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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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    Cited by:

    1. is not listed on IDEAS
    2. 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.
    3. Jinji, Naoto, 2014. "Comparative statics for oligopoly: A generalized result," Economics Letters, Elsevier, vol. 124(1), pages 79-82.

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    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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