Preserving coalitional rationality for non-balanced games
AbstractIn cooperative games, the core is one of the most popular solution concept since it ensures coalitional rationality. For non-balanced games however, the core is empty, and other solution concepts have to be found. We propose the use of general solutions, that is, to distribute the total worth of the game among groups rather than among individuals. In particular, the k-additive core proposed by Grabisch and Miranda is a general solution preserving coalitional rationality which distributes among coalitions of size at most k, and is never ampty for k ? 2. The extended core of Bejan and Gomez can also be viewed as a general solution, since it implies to give an amount to the grand coalition. The k-additive core being an unbounded set and therefore difficult to use in practice, we propose a subset of it called the minimal bargaining set. The idea is to select elements of the k-additive core minimizing the total amount given to coalitions of size greater than 1. Thus the minimum bargaining set naturally reduces to the core for balanced games. We study this set, giving properties and axiomatizations, as well as its relation to the extended core of Bejan and Gomez. We introduce also the notion of unstable coalition, and show how to find them using the minimum bargaining set. Lastly, we give a method of computing the minimum bargaining set.
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Bibliographic InfoPaper provided by Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne in its series Documents de travail du Centre d'Economie de la Sorbonne with number 12022.
Length: 27 pages
Date of creation: Apr 2012
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Cooperative game; core; balancedness; general solution.;
Other versions of this item:
- Stéphane Gonzalez & Michel Grabisch, 2012. "Preserving coalitional rationality for non-balanced games," UniversitÃ© Paris1 PanthÃ©on-Sorbonne (Post-Print and Working Papers) halshs-00718358, HAL.
- C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
This paper has been announced in the following NEP Reports:
- NEP-ALL-2012-05-02 (All new papers)
- NEP-GTH-2012-05-02 (Game Theory)
- NEP-HPE-2012-05-02 (History & Philosophy of Economics)
- NEP-MIC-2012-05-02 (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.:
- Michel Grabisch & Tong Li, 2011.
"On the set of imputations induced by the k-additive core,"
UniversitÃ© Paris1 PanthÃ©on-Sorbonne (Post-Print and Working Papers)
- Grabisch, Michel & Li, Tong, 2011. "On the set of imputations induced by the k-additive core," European Journal of Operational Research, Elsevier, vol. 214(3), pages 697-702, November.
- Pedro Miranda & Michel Grabisch, 2008.
"K-balanced games and capacities,"
Documents de travail du Centre d'Economie de la Sorbonne
b08079, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
- Pedro Miranda & Michel Grabisch, 2008. "K-balanced games and capacities," UniversitÃ© Paris1 PanthÃ©on-Sorbonne (Post-Print and Working Papers) halshs-00344809, HAL.
- Pedro Miranda & Michel Grabisch, 2010. "k-balanced games and capacities," UniversitÃ© Paris1 PanthÃ©on-Sorbonne (Post-Print and Working Papers) halshs-00445073, HAL.
- repec:hal:journl:hal-00321625 is not listed on IDEAS
- Camelia Bejan & Juan Gómez, 2009. "Core extensions for non-balanced TU-games," International Journal of Game Theory, Springer, vol. 38(1), pages 3-16, March.
- Chateauneuf, Alain & Jaffray, Jean-Yves, 1989. "Some characterizations of lower probabilities and other monotone capacities through the use of Mobius inversion," Mathematical Social Sciences, Elsevier, vol. 17(3), pages 263-283, June.
- Michel Grabisch & Pedro Miranda, 2008. "On the vertices of the k-additive core," UniversitÃ© Paris1 PanthÃ©on-Sorbonne (Post-Print and Working Papers) hal-00321625, HAL.
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