Axiomatizations of symmetrically weighted solutions
AbstractIf 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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Bibliographic InfoPaper provided by Department of Business and Economics, University of Southern Denmark in its series Discussion Papers of Business and Economics with number 3/2013.
Length: 18 pages
Date of creation: 01 Feb 2013
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TU game; nucleolus; kernel;
Other versions of this item:
- Kleppe, J. & Reijnierse, J.H. & Sudhölter, P., 2013. "Axiomatizations Of Symmetrically Weighted Solutions," Discussion Paper 2013-007, Tilburg University, Center for Economic Research.
- C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
This paper has been announced in the following NEP Reports:
- NEP-ALL-2013-03-09 (All new papers)
- NEP-GTH-2013-03-09 (Game Theory)
- NEP-HPE-2013-03-09 (History & Philosophy of Economics)
- NEP-MIC-2013-03-09 (Microeconomics)
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- Orshan, Gooni, 1993. "The Prenucleolus and the Reduced Game Property: Equal Treatment Replaces Anonymity," International Journal of Game Theory, Springer, vol. 22(3), pages 241-48.
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- Peleg, B, 1986. "On the Reduced Game Property and Its Converse," International Journal of Game Theory, Springer, vol. 15(3), pages 187-200.
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- Kleppe, J., 2010. "Modelling interactive behaviour, and solution concepts," Open Access publications from Tilburg University urn:nbn:nl:ui:12-3763576, Tilburg University.
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