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On the restricted cores and the bounded core of games on distributive lattices

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  • Grabisch, Michel

    ()
    (Paris School of Economics)

  • Sudhölter, Peter

    ()
    (Department of Business and Economics)

Abstract

We consider TU-games with restricted cooperation, where the set of feasible coalitions is a distributive lattice, hence generated by a partial order on the set of players. In such a situation, the core may be unbounded, and one has to select a bounded part of the core as a solution concept. The restricted core is obtained by imposing equality constraints in the core for sets belonging to so-called normal collections, resulting (if nonempty) in the selection of a bounded face of the core. The bounded core proves to be the union of all bounded faces (restricted cores). The paper aims at investigating in depth the relation between the bounded core and restricted cores, as well as the properties and structures of the restricted cores and normal collections. In particular, it is found that a game is convex if and only if all restricted cores corresponding to the minimal nested normal collections are nonempty. Moreover, in this case the union of these restricted cores already covers the bounded core.

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Bibliographic Info

Paper provided by Department of Business and Economics, University of Southern Denmark in its series Discussion Papers of Business and Economics with number 22/2012.

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Length: 17 pages
Date of creation: 31 Oct 2012
Date of revision:
Handle: RePEc:hhs:sdueko:2012_022

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Postal: Department of Business and Economics, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark
Phone: 65 50 32 33
Fax: 65 50 32 37
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Web page: http://www.sdu.dk/ivoe
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Keywords: TU-game; restricted cooperation; distributive lattice; core; extremal rays; faces of the core;

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References

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  1. Michel Grabisch & Lijue Xie, 2008. "The core of games on distributive lattices : how to share benefits in a hierarchy," Documents de travail du Centre d'Economie de la Sorbonne b08077, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne, revised Sep 2009.
  2. Béal, Sylvain & Rémila, Eric & Solal, Philippe, 2010. "Rooted-tree solutions for tree games," European Journal of Operational Research, Elsevier, vol. 203(2), pages 404-408, June.
  3. Michel Grabisch, 2010. "Ensuring the boundedness of the core of games with restricted cooperation," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00544134, HAL.
  4. 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.
  5. Pulido, Manuel A. & Sanchez-Soriano, Joaquin, 2006. "Characterization of the core in games with restricted cooperation," European Journal of Operational Research, Elsevier, vol. 175(2), pages 860-869, December.
  6. Michel Grabisch & Lijue Xie, 2008. "The core of games on distributive lattices : how to share benefits in a hierarchy," Post-Print halshs-00344802, HAL.
  7. Michel Grabisch, 2009. "The core of games on ordered structures and graphs," Post-Print halshs-00445171, HAL.
  8. Michel Grabisch, 2013. "The core of games on ordered structures and graphs," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-00803233, HAL.
  9. Ichiishi, Tatsuro, 1981. "Super-modularity: Applications to convex games and to the greedy algorithm for LP," Journal of Economic Theory, Elsevier, vol. 25(2), pages 283-286, October.
  10. Michel Grabisch & Peter Sudhölter, 2012. "The Bounded Core for Games with Precedence Constraints," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00673909, HAL.
  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. repec:hal:journl:halshs-00583868 is not listed on IDEAS
  13. Bilbao, J. M. & Lebron, E. & Jimenez, N., 1999. "The core of games on convex geometries," European Journal of Operational Research, Elsevier, vol. 119(2), pages 365-372, December.
  14. Michel Grabisch & Peter Sudhölter, 2012. "The Bounded Core for Games with Precedence Constraints," Post-Print halshs-00673909, HAL.
  15. Bilbao, J. M., 1998. "Axioms for the Shapley value on convex geometries," European Journal of Operational Research, Elsevier, vol. 110(2), pages 368-376, October.
  16. 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.
  17. 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.
  18. Peter Sudhölter & Yan-An Hwang, 2001. "Axiomatizations of the core on the universal domain and other natural domains," International Journal of Game Theory, Springer, vol. 29(4), pages 597-623.
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