IDEAS home Printed from https://ideas.repec.org/p/fth/minner/300.html
   My bibliography  Save this paper

Generic 4 x 4 Two Person Games Have at Most 15 Nash Equilibria

Author

Listed:
  • McLennan, A
  • Park, I-U

Abstract

The maximal generic number of Nash equilibria for two person games in which the two agents each have four pure strategies is shown to be 15. In contrast to Keiding (1995), who arrives at this result by computer enumeration, our argument is based on a collection of lemmas that constrain the set of equilibria. Several of these pertain to any common number d of pure strategies for the two agents.

Suggested Citation

  • McLennan, A & Park, I-U, 1997. "Generic 4 x 4 Two Person Games Have at Most 15 Nash Equilibria," Papers 300, Minnesota - Center for Economic Research.
    • Handle: RePEc:fth:minner:300
    as

    Download full text from publisher

    To our knowledge, this item is not available for download. To find whether it is available, there are three options:
    1. Check below whether another version of this item is available online.
    2. Check on the provider's web page whether it is in fact available.
    3. Perform a search for a similarly titled item that would be available.

    Other versions of this item:

    References listed on IDEAS

    as
    1. Faruk Gül & David Pearce & Ennio Stacchetti, 1993. "A Bound on the Proportion of Pure Strategy Equilibria in Generic Games," Mathematics of Operations Research, INFORMS, vol. 18(3), pages 548-552, August.
    2. McKelvey, Richard D. & McLennan, Andrew, 1997. "The Maximal Number of Regular Totally Mixed Nash Equilibria," Journal of Economic Theory, Elsevier, vol. 72(2), pages 411-425, February.
    3. McLennan, Andrew, 1997. "The Maximal Generic Number of Pure Nash Equilibria," Journal of Economic Theory, Elsevier, vol. 72(2), pages 408-410, February.
    4. Keiding, Hans, 1997. "On the Maximal Number of Nash Equilibria in ann x nBimatrix Game," Games and Economic Behavior, Elsevier, vol. 21(1-2), pages 148-160, October.
    5. Powers, Imelda Yeung, 1990. "Limiting Distributions of the Number of Pure Strategy Nash Equilibria in N-Person Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 19(3), pages 277-286.
    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. Jun Honda, 2015. "Games with the Total Bandwagon Property," Department of Economics Working Papers wuwp197, Vienna University of Economics and Business, Department of Economics.
    2. Conitzer, Vincent & Sandholm, Tuomas, 2008. "New complexity results about Nash equilibria," Games and Economic Behavior, Elsevier, vol. 63(2), pages 621-641, July.
    3. Rahul Savani & Bernhard von Stengel, 2016. "Unit vector games," International Journal of Economic Theory, The International Society for Economic Theory, vol. 12(1), pages 7-27, March.
    4. Philip V. Fellman & Jonathan Vos Post, 2007. "Quantum Nash Equilibria and Quantum Computing," Papers 0707.0324, arXiv.org.
    5. Jun Honda, 2018. "Games with the total bandwagon property meet the Quint–Shubik conjecture," International Journal of Game Theory, Springer;Game Theory Society, vol. 47(3), pages 893-912, September.
    6. Ɖura-Georg Granić & Johannes Kern, 2016. "Circulant games," Theory and Decision, Springer, vol. 80(1), pages 43-69, January.
    7. Sun, Ching-jen, 2020. "A sandwich theorem for generic n × n two person games," Games and Economic Behavior, Elsevier, vol. 120(C), pages 86-95.
    8. McLennan, Andrew & Berg, Johannes, 2005. "Asymptotic expected number of Nash equilibria of two-player normal form games," Games and Economic Behavior, Elsevier, vol. 51(2), pages 264-295, May.
    9. Ravi Kannan & Thorsten Theobald, 2010. "Games of fixed rank: a hierarchy of bimatrix games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 42(1), pages 157-173, January.
    10. M. Punniyamoorthy & Sarin Abraham & Jose Joy Thoppan, 2023. "A Method to Select Best Among Multi-Nash Equilibria," Studies in Microeconomics, , vol. 11(1), pages 101-127, April.
    11. Hwang, Sung-Ha & Rey-Bellet, Luc, 2020. "Strategic decompositions of normal form games: Zero-sum games and potential games," Games and Economic Behavior, Elsevier, vol. 122(C), pages 370-390.

