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On the computation of the nucleolus of a cooperative game

Author

Listed:
  • Jeroen Kuipers

    (Department of Mathematics, Maastricht University, P.O.Box 616, 6200 MD Maastricht, The Netherlands. E-mail address: kuipers@math.unimaas.nl)

  • Ulrich Faigle

    (Zentrum für Angewandte Informatik, Universität zu Köln, Weyertal 80, 50931 Köln, Germany. E-mail address: faigle@zpr.uni-koeln.de)

  • Walter Kern

    (Department of Applied Mathematics, University of Twente, P.O.Box 217, 7500 AE Enschede, The Netherlands. E-mail address: kern@math.utwente.nl)

Abstract

We consider classes of cooperative games. We show that we can efficiently compute an allocation in the intersection of the prekernel and the least core of the game if we can efficiently compute the minimum excess for any given allocation. In the case where the prekernel of the game contains exactly one core vector, our algorithm computes the nucleolus of the game. This generalizes both a recent result by Kuipers on the computation of the nucleolus for convex games and a classical result by Megiddo on the nucleolus of standard tree games to classes of more general minimum cost spanning tree games. Our algorithm is based on the ellipsoid method and Maschler's scheme for approximating the prekernel.

Suggested Citation

  • Jeroen Kuipers & Ulrich Faigle & Walter Kern, 2001. "On the computation of the nucleolus of a cooperative game," International Journal of Game Theory, Springer;Game Theory Society, vol. 30(1), pages 79-98.
    • Handle: RePEc:spr:jogath:v:30:y:2001:i:1:p:79-98
    Note: Received February 2000/Final version April 2001
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    Citations

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    Cited by:

    1. Cesco, Juan Carlos, 2008. "A general characterization for non-balanced games in terms of U-cycles," European Journal of Operational Research, Elsevier, vol. 191(2), pages 409-415, December.
    2. Simai He & Jiawei Zhang & Shuzhong Zhang, 2012. "Polymatroid Optimization, Submodularity, and Joint Replenishment Games," Operations Research, INFORMS, vol. 60(1), pages 128-137, February.
    3. Péter Biró & Walter Kern & Daniël Paulusma, 2012. "Computing solutions for matching games," International Journal of Game Theory, Springer;Game Theory Society, vol. 41(1), pages 75-90, February.
    4. Nguyen, Tri-Dung & Thomas, Lyn, 2016. "Finding the nucleoli of large cooperative games," European Journal of Operational Research, Elsevier, vol. 248(3), pages 1078-1092.
    5. Ulrich Faigle & Walter Kern & Jeroen Kuipers, 2006. "Computing an Element in the Lexicographic Kernel of a Game," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 63(3), pages 427-433, July.
    6. Elena Iñarra & Roberto Serrano & Ken-Ichi Shimomura, 2020. "The Nucleolus, the Kernel, and the Bargaining Set: An Update," Revue économique, Presses de Sciences-Po, vol. 71(2), pages 225-266.
    7. Kuipers, Jeroen & Mosquera, Manuel A. & Zarzuelo, José M., 2013. "Sharing costs in highways: A game theoretic approach," European Journal of Operational Research, Elsevier, vol. 228(1), pages 158-168.
    8. Kimms, Alf & Çetiner, Demet, 2012. "Approximate nucleolus-based revenue sharing in airline alliances," European Journal of Operational Research, Elsevier, vol. 220(2), pages 510-521.
    9. Guajardo, Mario & Jörnsten, Kurt, 2015. "Common mistakes in computing the nucleolus," European Journal of Operational Research, Elsevier, vol. 241(3), pages 931-935.
    10. Drechsel, J. & Kimms, A., 2010. "Computing core allocations in cooperative games with an application to cooperative procurement," International Journal of Production Economics, Elsevier, vol. 128(1), pages 310-321, November.
    11. A. Palestini & G. Pignataro, 2013. "A multi-factor inequality approach to a transfer scheme: the case of Common Agricultural Policy," Working Papers wp891, Dipartimento Scienze Economiche, Universita' di Bologna.
    12. Takuya Masuzawa, 2008. "Computing the cores of strategic games with punishment–dominance relations," International Journal of Game Theory, Springer;Game Theory Society, vol. 37(2), pages 185-201, June.
    13. Xiaotie Deng & Qizhi Fang & Xiaoxun Sun, 2009. "Finding nucleolus of flow game," Journal of Combinatorial Optimization, Springer, vol. 18(1), pages 64-86, July.
    14. Andreas S. Schulz & Nelson A. Uhan, 2010. "Sharing Supermodular Costs," Operations Research, INFORMS, vol. 58(4-part-2), pages 1051-1056, August.
    15. Bayati, Mohsen & Borgs, Christian & Chayes, Jennifer & Kanoria, Yash & Montanari, Andrea, 2015. "Bargaining dynamics in exchange networks," Journal of Economic Theory, Elsevier, vol. 156(C), pages 417-454.
    16. Granot, Daniel & Hamers, Herbert & Kuipers, Jeroen & Maschler, Michael, 2011. "On Chinese postman games where residents of each road pay the cost of their road," Games and Economic Behavior, Elsevier, vol. 72(2), pages 427-438, June.

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