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On Kakutani's Fixed Point Theorem, the K-K-M-S Theorem and the Core of a Balanced Game

Author

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  • Shapley, Lloyd
  • Vohra, Rajiv

Abstract

We provide elementary proofs of Scarf's theorem on the non-emptiness of the core and of the K-K-M-S theorem, based.on Kakutani's fixed point theorem. We also show how these proofs can be modified to apply a coincidence theorem of Fan instead of Kakutani's fixed point theorem, for some additional simplicity.

Suggested Citation

  • Shapley, Lloyd & Vohra, Rajiv, 1991. "On Kakutani's Fixed Point Theorem, the K-K-M-S Theorem and the Core of a Balanced Game," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 1(1), pages 108-116, January.
    • Handle: RePEc:spr:joecth:v:1:y:1991:i:1:p:108-16
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    Citations

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    Cited by:

    1. Kannai, Yakar & Wooders, Myrna H., 1999. "A Further Extension of the KKMS Theorem," Economic Research Papers 269259, University of Warwick - Department of Economics.
    2. Bonnisseau, Jean-Marc & Iehle, Vincent, 2007. "Payoff-dependent balancedness and cores," Games and Economic Behavior, Elsevier, vol. 61(1), pages 1-26, October.
    3. Takaaki Abe & Yukihiko Funaki, 2018. "The Unbinding Core for Coalitional Form Games," Working Papers 1805, Waseda University, Faculty of Political Science and Economics.
    4. 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.
    5. Bialek, William & DeWeese, Michael & Rieke, Fred & Warland, David, 1993. "Bits and brains: Information flow in the nervous system," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 200(1), pages 581-593.
    6. P. J. J. Herings & A. J. J. Talman, 1998. "Intersection Theorems with a Continuum of Intersection Points," Journal of Optimization Theory and Applications, Springer, vol. 96(2), pages 311-335, February.
    7. 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.
    8. Nizar Allouch & Arkadi Predtetchinski, 2008. "On the non-emptiness of the fuzzy core," International Journal of Game Theory, Springer;Game Theory Society, vol. 37(2), pages 203-210, June.
    9. Liu, Jiuqiang & Liu, Xiaodong, 2013. "A necessary and sufficient condition for an NTU fuzzy game to have a non-empty fuzzy core," Journal of Mathematical Economics, Elsevier, vol. 49(2), pages 150-156.
    10. Nizar Allouch & Monique Florenzano, 2004. "Edgeworth and Walras equilibria of an arbitrage-free exchange economy," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 23(2), pages 353-370, January.
    11. Jiuqiang Liu & Xiaodong Liu, 2014. "Existence of Edgeworth and competitive equilibria and fuzzy cores in coalition production economies," International Journal of Game Theory, Springer;Game Theory Society, vol. 43(4), pages 975-990, November.
    12. Azrieli, Yaron & Shmaya, Eran, 2014. "Rental harmony with roommates," Journal of Economic Theory, Elsevier, vol. 153(C), pages 128-137.
    13. Abe, Takaaki & Funaki, Yukihiko, 2021. "The unbinding core for coalitional form games," Mathematical Social Sciences, Elsevier, vol. 113(C), pages 39-42.
    14. Yakar Kannai, 2013. "Using oriented volume to prove Sperner’s lemma," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 1(1), pages 11-19, May.
    15. Charalambos Aliprantis & Kim Border & Owen Burkinshaw, 1996. "Market economies with many commodities," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 19(1), pages 113-185, March.
    16. Jean-Marc Bonnisseau & Vincent Iehlé, 2007. "Payoff-dependent balancedness and cores (revised version)," UFAE and IAE Working Papers 678.07, Unitat de Fonaments de l'Anàlisi Econòmica (UAB) and Institut d'Anàlisi Econòmica (CSIC).
    17. Nicolò, Antonio & Salmaso, Pietro & Sen, Arunava & Yadav, Sonal, 2023. "Stable sharing," Games and Economic Behavior, Elsevier, vol. 141(C), pages 337-363.

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