In this paper, we present a more simple and independent proof of Reny's theorem (1998), on the existence of a Nash equilibrium in discontinue game, with a better-reply secure game in a Hausdorff topological vector space stronger than Reny's one. We will get the equivalence if the payoff function is upper semi-continuous like in the second Reny's example. Our proof is based on a new version of the existence of maximal element of Fan-Browder given by Deguire and Lassonde (1995). Reny's proof used a lemma of approximation of payoff function by a continuous sequence and show the existence of Nash equilibrium by the existence of equilibrium in mixed strategy proved in continuous game by the classical result.
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Find related papers by JEL classification: C69 - Mathematical and Quantitative Methods - - Mathematical Methods and Programming - - - Other C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
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