IDEAS home Printed from https://ideas.repec.org/a/eee/gamebe/v145y2024icp510-525.html
   My bibliography  Save this article

Oddness of the number of Nash equilibria: The case of polynomial payoff functions

Author

Listed:
  • Bich, Philippe
  • Fixary, Julien

Abstract

In 1971, Wilson (1971) proved that “almost all” finite games have an odd number of mixed Nash equilibria. Since then, several other proofs have been given, but always for mixed extensions of finite games. In this paper, we present a new oddness theorem for large classes of polynomial payoff functions and semi-algebraic sets of strategies. Additionally, we provide some applications to recent models of games on networks such that Patacchini-Zenou's model about juvenile delinquency and conformism (Patacchini and Zenou, 2012), Calvó-Armengol-Patacchini-Zenou's model about social networks in education (Calvó-Armengol et al., 2009), Konig-Liu-Zenou's model about R&D networks (König et al., 2019), Helsley-Zenou's model about social networks and interactions in cities (Helsley and Zenou, 2014).

Suggested Citation

  • Bich, Philippe & Fixary, Julien, 2024. "Oddness of the number of Nash equilibria: The case of polynomial payoff functions," Games and Economic Behavior, Elsevier, vol. 145(C), pages 510-525.
    • Handle: RePEc:eee:gamebe:v:145:y:2024:i:c:p:510-525
    DOI: 10.1016/j.geb.2024.04.005
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S089982562400054X
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.geb.2024.04.005?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. Bich, Philippe & Fixary, Julien, 2022. "Network formation and pairwise stability: A new oddness theorem," Journal of Mathematical Economics, Elsevier, vol. 103(C).
    2. Pimienta, Carlos, 2009. "Generic determinacy of Nash equilibrium in network-formation games," Games and Economic Behavior, Elsevier, vol. 66(2), pages 920-927, July.
    3. Helsley, Robert W. & Zenou, Yves, 2014. "Social networks and interactions in cities," Journal of Economic Theory, Elsevier, vol. 150(C), pages 426-466.
    4. Michael D. König & Xiaodong Liu & Yves Zenou, 2019. "R&D Networks: Theory, Empirics, and Policy Implications," The Review of Economics and Statistics, MIT Press, vol. 101(3), pages 476-491, July.
    5. Blume, Lawrence E & Zame, William R, 1994. "The Algebraic Geometry of Perfect and Sequential Equilibrium," Econometrica, Econometric Society, vol. 62(4), pages 783-794, July.
    6. Mas-Colell, Andreu, 2010. "Generic finiteness of equilibrium payoffs for bimatrix games," Journal of Mathematical Economics, Elsevier, vol. 46(4), pages 382-383, July.
    7. Govindan, Srihari & Wilson, Robert, 2001. "Direct Proofs of Generic Finiteness of Nash Equilibrium Outcomes," Econometrica, Econometric Society, vol. 69(3), pages 765-769, May.
    8. Philippe Bich & Julien Fixary, 2022. "Network formation and pairwise stability: A new oddness theorem," Post-Print hal-03969600, HAL.
    9. Antoni Calvó-Armengol & Eleonora Patacchini & Yves Zenou, 2009. "Peer Effects and Social Networks in Education," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 76(4), pages 1239-1267.
    10. Philippe Bich & Julien Fixary, 2022. "Network formation and pairwise stability: A new oddness theorem," PSE-Ecole d'économie de Paris (Postprint) hal-03969600, HAL.
    11. Eleonora Patacchini & Yves Zenou, 2012. "Juvenile Delinquency and Conformism," The Journal of Law, Economics, and Organization, Oxford University Press, vol. 28(1), pages 1-31.
    12. Predtetchinski, Arkadi, 2009. "A general structure theorem for the Nash equilibrium correspondence," Games and Economic Behavior, Elsevier, vol. 66(2), pages 950-958, July.
    13. Philippe Bich & Julien Fixary, 2022. "Network formation and pairwise stability: A new oddness theorem," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-03969600, HAL.
    14. Govindan, Srihari & McLennan, Andrew, 2001. "On the Generic Finiteness of Equilibrium Outcome Distributions in Game Forms," Econometrica, Econometric Society, vol. 69(2), pages 455-471, March.
    15. Kohlberg, Elon & Mertens, Jean-Francois, 1986. "On the Strategic Stability of Equilibria," Econometrica, Econometric Society, vol. 54(5), pages 1003-1037, September.
    16. P. Jean-Jacques Herings & Ronald J.A.P. Peeters, 2001. "symposium articles: A differentiable homotopy to compute Nash equilibria of n -person games," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 18(1), pages 159-185.
