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

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  • Philippe Bich

    (UP1 - Université Paris 1 Panthéon-Sorbonne, CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique, PSE - Paris School of Economics - UP1 - Université Paris 1 Panthéon-Sorbonne - ENS-PSL - École normale supérieure - Paris - PSL - Université Paris Sciences et Lettres - EHESS - École des hautes études en sciences sociales - ENPC - École des Ponts ParisTech - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement)

  • Julien Fixary

    (UP1 - Université Paris 1 Panthéon-Sorbonne, CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

Abstract

In 1971, Robert Wilson ([19]) proved that "almost all" finite games have an odd number of mixed Nash equilibria (oddness theorem). Since then, several other proofs have been given, but always for mixed extensions of finite games. In this paper, we prove oddness theorem for large classes of polynomial payo↵ functions and semi-algebraic sets of strategies, and we provide some applications to recent models.

Suggested Citation

  • 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.
    • Handle: RePEc:hal:cesptp:halshs-03354269
    Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-03354269
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    References listed on IDEAS

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    1. 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.
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    8. Philippe Bich & Julien Fixary, 2021. "Structure and oddness theorems for pairwise stable networks," Documents de travail du Centre d'Economie de la Sorbonne 21016, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    9. 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.
    10. Helsley, Robert W. & Zenou, Yves, 2014. "Social networks and interactions in cities," Journal of Economic Theory, Elsevier, vol. 150(C), pages 426-466.
    11. Jackson, Matthew O. & Zenou, Yves, 2015. "Games on Networks," Handbook of Game Theory with Economic Applications,, Elsevier.
    12. 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.
    13. Philippe Bich & Julien Fixary, 2021. "Structure and oddness theorems for pairwise stable networks," Post-Print halshs-03287524, HAL.
    14. 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.
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    Cited by:

    1. Bich, Philippe & Fixary, Julien, 2022. "Network formation and pairwise stability: A new oddness theorem," Journal of Mathematical Economics, Elsevier, vol. 103(C).

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