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A New Existence and Uniqueness Theorem for Continuous Games

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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.

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File URL: http://www.econ.canterbury.ac.nz/RePEc/cbt/econwp/1059.pdf
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Paper provided by University of Canterbury, Department of Economics and Finance in its series Working Papers in Economics with number 10/59.

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Length: 28 pages
Date of creation: 01 Oct 2010
Handle: RePEc:cbt:econwp:10/59
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  1. Gérard Gaudet & Stephen W. Salant, 1991. "Uniqueness of Cournot Equilibrium: New Results From Old Methods," Review of Economic Studies, Oxford University Press, vol. 58(2), pages 399-404.
  2. Drew Fudenberg & Jean Tirole, 1991. "Game Theory," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262061414, July.
  3. 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.
  4. Charles D. Kolstad & Lars Mathiesen, 1987. "Necessary and Sufficient Conditions for Uniqueness of a Cournot Equilibrium," Review of Economic Studies, Oxford University Press, vol. 54(4), pages 681-690.
  5. Paul A. Samuelson, 1953. "Prices of Factors and Goods in General Equilibrium," Review of Economic Studies, Oxford University Press, vol. 21(1), pages 1-20.
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