    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. Philippe Bich & Julien Fixary, 2021. "Oddness of the number of Nash equilibria: the case of polynomial payoff functions," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-03354269, HAL.
    2. Philippe Bich & Julien Fixary, 2021. "Oddness of the number of Nash equilibria: the Case of Polynomial Payoff Functions," Documents de travail du Centre d'Economie de la Sorbonne 21027, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    3. Philippe Bich & Julien Fixary, 2021. "Oddness of the number of Nash equilibria: the case of polynomial payoff functions," Post-Print halshs-03354269, HAL.
    4. Bich, Philippe & Fixary, Julien, 2022. "Network formation and pairwise stability: A new oddness theorem," Journal of Mathematical Economics, Elsevier, vol. 103(C).
    5. Philippe Bich & Julien Fixary, 2021. "Structure and oddness theorems for pairwise stable networks," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-03287524, HAL.
    6. Philippe Bich & Julien Fixary, 2021. "Structure and oddness theorems for pairwise stable networks," Post-Print halshs-03287524, HAL.
    7. Pimienta, Carlos, 2010. "Generic finiteness of outcome distributions for two-person game forms with three outcomes," Mathematical Social Sciences, Elsevier, vol. 59(3), pages 364-365, May.
    8. Meroni, Claudia & Pimienta, Carlos, 2017. "The structure of Nash equilibria in Poisson games," Journal of Economic Theory, Elsevier, vol. 169(C), pages 128-144.
    9. Jackson, Matthew O. & Zenou, Yves, 2015. "Games on Networks," Handbook of Game Theory with Economic Applications,, Elsevier.
    10. Topa, Giorgio & Zenou, Yves, 2015. "Neighborhood and Network Effects," Handbook of Regional and Urban Economics, in: Gilles Duranton & J. V. Henderson & William C. Strange (ed.), Handbook of Regional and Urban Economics, edition 1, volume 5, chapter 0, pages 561-624, Elsevier.
    11. Hiller, Timo, 2022. "A simple model of network formation with competition effects," Journal of Mathematical Economics, Elsevier, vol. 99(C).
    12. Xiao Luo & Xuewen Qian & Yang Sun, 2021. "The algebraic geometry of perfect and sequential equilibrium: an extension," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 71(2), pages 579-601, March.
    13. Timo Hiller, 2017. "Too Big to Jail? Key-Player Policies When the Network is Endogenous," Bristol Economics Discussion Papers 17/693, School of Economics, University of Bristol, UK.
    14. Hahn, Youjin & Islam, Asadul & Patacchini, Eleonora & Zenou, Yves, 2015. "Network Structure and Education Outcomes: Evidence from a Field Experiment in Bangladesh," IZA Discussion Papers 8872, Institute of Labor Economics (IZA).
    15. Kim, Jun Sung & Patacchini, Eleonora & Picard, Pierre M. & Zenou, Yves, 2017. "Urban Interactions," Working Paper Series 1192, Research Institute of Industrial Economics.
    16. Matthew O. Jackson & Brian W. Rogers & Yves Zenou, 2017. "The Economic Consequences of Social-Network Structure," Journal of Economic Literature, American Economic Association, vol. 55(1), pages 49-95, March.
    17. Takahashi, Satoru & Tercieux, Olivier, 2020. "Robust equilibrium outcomes in sequential games under almost common certainty of payoffs," Journal of Economic Theory, Elsevier, vol. 188(C).
    18. Ushchev, Philip & Zenou, Yves, 2020. "Social norms in networks," Journal of Economic Theory, Elsevier, vol. 185(C).
    19. Predtetchinski, Arkadi, 2009. "A general structure theorem for the Nash equilibrium correspondence," Games and Economic Behavior, Elsevier, vol. 66(2), pages 950-958, July.
    20. Srihari Govindan & Robert Wilson, 2008. "Metastable Equilibria," Mathematics of Operations Research, INFORMS, vol. 33(4), pages 787-820, November.

    More about this item

    Keywords

    Nash equilibrium; Polynomial payoff functions; Generic oddness; Network games;
    All these keywords.

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • D85 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Network Formation

    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:eee:gamebe:v:145:y:2024:i:c:p:510-525. 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: http://www.elsevier.com/locate/inca/622836 .

    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.