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Preserving coalitional rationality for non-balanced games

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Abstract

In 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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Paper 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.

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Length: 27 pages
Date of creation: Apr 2012
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Handle: RePEc:mse:cesdoc:12022

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Keywords: Cooperative game; core; balancedness; general solution.;

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  1. Thomson, William, 2003. "Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: a survey," Mathematical Social Sciences, Elsevier, vol. 45(3), pages 249-297, July.
  2. repec:hal:cesptp:hal-00321625 is not listed on IDEAS
  3. 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.
  4. 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.
  5. repec:hal:cesptp:halshs-00445073 is not listed on IDEAS
  6. 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.
  7. repec:hal:journl:halshs-00445073 is not listed on IDEAS
  8. 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.
  9. repec:hal:cesptp:hal-00625339 is not listed on IDEAS
  10. 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.
  11. Jean Derks & Hans Haller & Hans Peters, 2000. "The selectope for cooperative games," International Journal of Game Theory, Springer, vol. 29(1), pages 23-38.
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Cited by:
  1. Stéphane Gonzalez & Michel Grabisch, 2013. "Multicoalitional solutions," Documents de travail du Centre d'Economie de la Sorbonne 13062, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
  2. Stéphane Gonzalez & Michel Grabisch, 2013. "Multicoalitional solutions," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00881108, HAL.

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