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Ensuring the boundedness of the core of games with restricted cooperation

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  • Michel Grabisch

    ()
    (CES - Centre d'économie de la Sorbonne - CNRS : UMR8174 - Université Paris 1 - Panthéon-Sorbonne, EEP-PSE - Ecole d'Économie de Paris - Paris School of Economics - Ecole d'Économie de Paris)

Abstract

The core of a cooperative game on a set of players $N$ is one of the most popular concepts of solution. When cooperation is restricted (feasible coalitions form a subcollection $\cF$ of $2^N$), the core may become unbounded, which makes its 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 $\cF$ 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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Bibliographic Info

Paper provided by HAL in its series Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) with number hal-00650964.

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Date of creation: 2011
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Publication status: Published, Annals of Operations Research, 2011, 137-154
Handle: RePEc:hal:cesptp:hal-00650964

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  1. 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.
  2. Lange, Fabien & Grabisch, Michel, 2009. "Values on regular games under Kirchhoff's laws," Mathematical Social Sciences, Elsevier, vol. 58(3), pages 322-340, November.
  3. Borm, P.E.M. & Algaba, A. & Bilbao, J.M. & Lopez, J., 2002. "The Myerson value for union stable systems," Open Access publications from Tilburg University urn:nbn:nl:ui:12-90189, Tilburg University.
  4. 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.
  5. 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.
  6. Faigle, U. & Grabisch, M. & Heyne, M., 2010. "Monge extensions of cooperation and communication structures," European Journal of Operational Research, Elsevier, vol. 206(1), pages 104-110, October.
  7. 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.
  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 & 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.
  10. Christophe Labreuche & Michel Grabisch, 2008. "A value for bi-cooperative games," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00308738, HAL.
  11. 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.
  12. Michel Grabisch, 2009. "The core of games on ordered structures and graphs," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00445171, HAL.
  13. 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.
  14. 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.
  15. 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.
  16. 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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Cited by:
  1. Michel Grabisch & Peter Sudhölter, 2012. "The Bounded Core for Games with Precedence Constraints," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-00759893, HAL.
  2. Michel Grabisch & Peter Sudhölter, 2014. "On the restricted cores and the bounded core of games on distributive lattices," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00950109, HAL.
  3. Michel Grabisch & Peter Sudhölter, 2012. "The Bounded Core for Games with Precedence Constraints," Post-Print halshs-00673909, HAL.

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