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Ensuring the boundedness of the core of games with restricted cooperation

The core of a cooperative game on a set of players N is one of the most popular concept of solution. When cooperation is restricted (feasible coalitions form a subcollection F of 2N), the core may become unbounded, which makes it usage questionable in practice. Our proposal is to make the core bounded by turning some of the inequalities defining the core into equalities (additional efficiency constraints). We address the following mathematical problem : can we find a minimal set of inequalities in the core such that, if turned into equalities, the core becomes bounded ? The new core obtained is called the restricted core. We completely solve the question when F is a distributive lattice, introducing also the notion of restricted Weber set. We show that the case of regular set systems amounts more or less to the case of distributive lattices. We also study the case of weakly union-closed systems and give some results for the general case.

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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 10093.

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Length: 19 pages
Date of creation: Nov 2010
Date of revision:
Handle: RePEc:mse:cesdoc:10093
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  1. Honda, Aoi & Grabisch, Michel, 2008. "An axiomatization of entropy of capacities on set systems," European Journal of Operational Research, Elsevier, vol. 190(2), pages 526-538, October.
  2. 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.
  3. Ulrich Faigle & Michel Grabisch & Maximilian Heyne, 2010. "Monge extensions of cooperation and communication structures," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-00625336, HAL.
  4. 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.
  5. 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.
  6. Fabien Lange & Michel Grabisch, 2006. "Values on regular games under Kirchhoff's laws," Cahiers de la Maison des Sciences Economiques b06087, Université Panthéon-Sorbonne (Paris 1).
  7. 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.
  8. Christophe Labreuche & Michel Grabisch, 2008. "A value for bi-cooperative games," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00308738, HAL.
  9. Suijs, J.P.M. & Borm, P.E.M. & Hamers, H.J.M. & Koster, M.A.L. & Quant, M., 2001. "Communications and Cooperation in Public Network Situations," Discussion Paper 2001-44, Tilburg University, Center for Economic Research.
  10. 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.
  11. Jesús Mario Bilbao & Julio R. Fernández & Nieves Jiménez & Jorge Jesús López, 2004. "The Shapley value for bicooperative games," Economic Working Papers at Centro de Estudios Andaluces E2004/56, Centro de Estudios Andaluces.
  12. 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.
  13. 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.
  14. Michel Grabisch, 2009. "The core of games on ordered structures and graphs," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00445171, HAL.
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