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Convexity on Nash Equilibria without Linear Structure

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  • Francesco Ciardiello

Abstract

To give sucient conditions for Nash Equilibrium existence in a continuous game is a central problem in Game Theory. In this paper, we present two games in which we show how the continuity and quasi-concavity hypotheses are unconnected one to each other. Then, we relax the quasiconcavity assumption by exploiting the multiconnected convexity's concept (Mechaiekh & Others, 1998) in spaces without any linear structure. These results will be applied to two non-zero-sum games lacking the classical assumptions and more recent improvements (Ziad, 1997), (Abalo & Kostreva, 2004). As a minor result, some counterexamples about relationship between some continuity conditions due to Lignola (1997), Reny (1999) and Simon (1995) for Nash equilibria existence are obtained.

Suggested Citation

  • Francesco Ciardiello, 2007. "Convexity on Nash Equilibria without Linear Structure," Quaderni DSEMS 15-2007, Dipartimento di Scienze Economiche, Matematiche e Statistiche, Universita' di Foggia.
    • Handle: RePEc:ufg:qdsems:15-2007
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    References listed on IDEAS

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    1. Tian, Guoqiang & Zhou, Jianxin, 1995. "Transfer continuities, generalizations of the Weierstrass and maximum theorems: A full characterization," Journal of Mathematical Economics, Elsevier, vol. 24(3), pages 281-303.
    2. Ziad, Abderrahmane, 1999. "Pure strategy Nash equilibria of non-zero-sum two-person games: non-convex case," Economics Letters, Elsevier, vol. 62(3), pages 307-310, March.
    3. Partha Dasgupta & Eric Maskin, 1986. "The Existence of Equilibrium in Discontinuous Economic Games, I: Theory," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 53(1), pages 1-26.
    4. M. B. Lignola, 1997. "Ky Fan Inequalities and Nash Equilibrium Points without Semicontinuity and Compactness," Journal of Optimization Theory and Applications, Springer, vol. 94(1), pages 137-145, July.
    5. 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.
    6. Leo K. Simon, 1987. "Games with Discontinuous Payoffs," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 54(4), pages 569-597.
    7. Llinares, Juan-Vicente, 1998. "Unified treatment of the problem of existence of maximal elements in binary relations: a characterization," Journal of Mathematical Economics, Elsevier, vol. 29(3), pages 285-302, April.
    8. Eaton, B Curtis & Lipsey, Richard G, 1976. "The Non-Uniqueness of Equilibrium in the Loschian Location Model," American Economic Review, American Economic Association, vol. 66(1), pages 71-93, March.
    9. Michael R. Baye & Guoqiang Tian & Jianxin Zhou, 1993. "Characterizations of the Existence of Equilibria in Games with Discontinuous and Non-quasiconcave Payoffs," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 60(4), pages 935-948.
    10. Reny, Philip J., 1995. "Local Payoff Security and the Existence of Nash Equilibrium in Discontinuous Games," Working Paper Series 435, Research Institute of Industrial Economics.
    11. Nishimura, Kazuo & Friedman, James, 1981. "Existence of Nash Equilibrium in n Person Games without Quasi-Concavity," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 22(3), pages 637-648, October.
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    More about this item

    Keywords

    Nash Equilibria Existence; Fixed Point Theorem; Generalized Convexity; 2 Person Game; 3 Person Game; Symmetric Game; Generalized Continuity.;
    All these keywords.

    JEL classification:

    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium

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