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An extremely simple proof of the K-K-M-S Theorem

Author

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  • P. Jean-Jacques Herings

    (Department of Econometrics and CentER, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, THE NETHERLANDS)

Abstract

An extremely simple proof of the K-K-M-S Theorem is given involving only Brouwer's fixed point theorem and some elementary calculus. A function is explicitly given such that a fixed point of it yields an intersection point of a balanced collection of sets together with balancing weights. Moreover, any intersection point of a balanced collection of sets together with balancing weights corresponds to a fixed point of the function. Furthermore, the proof can be used to show $\pi $-balanced versions of the K-K-M-S Theorem, with $\pi $-balancedness as introduced in Billera (1970). The proof makes clear that the conditions made with respect to $\pi $ by Billera can be even weakened.

Suggested Citation

  • P. Jean-Jacques Herings, 1997. "An extremely simple proof of the K-K-M-S Theorem," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 10(2), pages 361-367.
    • Handle: RePEc:spr:joecth:v:10:y:1997:i:2:p:361-367
    Note: Received: January 22, 1996; revised version June 9, 1996
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    Cited by:

    1. P. Herings & A. Predtetchinski & A. Perea, 2006. "The Weak Sequential Core for Two-Period Economies," International Journal of Game Theory, Springer;Game Theory Society, vol. 34(1), pages 55-65, April.
    2. P. Jean-Jacques Herings & Gerard van der Laan & Dolf Talman, 2000. "Cooperative Games in Graph Structure," Tinbergen Institute Discussion Papers 00-072/1, Tinbergen Institute.
    3. Jean Guillaume Forand & Metin Uyanık, 2019. "Fixed-point approaches to the proof of the Bondareva–Shapley Theorem," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 7(1), pages 117-124, May.
    4. Herings, P.J.J. & van der Laan, G. & Talman, A.J.J., 2003. "Socially Structured Games and their Applications," Discussion Paper 2003-40, Tilburg University, Center for Economic Research.
    5. Predtetchinski, Arkadi & Jean-Jacques Herings, P., 2004. "A necessary and sufficient condition for non-emptiness of the core of a non-transferable utility game," Journal of Economic Theory, Elsevier, vol. 116(1), pages 84-92, May.
    6. Liu, Jiuqiang & Tian, Hai-Yan, 2014. "Existence of fuzzy cores and generalizations of the K–K–M–S theorem," Journal of Mathematical Economics, Elsevier, vol. 52(C), pages 148-152.
    7. P. Herings & Gerard Laan & Dolf Talman, 2007. "Socially Structured Games," Theory and Decision, Springer, vol. 62(1), pages 1-29, February.
    8. Azrieli, Yaron & Shmaya, Eran, 2014. "Rental harmony with roommates," Journal of Economic Theory, Elsevier, vol. 153(C), pages 128-137.

    More about this item

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games

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