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Axiomatizations Of Symmetrically Weighted Solutions

Author

Listed:
  • Kleppe, J.

    (Tilburg University, Center For Economic Research)

  • Reijnierse, J.H.

    (Tilburg University, Center For Economic Research)

  • Sudhölter, P.

Abstract

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.
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Suggested Citation

  • Kleppe, J. & Reijnierse, J.H. & Sudhölter, P., 2013. "Axiomatizations Of Symmetrically Weighted Solutions," Discussion Paper 2013-007, Tilburg University, Center for Economic Research.
    • Handle: RePEc:tiu:tiucen:98930f2c-667e-4a4a-b76e-85dd1d783896
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    References listed on IDEAS

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    Cited by:

    1. Tamás Solymosi, 2019. "Weighted nucleoli and dually essential coalitions," International Journal of Game Theory, Springer;Game Theory Society, vol. 48(4), pages 1087-1109, December.
    2. Natalia I. Naumova, 2022. "Some solutions for generalized games with restricted cooperation," Annals of Operations Research, Springer, vol. 318(2), pages 1077-1093, November.
    3. Calleja, Pedro & Llerena, Francesc & Sudhölter, Peter, 2018. "Monotonicity and weighted prenucleoli: A characterization without consistency," Discussion Papers on Economics 4/2018, University of Southern Denmark, Department of Economics.
    4. 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.
    5. Tamás Solymosi, 2019. "Weighted nucleoli and dually essential coalitions (extended version)," CERS-IE WORKING PAPERS 1914, Institute of Economics, Centre for Economic and Regional Studies.
    6. 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.
    7. Pedro Calleja & Francesc Llerena & Peter Sudhölter, 2020. "Monotonicity and Weighted Prenucleoli: A Characterization Without Consistency," Mathematics of Operations Research, INFORMS, vol. 45(3), pages 1056-1068, August.

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    More about this item

    Keywords

    TU game · Nucleolus · Kernel;

    JEL classification:

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

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