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On the Tikhonov Well-Posedness of Concave Games and Cournot Oligopoly Games

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  • M. Margiocco

    (University of Genova)

  • F. Patrone

    (University of Genova)

  • L. Pusillo

    (University of Genova)

Abstract

The purpose of this paper is to investigate whether theorems known to guarantee the existence and uniqueness of Nash equilibria, provide also sufficient conditions for the Tikhonov well-posedness (T-wp). We consider several hypotheses that ensure the existence and uniqueness of a Nash equilibrium (NE), such as strong positivity of the Jacobian of the utility function derivatives (Ref. 1), pseudoconcavity, and strict diagonal dominance of the Jacobian of the best reply functions in implicit form (Ref. 2). The aforesaid assumptions imply the existence and uniqueness of NE. We show that the hypotheses in Ref. 2 guarantee also the T-wp property of the Nash equilibrium. As far as the hypotheses in Ref. 1 are concerned, the result is true for quadratic games and zero-sum games. A standard way to prove the T-wp property is to show that the sets of ∈-equilibria are compact. This last approach is used to demonstrate directly the T-wp property for the Cournot oligopoly model given in Ref. 3. The compactness of ∈-equilibria is related also to the condition that the best reply surfaces do not approach each other near infinity.

Suggested Citation

  • M. Margiocco & F. Patrone & L. Pusillo, 2002. "On the Tikhonov Well-Posedness of Concave Games and Cournot Oligopoly Games," Journal of Optimization Theory and Applications, Springer, vol. 112(2), pages 361-379, February.
    • Handle: RePEc:spr:joptap:v:112:y:2002:i:2:d:10.1023_a:1013658023971
    DOI: 10.1023/A:1013658023971
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    References listed on IDEAS

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    1. Watts, Alison, 1996. "On the Uniqueness of Equilibrium in Cournot Oligopoly and Other Games," Games and Economic Behavior, Elsevier, vol. 13(2), pages 269-285, April.
    2. Szidarovszky, F & Yakowitz, S, 1977. "A New Proof of the Existence and Uniqueness of the Cournot Equilibrium," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 18(3), pages 787-789, October.
    3. M. Margiocco & F. Patrone & L. Pusillo Chicco, 1999. "Metric Characterizations of Tikhonov Well-Posedness in Value," Journal of Optimization Theory and Applications, Springer, vol. 100(2), pages 377-387, February.
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    Cited by:

    1. San-hua Wang & Nan-jing Huang & Donal O’Regan, 2013. "Well-posedness for generalized quasi-variational inclusion problems and for optimization problems with constraints," Journal of Global Optimization, Springer, vol. 55(1), pages 189-208, January.
    2. Yi-bin Xiao & Nan-jing Huang, 2011. "Well-posedness for a Class of Variational–Hemivariational Inequalities with Perturbations," Journal of Optimization Theory and Applications, Springer, vol. 151(1), pages 33-51, October.
    3. Francisco Facchinei & Christian Kanzow, 2010. "Generalized Nash Equilibrium Problems," Annals of Operations Research, Springer, vol. 175(1), pages 177-211, March.
    4. Fang, Ya-Ping & Huang, Nan-Jing & Yao, Jen-Chih, 2010. "Well-posedness by perturbations of mixed variational inequalities in Banach spaces," European Journal of Operational Research, Elsevier, vol. 201(3), pages 682-692, March.

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