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A Further Extension of the KKMS Theorem

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  • Kannai, Yakar
  • Wooders, Myrna H.

Abstract

Recently Reny and Wooders ([23]) showed that there is some point in the intersection of sets in Shapley's ([24]) generalization of the Knaster-Kuratowski-Mazurkiwicz Theorem with the property that the collection of all sets containing that point is partnered as well as balanced. In this paper we provide a further extension by showing that the collection of all such sets can be chosen to be strictly balanced, implying the Reny-Wooders result. Our proof is topological, based on the Eilenberg-Montgomery ¯xed point Theorem. Reny and Wooders ([23]) also show that if the collection of partnered points in the intersection is countable, then at least one of them is minimally partnered. Applying degree theory for correspondences, we show that if this collection is only assumed to be zero dimensional (or if the set of partnered and strictly balanced points is of dimension zero), then there is at least one strictly balanced and minimally partnered point in the intersection. The approach presented in this paper sheds a new geometric-topological light on the Reny-Wooders results.

Suggested Citation

  • Kannai, Yakar & Wooders, Myrna H., 1999. "A Further Extension of the KKMS Theorem," Economic Research Papers 269259, University of Warwick - Department of Economics.
    • Handle: RePEc:ags:uwarer:269259
    DOI: 10.22004/ag.econ.269259
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    References listed on IDEAS

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    1. McLennan, Andrew, 1989. "Fixed Points of Contractible Valued Correspondences," International Journal of Game Theory, Springer;Game Theory Society, vol. 18(2), pages 175-184.
    2. Philip Reny & Eyal Winter & Myrna Wooders, 2012. "The partnered core of a game with side payments," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 39(2), pages 521-536, July.
    3. Keiding, Hans, 1985. "On the existence of equilibrium in social systems with coordination," Journal of Mathematical Economics, Elsevier, vol. 14(2), pages 105-111, April.
    4. Reny, Philip J. & Holtz Wooders, Myrna, 1996. "The Partnered Core of a Game without Side Payments," Journal of Economic Theory, Elsevier, vol. 70(2), pages 298-311, August.
    5. Reny, Philip J. & Holtz Wooders, Myrna, 1998. "An extension of the KKMS theorem," Journal of Mathematical Economics, Elsevier, vol. 29(2), pages 125-134, March.
    6. Bennett, Elaine & Zame, William R, 1988. "Bargaining in Cooperative Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 17(4), pages 279-300.
    7. McLennan, Andrew, 1989. "Consistent Conditional Systems in Noncooperative Game Theory," International Journal of Game Theory, Springer;Game Theory Society, vol. 18(2), pages 141-174.
    8. Page Jr., Frank H. & Wooders, Myrna Holtz, 1996. "The Partnered Core of an Economy and the Partnered Competitive Equilibrium," Economics Letters, Elsevier, vol. 52(2), pages 143-152, August.
    9. 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.
    10. Kannai, Yakar, 1992. "The core and balancedness," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 1, chapter 12, pages 355-395, Elsevier.
    11. Mas-Colell, Andrew, 1974. "An equilibrium existence theorem without complete or transitive preferences," Journal of Mathematical Economics, Elsevier, vol. 1(3), pages 237-246, December.
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    Cited by:

    1. Reny, Philip J. & Holtz Wooders, Myrna, 1998. "An extension of the KKMS theorem," Journal of Mathematical Economics, Elsevier, vol. 29(2), pages 125-134, March.
    2. 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).
    3. Philip Reny & Eyal Winter & Myrna Wooders, 2012. "The partnered core of a game with side payments," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 39(2), pages 521-536, July.

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