IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v438y2023ics0096300322006270.html
   My bibliography  Save this article

Extended matrix norm method: Applications to bimatrix games and convergence results

Author

Listed:
  • İzgi, Burhaneddin
  • Özkaya, Murat
  • Üre, Nazım Kemal
  • Perc, Matjaž

Abstract

In this paper, we extend and apply the Matrix Norm (MN) approach to the nonzero-sum bimatrix games. We present preliminary results regarding the convergence of the MN approaches. We provide a notation for expressing nonzero-sum bimatrix games in terms of two matrix games using the idea of separation of a bimatrix game into two different matrix games. Next, we prove theorems regarding boundaries of the game value depending on only norms of the payoff matrix for each player of the nonzero-sum bimatrix game. In addition to these, we refine the boundaries of the game value for the zero/nonzero sum matrix games. Therefore, we succeed to find an improved interval for the game value, which is a crucial improvement for both nonzero and zero-sum matrix games. As a consequence, we can solve a nonzero-sum bimatrix game for each player approximately without solving any equations. Moreover, we modify the inequalities for the extrema of the strategy set for the nonzero-sum bimatrix games. Furthermore, we adapt the min-max theorem of the MN approach for the nonzero-sum bimatrix games. Finally, we consider various bimatrix game examples from the literature, including the famous battle of sexes, to demonstrate the consistency of our approaches. We also show that the repeated applications of Extended Matrix Norm (EMN) methods work well to obtain a better-estimated game value in view of the obtained convergence results.

Suggested Citation

  • İzgi, Burhaneddin & Özkaya, Murat & Üre, Nazım Kemal & Perc, Matjaž, 2023. "Extended matrix norm method: Applications to bimatrix games and convergence results," Applied Mathematics and Computation, Elsevier, vol. 438(C).
  • Handle: RePEc:eee:apmaco:v:438:y:2023:i:c:s0096300322006270
    DOI: 10.1016/j.amc.2022.127553
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300322006270
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.amc.2022.127553?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. Sankar Kumar Roy & Prasanta Mula, 2016. "Solving matrix game with rough payoffs using genetic algorithm," Operational Research, Springer, vol. 16(1), pages 117-130, April.
    2. L. D. Berkovitz & Melvin Dresher, 1959. "A Game-Theory Analysis of Tactical Air War," Operations Research, INFORMS, vol. 7(5), pages 599-620, October.
    3. Lensberg, Terje & Schenk-Hoppé, Klaus Reiner, 2021. "Cold play: Learning across bimatrix games," Journal of Economic Behavior & Organization, Elsevier, vol. 185(C), pages 419-441.
    4. Nash, John, 1950. "The Bargaining Problem," Econometrica, Econometric Society, vol. 18(2), pages 155-162, April.
    5. repec:wsi:jeapmx:v:20:y:2018:i:04:n:s0219198918500081 is not listed on IDEAS
    6. Ryle S. Perera, 2018. "An Evolutionary Game Theory Strategy for Carbon Emission Reduction in the Electricity Market," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 20(04), pages 1-20, December.
    7. Deng, Xinyang & Jiang, Wen & Wang, Zhen, 2019. "Zero-sum polymatrix games with link uncertainty: A Dempster-Shafer theory solution," Applied Mathematics and Computation, Elsevier, vol. 340(C), pages 101-112.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Han, Dun & He, Youxin, 2023. "The impact of labor subsidy, taxation and corruption on individual behavior," Applied Mathematics and Computation, Elsevier, vol. 458(C).
    2. Molnar, Grant & Hammond, Caroline & Fu, Feng, 2023. "Reactive means in the iterated Prisoner’s dilemma," Applied Mathematics and Computation, Elsevier, vol. 458(C).
    3. İzgi, Burhaneddin & Özkaya, Murat & Üre, Nazım Kemal & Perc, Matjaž, 2024. "Matrix norm based hybrid Shapley and iterative methods for the solution of stochastic matrix games," Applied Mathematics and Computation, Elsevier, vol. 473(C).

