these-boulogne - TH ` ESE DE DOCTORAT DE L’UNIVERSIT ´ E...

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Unformatted text preview: TH ` ESE DE DOCTORAT DE L’UNIVERSIT ´ E PARIS 6 Sp´ ecialit´ e : MATH ´ EMATIQUES APPLIQU ´ EES pr´ esent´ ee par Thomas BOULOGNE pour obtenir le grade de Docteur de l’UNIVERSIT ´ E PARIS 6 Jeux strat´ egiques non-atomiques et applications aux r´ eseaux Soutenue le 15 d´ ecembre 2004 devant le jury compos´ e de : MM. Eitan Altman Directeur Guillaume Carlier Examinateur Jean Fonlupt Examinateur St´ ephane Gaubert Examinateur Sylvain Sorin Directeur Nicolas Vieille Rapporteur Bernhard Von Stengel Rapporteur Table des mati` eres / Contents Introduction 7 Bibliographie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 I Nonatomic strategic games 15 Notations 17 1 Two models of nonatomic games 19 1.1 The assumption of nonatomicity . . . . . . . . . . . . . . . . . . . . . 19 1.2 S-games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.2.1 Payoff functions defined on S × F S . . . . . . . . . . . . . . . 20 1.2.2 Payoff functions defined on S × co ( S ) . . . . . . . . . . . . . . 24 1.3 M-games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.4 Relations between S-games and M-games . . . . . . . . . . . . . . . 29 1.4.1 From S-games to M-games . . . . . . . . . . . . . . . . . . . 30 1.4.2 From M-games to S-games . . . . . . . . . . . . . . . . . . . 32 2 Approximation of large games by nonatomic games 37 2.1 S-games and λ-convergence . . . . . . . . . . . . . . . . . . . . . . . 38 2.2 S-games and convergence in distribution . . . . . . . . . . . . . . . . 45 2.3 M-games and weak convergence . . . . . . . . . . . . . . . . . . . . . 47 3 Extensions and variations 53 3.1 Extensions of S-games . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.2 Extension of M-games . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.3 Population games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.4 Potential games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.4.1 Infinite potential games . . . . . . . . . . . . . . . . . . . . . 61 3.4.2 Potential games as limits of finite player games . . . . . . . . 62 4 Applications 69 4.1 Routing games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.1.1 Nonatomic routing games . . . . . . . . . . . . . . . . . . . . 69 4.1.2 Approximation of Wardrop equilibria by Nash equilibria . . . 72 4.2 Crowding games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.2.1 Large crowding games . . . . . . . . . . . . . . . . . . . . . . 75 iii iv TABLE DES MATI ` ERES / CONTENTS 4.2.2 Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.3 Evolutionary games . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.3.1 From Nash to Maynard Smith: different interpretations of Nash equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.3.2 Games with a single population . . . . . . . . . . . . . . . . . 79 4.3.3 Stability in games with n-populations . . . . . . . . . . . . . . 84-populations ....
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This note was uploaded on 09/27/2010 for the course EE 229 taught by Professor R.srikant during the Spring '09 term at University of Illinois, Urbana Champaign.

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these-boulogne - TH ` ESE DE DOCTORAT DE L’UNIVERSIT ´ E...

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