    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. Sun, Ching-jen, 2020. "A sandwich theorem for generic n × n two person games," Games and Economic Behavior, Elsevier, vol. 120(C), pages 86-95.
    2. Ɖura-Georg Granić & Johannes Kern, 2016. "Circulant games," Theory and Decision, Springer, vol. 80(1), pages 43-69, January.
    3. Tom Johnston & Michael Savery & Alex Scott & Bassel Tarbush, 2023. "Game Connectivity and Adaptive Dynamics," Papers 2309.10609, arXiv.org, revised Oct 2024.
    4. Fabrizio Germano, 2006. "On some geometry and equivalence classes of normal form games," International Journal of Game Theory, Springer;Game Theory Society, vol. 34(4), pages 561-581, November.
    5. McLennan, Andrew, 1997. "The Maximal Generic Number of Pure Nash Equilibria," Journal of Economic Theory, Elsevier, vol. 72(2), pages 408-410, February.
    6. McLennan, Andrew & Berg, Johannes, 2005. "Asymptotic expected number of Nash equilibria of two-player normal form games," Games and Economic Behavior, Elsevier, vol. 51(2), pages 264-295, May.
    7. Eraslan, Hülya & McLennan, Andrew, 2013. "Uniqueness of stationary equilibrium payoffs in coalitional bargaining," Journal of Economic Theory, Elsevier, vol. 148(6), pages 2195-2222.
    8. Arieli, Itai & Babichenko, Yakov, 2016. "Random extensive form games," Journal of Economic Theory, Elsevier, vol. 166(C), pages 517-535.
    9. Heinrich, Torsten & Wiese, Samuel, 2020. "The Frequency of Convergent Games under Best-Response Dynamics," INET Oxford Working Papers 2020-24, Institute for New Economic Thinking at the Oxford Martin School, University of Oxford.
    10. Michael R. Powers & Martin Shubik & Wen Wang, 2016. "Expected Worth for 2 � 2 Matrix Games with Variable Grid Sizes," Cowles Foundation Discussion Papers 2039, Cowles Foundation for Research in Economics, Yale University.
    11. Demichelis, Stefano & Ritzberger, Klaus, 2003. "From evolutionary to strategic stability," Journal of Economic Theory, Elsevier, vol. 113(1), pages 51-75, November.
    12. S. Mishra & T. K. Kumar, 1997. "On the Probability of Existence of Pure Equilibria in Matrix Games," Journal of Optimization Theory and Applications, Springer, vol. 94(3), pages 765-770, September.
    13. Porter, Ryan & Nudelman, Eugene & Shoham, Yoav, 2008. "Simple search methods for finding a Nash equilibrium," Games and Economic Behavior, Elsevier, vol. 63(2), pages 642-662, July.
    14. DeMichelis, Stefano & Germano, Fabrizio, 2000. "On the Indices of Zeros of Nash Fields," Journal of Economic Theory, Elsevier, vol. 94(2), pages 192-217, October.
    15. Torsten Heinrich & Yoojin Jang & Luca Mungo & Marco Pangallo & Alex Scott & Bassel Tarbush & Samuel Wiese, 2021. "Best-response dynamics, playing sequences, and convergence to equilibrium in random games," Papers 2101.04222, arXiv.org, revised Nov 2022.
    16. Govindan, Srihari & Laraki, Rida & Pahl, Lucas, 2023. "On sustainable equilibria," Journal of Economic Theory, Elsevier, vol. 213(C).
    17. Elizabeth Baldwin & Paul Klemperer, 2019. "Understanding Preferences: “Demand Types”, and the Existence of Equilibrium With Indivisibilities," Econometrica, Econometric Society, vol. 87(3), pages 867-932, May.
    18. Ben Amiet & Andrea Collevecchio & Marco Scarsini & Ziwen Zhong, 2021. "Pure Nash Equilibria and Best-Response Dynamics in Random Games," Mathematics of Operations Research, INFORMS, vol. 46(4), pages 1552-1572, November.
    19. Balkenborg, Dieter & Vermeulen, Dries, 2014. "Universality of Nash components," Games and Economic Behavior, Elsevier, vol. 86(C), pages 67-76.
    20. Rinott, Yosef & Scarsini, Marco, 2000. "On the Number of Pure Strategy Nash Equilibria in Random Games," Games and Economic Behavior, Elsevier, vol. 33(2), pages 274-293, November.

    More about this item

    Keywords

    ECONOMETRICS;

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games

    Statistics

    Access and download statistics

    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:fth:minner:300. 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: Thomas Krichel (email available below). General contact details of provider: https://edirc.repec.org/data/deumnus.html .

    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.