IDEAS home Printed from
   My bibliography  Save this paper

Axiomatizations of symmetrically weighted solutions


  • Kleppe, John

    () (CentER and Department of Econometrics and Operations Research)

  • Reijnierse, Hans

    () (CentER and Department of Econometrics and Operations Research)

  • Sudhölter, Peter

    () (Department of Business and Economics)


If the excesses of the coalitions in a transferable utility game are weighted, then we show that the arising weighted modifications of the well-known (pre)nucleolus and (pre)kernel satisfy the equal treatment property if and only if the weight system is symmetric in the sense that the weight of a subcoalition of a grand coalition may only depend on the grand coalition and the size of the subcoalition. Hence, the symmetrically weighted versions of the (pre)nucleolus and the (pre)kernel are symmetric, i.e., invariant under symmetries of a game. They may, however, violate anonymity, i.e., they may depend on the names of the players. E.g., a symmetrically weighted nucleolus may assign the classical nucleolus to one game and the per capita nucleolus to another game. We generalize Sobolev's axiomatization of the prenucleolus and its modification for the nucleolus as well as Peleg's axiomatization of the prekernel to the symmetrically weighted versions. Only the reduced games have to be replaced by suitably modified reduced games whose definitions may depend on the weight system. Moreover, it is shown that a solution may only satisfy the mentioned sets of modified axioms if the weight system is symmetric.

Suggested Citation

  • Kleppe, John & Reijnierse, Hans & Sudhölter, Peter, 2013. "Axiomatizations of symmetrically weighted solutions," Discussion Papers of Business and Economics 3/2013, University of Southern Denmark, Department of Business and Economics.
  • Handle: RePEc:hhs:sdueko:2013_003

    Download full text from publisher

    File URL:
    Download Restriction: no

    Other versions of this item:

    References listed on IDEAS

    1. 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.
    2. 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.
    3. Potters, J.A.M. & Tijs, S.H., 1992. "The nucleolus of a matrix game and other nucleoli," Other publications TiSEM ae3402e7-bd19-494b-b0a1-f, Tilburg University, School of Economics and Management.
    4. Hans Haller & Jean Derks, 1999. "Weighted nucleoli," International Journal of Game Theory, Springer;Game Theory Society, vol. 28(2), pages 173-187.
    5. 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.
    Full references (including those not matched with items on IDEAS)


    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.

    Cited by:

    1. Meinhardt, Holger Ingmar, 2015. "The Incorrect Usage of Propositional Logic in Game Theory: The Case of Disproving Oneself," MPRA Paper 66637, University Library of Munich, Germany.

    More about this item


    TU game; nucleolus; kernel;

    JEL classification:

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

    NEP fields

    This paper has been announced in the following NEP Reports:


    Access and download statistics


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:hhs:sdueko:2013_003. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Lene Holbæk). General contact details of provider: .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.