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Strongly essential coalitions and the nucleolus of peer group games

  • Rodica Brânzei

    ()

  • Tamás Solymosi

    ()

  • Stef Tijs

    ()

Most of the known efficient algorithms designed to compute the nucleolus for special classes of balanced games are based on two facts: (i) in any balanced game, the coalitions which actually determine the nucleolus are essential; and (ii) all essential coalitions in any of the games in the class belong to a prespeci ed collection of size polynomial in the number of players.We consider a subclass of essential coalitions, called strongly essential coalitions, and show that in any game, the collection of strongly essential coalitions contains all the coalitions which actually determine the core, and in case the core is not empty, the nucleolus and the kernelcore.As an application, we consider peer group games, and show that they admit at most 2n - 1 strongly essential coalitions, whereas the number of essential coalitions could be as much as 2n-1. We propose an algorithm that computes the nucleolus of an n-player peer group game in O(n2) time directly from the data of the underlying peer group situation.

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File URL: http://hdl.handle.net/10.1007/s00182-005-0213-9
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Article provided by Springer in its journal International Journal of Game Theory.

Volume (Year): 33 (2005)
Issue (Month): 3 (09)
Pages: 447-460

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Handle: RePEc:spr:jogath:v:33:y:2005:i:3:p:447-460
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  1. Brânzei, R. & Fragnelli, V. & Tijs, S.H., 2002. "Tree-connected peer group situations and peer group games," Other publications TiSEM f4601b66-2e29-4969-85ca-0, Tilburg University, School of Economics and Management.
  2. D. Granot & F. Granot & W. R. Zhu, 1998. "Characterization sets for the nucleolus," International Journal of Game Theory, Springer, vol. 27(3), pages 359-374.
  3. Brânzei, R. & Fragnelli, V. & Tijs, S.H., 2000. "On the computation of the nucleolus of line-graph peer group games," Other publications TiSEM fd889ce3-d034-47a2-9f6b-2, Tilburg University, School of Economics and Management.
  4. Maschler, M & Potters, J A M & Tijs, S H, 1992. "The General Nucleolus and the Reduced Game Property," International Journal of Game Theory, Springer, vol. 21(1), pages 85-106.
  5. Solymosi, Tamas & Raghavan, T. E. S. & Tijs, Stef, 2005. "Computing the nucleolus of cyclic permutation games," European Journal of Operational Research, Elsevier, vol. 162(1), pages 270-280, April.
  6. Le Breton,Michel & Owen,Guillermo & Weber,Shlomo, 1991. "Strongly balanced cooperative games," Discussion Paper Serie A 338, University of Bonn, Germany.
  7. Reijnierse, Hans & Potters, Jos, 1998. "The -Nucleolus of TU-Games," Games and Economic Behavior, Elsevier, vol. 24(1-2), pages 77-96, July.
  8. Maschler, M. & Potters, J.A.M. & Tijs, S.H., 1992. "The general nucleolus and the reduced game property," Other publications TiSEM ab187dab-1b5b-40c3-a673-8, Tilburg University, School of Economics and Management.
  9. Vincent Feltkamp & Javier Arin, 1997. "The Nucleolus and Kernel of Veto-Rich Transferable Utility Games," International Journal of Game Theory, Springer, vol. 26(1), pages 61-73.
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