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The restricted core of games on distributive lattices: how to share benefits in a hierarchy

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  • Michel Grabisch

    ()
    (CES - Centre d'économie de la Sorbonne - CNRS : UMR8174 - Université Paris 1 - Panthéon-Sorbonne)

  • Lijue Xie

    (China - chine)

Abstract

Finding a solution concept is one of the central problems in cooperative game theory, and the notion of core is the most popular solution concept since it is based on some rationality condition. In many real situations, not all possible coalitions can form, so that classical TU-games cannot be used. An interesting case is when possible coalitions are defined through a partial ordering of the players (or hierarchy). Then feasible coalitions correspond to teams of players, that is, one or several players with all their subordinates. In these situations, the core in its usual formulation may be unbounded, making its use difficult in practice. We propose a new notion of core, called the restricted core, which imposes efficiency of the allocation at each level of the hierarchy, is always bounded, and answers the problem of sharing benefits in a hierarchy. We show that the core we defined has properties very close to the classical case, with respect to marginal vectors, the Weber set, and balancedness.

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Bibliographic Info

Paper provided by HAL in its series Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) with number halshs-00583868.

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Date of creation: 2011
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Publication status: Published, Mathematical Methods of Operations Research, 2011, 73, 2, 189-208
Handle: RePEc:hal:cesptp:halshs-00583868

Note: View the original document on HAL open archive server: http://halshs.archives-ouvertes.fr/halshs-00583868
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Keywords: cooperative game; feasible coalition; core; hierarchy;

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  1. Michel Grabisch & Lijue Xie, 2008. "The core of games on distributive lattices : how to share benefits in a hierarchy," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00344802, HAL.
  2. Michel Grabisch & Lijue Xie, 2007. "A new approach to the core and Weber set of multichoice games," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00267933, HAL.
  3. Ambec, S. & Sprumont, Y., 2000. "Sharing a River," Cahiers de recherche 2000-08, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
  4. Gabrielle Demange, 2004. "On Group Stability in Hierarchies and Networks," Journal of Political Economy, University of Chicago Press, vol. 112(4), pages 754-778, August.
  5. René van den Brink & Gerard van der Laan & Valeri Vasil'ev, 2007. "Distributing Dividends in Games with Ordered Players," Tinbergen Institute Discussion Papers 06-114/1, Tinbergen Institute.
  6. Gilles, R.P. & Owen, G. & Brink, J.R. van den, 1991. "Games with permission structures: The conjunctive approach," Discussion Paper 1991-14, Tilburg University, Center for Economic Research.
  7. Michel Grabisch, 2013. "The core of games on ordered structures and graphs," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-00803233, HAL.
  8. Hans Reijnierse & Jean Derks, 1998. "Note On the core of a collection of coalitions," International Journal of Game Theory, Springer, vol. 27(3), pages 451-459.
  9. Michel Grabisch & Fabien Lange, 2007. "Games on lattices, multichoice games and the Shapley value: a new approach," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00178916, HAL.
  10. René Brink & Gerard Laan & Valeri Vasil’ev, 2007. "Component efficient solutions in line-graph games with applications," Economic Theory, Springer, vol. 33(2), pages 349-364, November.
  11. Algaba, A. & Bilbao, J.M. & Brink, J.R. van den & Jiménez-Losada, A., 2000. "Cooperative Games on Antimatroids," Discussion Paper 2000-124, Tilburg University, Center for Economic Research.
  12. Faigle, U & Kern, W, 1992. "The Shapley Value for Cooperative Games under Precedence Constraints," International Journal of Game Theory, Springer, vol. 21(3), pages 249-66.
  13. Derks, Jean J M & Gilles, Robert P, 1995. "Hierarchical Organization Structures and Constraints on Coalition Formation," International Journal of Game Theory, Springer, vol. 24(2), pages 147-63.
  14. Hsiao Chih-Ru & Raghavan T. E. S., 1993. "Shapley Value for Multichoice Cooperative Games, I," Games and Economic Behavior, Elsevier, vol. 5(2), pages 240-256, April.
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