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A fixed point theorem for discontinuous functions

Author

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  • HERINGS, P. Jean-Jacques
  • van der LAAN, Gerard
  • TALMAN, Dolf
  • YANG, Zaifu

Abstract

Any function from a non-empty polytope into itself that is locally gross direction preserving is shown to have the fixed point property. Brouwer's fixed point theorem for continuous functions is a special case. We discuss the application of the result in the area of non-cooperative game theory. This discussion paper has resulted in a publication in Operations Research Letters . (2008, vol. 36, issue 1, pp.89-93.)
(This abstract was borrowed from another version of this item.)

Suggested Citation

  • HERINGS, P. Jean-Jacques & van der LAAN, Gerard & TALMAN, Dolf & YANG, Zaifu, 2009. "A fixed point theorem for discontinuous functions," CORE Discussion Papers RP 2154, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  • Handle: RePEc:cor:louvrp:2154
    Note: In : Operations Research Letters, 36, 89-93, 2008
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    File URL: http://dx.doi.org/10.1016/j.orl.2007.03.008
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    References listed on IDEAS

    as
    1. Drew Fudenberg & Jean Tirole, 1991. "Game Theory," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262061414, January.
    2. P. Jean-Jacques Herings, 2000. "Two simple proofs of the feasibility of the linear tracing procedure," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 15(2), pages 485-490.
    3. Dai, Y. & van der Laan, G. & Talman, A.J.J. & Yamamoto, Y., 1989. "A simplicial algorithm for the nonlinear stationary point problem on an unbounded polyhedron," Discussion Paper 1989-52, Tilburg University, Center for Economic Research.
    4. Talman, A.J.J. & Yamamoto, Y., 1989. "A simplicial algorithm for stationary point problems on polytopes," Other publications TiSEM 0d6b2de0-17c0-4d5e-963f-5, Tilburg University, School of Economics and Management.
    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. Shoven,John B. & Whalley,John, 1992. "Applying General Equilibrium," Cambridge Books, Cambridge University Press, number 9780521266550, May.
    7. Gérard Debreu (ed.), 1996. "General Equilibrium Theory," Books, Edward Elgar Publishing, volume 0, number 548.
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    Citations

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    Cited by:

    1. Philippe Bich, 2006. "Some fixed point theorems for discontinuous mappings," Cahiers de la Maison des Sciences Economiques b06066, Université Panthéon-Sorbonne (Paris 1).
    2. Takao Fujimoto, 2013. "Fixed Point Theorems for Discontinuous Maps on a Non-convex Domain," Metroeconomica, Wiley Blackwell, vol. 64(3), pages 547-572, July.

    More about this item

    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques

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