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A constructive and elementary proof of Reny's theorem

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Abstract

In a recent but well known paper, Reny proved the existence of Nash equilibria for better-reply-secure games, with possibly discontinuous payoff functions. Reny's proof is purely existential, and is similar to a contradiction proof : it gives non hint of a method to compute a Nash equilibrium in the class of games considered. In this paper, we adapt the arguments of Reny in order to obtain, for better-reply-secure games : an elementary proof of Nash equilibria existence, which is a consequence of Kakutani's theorem, and a " constructive " proof, in the sense that we obtain Nash equilibria as limits of fixed-point of well chosen correspondences.

Suggested Citation

  • Philippe Bich, 2006. "A constructive and elementary proof of Reny's theorem," Cahiers de la Maison des Sciences Economiques b06001, Université Panthéon-Sorbonne (Paris 1).
  • Handle: RePEc:mse:wpsorb:b06001
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    File URL: ftp://mse.univ-paris1.fr/pub/mse/cahiers2006/B06001.pdf
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    1. Jean-Marc Bonnisseau & Pascal Gourdel & Hakim Hammami, 2005. "Existence d'un équilibre de Nash dans un jeu discontinu," Cahiers de la Maison des Sciences Economiques b05099, Université Panthéon-Sorbonne (Paris 1).
    2. Philip J. Reny, 1999. "On the Existence of Pure and Mixed Strategy Nash Equilibria in Discontinuous Games," Econometrica, Econometric Society, vol. 67(5), pages 1029-1056, September.
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    Keywords

    Reny's theorem; discontinuous payoffs.;

    JEL classification:

    • C7 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory

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