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Nonsymmetric variants of the prekernel and the prenucleolus

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  • Guni Orshan
  • Peter Sudhölter

Abstract

A solution on a class of TU games that satisfies the axioms of the pre-nucleolus or -kernel except the equal treatment property and is single valued for two-person games, is a nonsymmetric pre-nucleolus (NSPN) or -kernel (NSPK). We investigate the NSPKs and NSPNs and their relations to the positive prekernel and to the positive core. It turns out that any NSPK is a subsolution of the positive prekernel. Moreover, we show that an arbitrary NSPK, when applied to a TU game, intersects the set of preimputations whose dissatisfactions coincide with the dissatisfactions of an arbitrary element of any other NSPK applied to this game. This result also provides a new proof of sufficiency of the characterizing conditions for NSPKs introduced by Orshan (Non-symmetric prekernels, discussion paper 60. Center for Rationality, The Hebrew University of Jerusalem, 1994 ). Any NSPN belongs to “its” NSPK. Several classes of NSPNs are presented, all of them being subsolutions of the positive core. We show that any NSPN is a subsolution of the positive core provided that it satisfies the equal treatment property on an infinite subset of the universe of potential players. Moreover, we prove that, for any game whose prenucleolus is in its anticore, any NSPN coincides with the prenucleolus. Copyright Springer-Verlag 2012

Suggested Citation

  • Guni Orshan & Peter Sudhölter, 2012. "Nonsymmetric variants of the prekernel and the prenucleolus," International Journal of Game Theory, Springer;Game Theory Society, vol. 41(4), pages 809-828, November.
    • Handle: RePEc:spr:jogath:v:41:y:2012:i:4:p:809-828
    DOI: 10.1007/s00182-011-0294-6
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    References listed on IDEAS

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    1. Orshan, Gooni, 1993. "The Prenucleolus and the Reduced Game Property: Equal Treatment Replaces Anonymity," International Journal of Game Theory, Springer;Game Theory Society, vol. 22(3), pages 241-248.
    2. SCHMEIDLER, David, 1969. "The nucleolus of a characteristic function game," LIDAM Reprints CORE 44, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Aumann, Robert J. & Maschler, Michael, 1985. "Game theoretic analysis of a bankruptcy problem from the Talmud," Journal of Economic Theory, Elsevier, vol. 36(2), pages 195-213, August.
    4. Peleg, B, 1986. "On the Reduced Game Property and Its Converse," International Journal of Game Theory, Springer;Game Theory Society, vol. 15(3), pages 187-200.
    5. Peter Sudhölter & Bezalel Peleg, 2000. "The Positive Prekernel Of A Cooperative Game," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 2(04), pages 287-305.
    6. Guni Orshan & Peter Sudhölter, 2003. "Reconfirming the Prenucleolus," Mathematics of Operations Research, INFORMS, vol. 28(2), pages 283-293, May.
    7. Bezalel Peleg & Peter Sudhölter, 2007. "Introduction to the Theory of Cooperative Games," Theory and Decision Library C, Springer, edition 0, number 978-3-540-72945-7, December.
    8. Guni Orshan & Peter Sudhölter, 2010. "The positive core of a cooperative game," International Journal of Game Theory, Springer;Game Theory Society, vol. 39(1), pages 113-136, March.
    9. Peter Sudhölter & Yan-An Hwang, 2001. "Axiomatizations of the core on the universal domain and other natural domains," International Journal of Game Theory, Springer;Game Theory Society, vol. 29(4), pages 597-623.
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    Cited by:

    1. Elena Iñarra & Roberto Serrano & Ken-Ichi Shimomura, 2020. "The Nucleolus, the Kernel, and the Bargaining Set: An Update," Revue économique, Presses de Sciences-Po, vol. 71(2), pages 225-266.
    2. Michel Grabisch & Hervé Moulin & José Manuel Zarzuelo, 2024. "Professor Peter Sudhölter (1957–2024)," International Journal of Game Theory, Springer;Game Theory Society, vol. 53(2), pages 289-294, June.

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    More about this item

    Keywords

    TU game; Solution concept; Kernel; Nucleolus; Core; Equal treatment; C71;
    All these keywords.

    JEL classification:

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

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