    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. Maurizio Zanardi, 2004. "Antidumping law as a collusive device," Canadian Journal of Economics, Canadian Economics Association, vol. 37(1), pages 95-122, February.
    2. M. Hinojosa & A. Mármol & J. Zarzuelo, 2008. "Inequality averse multi-utilitarian bargaining solutions," International Journal of Game Theory, Springer;Game Theory Society, vol. 37(4), pages 597-618, December.
    3. Matsui, Kenji, 2020. "Optimal bargaining timing of a wholesale price for a manufacturer with a retailer in a dual-channel supply chain," European Journal of Operational Research, Elsevier, vol. 287(1), pages 225-236.
    4. Simon G�chter & Arno Riedl, "undated". "Moral Property Rights in Bargaining," IEW - Working Papers 113, Institute for Empirical Research in Economics - University of Zurich.
    5. H Peyton Young, 2014. "The Evolution of Social Norms," Economics Series Working Papers 726, University of Oxford, Department of Economics.
    6. Erkki Koskela & Ronnie Schöb, 2002. "Alleviating Unemployment: The Case for Green Tax Reforms," Chapters, in: Lawrence H. Goulder (ed.), Environmental Policy Making in Economies with Prior Tax Distortions, chapter 20, pages 355-378, Edward Elgar Publishing.
    7. Magni, Carlo Alberto, 2009. "Splitting up value: A critical review of residual income theories," European Journal of Operational Research, Elsevier, vol. 198(1), pages 1-22, October.
    8. Stefano Vannucci, 2015. "La teoria dei giochi e John Nash," Department of Economics University of Siena 722, Department of Economics, University of Siena.
    9. Roland Kirstein & Matthias Peiss, 2013. "Quantitative Machtkonzepte in der Ökonomik," FEMM Working Papers 130004, Otto-von-Guericke University Magdeburg, Faculty of Economics and Management.
    10. Michael Carter & Julian Wright, 1999. "Interconnection in Network Industries," Review of Industrial Organization, Springer;The Industrial Organization Society, vol. 14(1), pages 1-25, February.
    11. Laruelle, Annick & Valenciano, Federico, 2008. "Noncooperative foundations of bargaining power in committees and the Shapley-Shubik index," Games and Economic Behavior, Elsevier, vol. 63(1), pages 341-353, May.
    12. Thomas M. Humphrey, 1996. "The early history of the box diagram," Economic Quarterly, Federal Reserve Bank of Richmond, issue Win, pages 37-75.
    13. Omer F. Baris, 2018. "Timing effect in bargaining and ex ante efficiency of the relative utilitarian solution," Theory and Decision, Springer, vol. 84(4), pages 547-556, June.
    14. Izat B. Baybusinov & Enrico Maria Fenoaltea & Yi-Cheng Zhang, 2022. "Negotiation problem," Papers 2201.12619, arXiv.org.
    15. Bergantiños, Gustavo & Moreno-Ternero, Juan D., 2022. "Monotonicity in sharing the revenues from broadcasting sports leagues," European Journal of Operational Research, Elsevier, vol. 297(1), pages 338-346.
    16. Yoshihara, Naoki, 2003. "Characterizations of bargaining solutions in production economies with unequal skills," Journal of Economic Theory, Elsevier, vol. 108(2), pages 256-285, February.
    17. Yan, Ruiliang & Wang, John & Zhou, Bin, 2010. "Channel integration and profit sharing in the dynamics of multi-channel firms," Journal of Retailing and Consumer Services, Elsevier, vol. 17(5), pages 430-440.
    18. S. S. Askar & A. Al-khedhairi, 2019. "Analysis of a Four-Firm Competition Based on a Generalized Bounded Rationality and Different Mechanisms," Complexity, Hindawi, vol. 2019, pages 1-12, May.
    19. Marc Fleurbaey, 2000. "Choix social : une difficulté et de multiples possibilités," Revue Économique, Programme National Persée, vol. 51(5), pages 1215-1232.
    20. Juarez, Ruben & Ko, Chiu Yu & Xue, Jingyi, 2018. "Sharing sequential values in a network," Journal of Economic Theory, Elsevier, vol. 177(C), pages 734-779.

    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:eee:apmaco:v:438:y:2023:i:c:s0096300322006270. 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: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    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.