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

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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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 12067.

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Length: 18 pages
Date of creation: Oct 2012
Date of revision:
Handle: RePEc:mse:cesdoc:12067
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  1. 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.
  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.
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  4. 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.
  5. Peter Sudhölter & Yan-An Hwang, 2001. "Axiomatizations of the core on the universal domain and other natural domains," International Journal of Game Theory, Springer;Game Theory Society, vol. 29(4), pages 597-623.
  6. 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.
  7. 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.
  8. Csóka Péter & Herings P. Jean-Jacques & Kóczy László Á., 2007. "Balancedness Conditions for Exact Games," Research Memorandum 039, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
  9. 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.
  10. Michel Grabisch, 2013. "The core of games on ordered structures and graphs," PSE - Labex "OSE-Ouvrir la Science Economique" hal-00803233, HAL.
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  12. Michel Grabisch & Lijue Xie, 2011. "The restricted core of games on distributive lattices: how to share benefits in a hierarchy," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 73(2), pages 189-208, April.
  13. repec:hal:journl:halshs-00673909 is not listed on IDEAS
  14. repec:spr:compst:v:74:y:2011:i:1:p:41-52 is not listed on IDEAS
  15. repec:spr:compst:v:73:y:2011:i:2:p:189-208 is not listed on IDEAS
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  17. Derks, Jean J M & Gilles, Robert P, 1995. "Hierarchical Organization Structures and Constraints on Coalition Formation," International Journal of Game Theory, Springer;Game Theory Society, vol. 24(2), pages 147-63.
  18. 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.
  19. repec:hal:journl:halshs-00344802 is not listed on IDEAS
  20. 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.
  21. Faigle, U & Kern, W, 1992. "The Shapley Value for Cooperative Games under Precedence Constraints," International Journal of Game Theory, Springer;Game Theory Society, vol. 21(3), pages 249-66.
  22. 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.
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