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A New Existence and Uniqueness Theorem for Continuous Games

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Abstract

This paper derives a general sufficient condition for existence and uniqueness in continuous games using a variant of the contraction mapping theorem applied to mapping from a subset of the real line on to itself. We first prove this contraction mapping variant, and then show how the existence of a unique equilibrium in the general game can be shown by proving the existence of a unique equilibrium in an iterative sequence of games involving such R-to-R mappings. Finally, we show how a general condition for this to occur is that a matrix derived from the Jacobean matrix of best-response functions be have positive leading principal minors, and how this condition generalises some existing uniqueness theorems for particular games.

Suggested Citation

  • Seamus Hogan, 2010. "A New Existence and Uniqueness Theorem for Continuous Games," Working Papers in Economics 10/59, University of Canterbury, Department of Economics and Finance.
    • Handle: RePEc:cbt:econwp:10/59
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    File URL: https://repec.canterbury.ac.nz/cbt/econwp/1059.pdf
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    References listed on IDEAS

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    1. Drew Fudenberg & Jean Tirole, 1991. "Game Theory," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262061414, 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. Van Long, Ngo & Soubeyran, Antoine, 2000. "Existence and uniqueness of Cournot equilibrium: a contraction mapping approach," Economics Letters, Elsevier, vol. 67(3), pages 345-348, June.
    4. Charles D. Kolstad & Lars Mathiesen, 1987. "Necessary and Sufficient Conditions for Uniqueness of a Cournot Equilibrium," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 54(4), pages 681-690.
    5. Paul A. Samuelson, 1953. "Prices of Factors and Goods in General Equilibrium," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 21(1), pages 1-20.
    6. Gérard Gaudet & Stephen W. Salant, 1991. "Uniqueness of Cournot Equilibrium: New Results From Old Methods," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 58(2), pages 399-404.
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    Cited by:

    1. Klis Anna A., 2019. "On the Openness of Unique Pure-Strategy Nash Equilibrium," The B.E. Journal of Theoretical Economics, De Gruyter, vol. 19(1), pages 1-9, January.
    2. Jinji, Naoto, 2014. "Comparative statics for oligopoly: A generalized result," Economics Letters, Elsevier, vol. 124(1), pages 79-82.

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    More about this item

    Keywords

    Existence; Uniqueness; Continuous Games; Contraction Mapping Theorem;
    All these keywords.

    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • C72 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Noncooperative Games
    • D43 - Microeconomics - - Market Structure, Pricing, and Design - - - Oligopoly and Other Forms of Market Imperfection

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