feijoo, Felipe & Das, Tapas K., 2014. "Design of Pareto optimal CO2 cap-and-trade policies for deregulated electricity networks," Applied Energy, Elsevier, vol. 119(C), pages 371-383.
    5. D. Dorsch & H. T. Jongen & V. Shikhman, 2013. "On Intrinsic Complexity of Nash Equilibrium Problems and Bilevel Optimization," Journal of Optimization Theory and Applications, Springer, vol. 159(3), pages 606-634, December.
    6. S. Siddiqui & S. Gabriel, 2013. "An SOS1-Based Approach for Solving MPECs with a Natural Gas Market Application," Networks and Spatial Economics, Springer, vol. 13(2), pages 205-227, June.
    7. Ludovic A. Julien, 2021. "Noncooperative oligopoly equilibrium in markets with hierarchical competition," EconomiX Working Papers 2021-14, University of Paris Nanterre, EconomiX.
    8. Jian Yao & Ilan Adler & Shmuel S. Oren, 2008. "Modeling and Computing Two-Settlement Oligopolistic Equilibrium in a Congested Electricity Network," Operations Research, INFORMS, vol. 56(1), pages 34-47, February.
    9. Aghamohammadloo, Hossein & Talaeizadeh, Valiollah & Shahanaghi, Kamran & Aghaei, Jamshid & Shayanfar, Heidarali & Shafie-khah, Miadreza & Catalão, João P.S., 2021. "Integrated Demand Response programs and energy hubs retail energy market modelling," Energy, Elsevier, vol. 234(C).
    10. Giorgia Oggioni & Yves Smeers & Elisabetta Allevi & Siegfried Schaible, 2012. "A Generalized Nash Equilibrium Model of Market Coupling in the European Power System," Networks and Spatial Economics, Springer, vol. 12(4), pages 503-560, December.
    11. Ciarcià, Carla & Daniele, Patrizia, 2016. "New existence theorems for quasi-variational inequalities and applications to financial models," European Journal of Operational Research, Elsevier, vol. 251(1), pages 288-299.
    12. Zhang, Fang & Lu, Jian & Hu, Xiaojian & Meng, Qiang, 2023. "Integrated deployment of dedicated lane and roadside unit considering uncertain road capacity under the mixed-autonomy traffic environment," Transportation Research Part B: Methodological, Elsevier, vol. 174(C).
    13. Andreas Ehrenmann & Karsten Neuhoff, 2009. "A Comparison of Electricity Market Designs in Networks," Operations Research, INFORMS, vol. 57(2), pages 274-286, April.
    14. Contreras, Javier & Krawczyk, Jacek & Zuccollo, James, 2008. "The invisible polluter: Can regulators save consumer surplus?," MPRA Paper 9890, University Library of Munich, Germany.
    15. Gui-Hua Lin & Mei-Ju Luo & Jin Zhang, 2016. "Smoothing and SAA method for stochastic programming problems with non-smooth objective and constraints," Journal of Global Optimization, Springer, vol. 66(3), pages 487-510, November.
    16. Lei Guo & Gui-Hua Lin & Jane J. Ye, 2015. "Solving Mathematical Programs with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 166(1), pages 234-256, July.
    17. Tao Tan & Yanyan Li & Xingsi Li, 2011. "A Smoothing Method for Zero–One Constrained Extremum Problems," Journal of Optimization Theory and Applications, Springer, vol. 150(1), pages 65-77, July.
    18. Jiang, Zhoutong & Lei, Chao & Ouyang, Yanfeng, 2020. "Optimal investment and management of shared bikes in a competitive market," Transportation Research Part B: Methodological, Elsevier, vol. 135(C), pages 143-155.
    19. Alexey Izmailov & Mikhail Solodov, 2014. "On error bounds and Newton-type methods for generalized Nash equilibrium problems," Computational Optimization and Applications, Springer, vol. 59(1), pages 201-218, October.
    20. J. Contreras & J. B. Krawczyk & J. Zuccollo, 2016. "Economics of collective monitoring: a study of environmentally constrained electricity generators," Computational Management Science, Springer, vol. 13(3), pages 349-369, July.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:180:y:2019:i:2:d:10.1007_s10957-018-1391-5. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.