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An Algorithm for Computing the Nucleolus of Disjunctive Additive Games with An Acyclic Permission Structure

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  • René van den Brink

    ()
    (VU University Amsterdam)

  • Ilya Katsev

    (St. Petersburg Institute for Economics and Mathematics, Russian Academy of Sciences)

  • Gerard van der Laan

    ()
    (VU University Amsterdam)

Abstract

A situation in which a finite set of players can obtain certain payoffs by cooperation can be described by a cooperative game with transferable utility, or simply a TU-game. A (single-valued) solution for TU-games assigns a payoff distribution to every TU-game. A well-known solution is the nucleolus. A cooperative game with a permission structure describes a situation in which players in a cooperative TU-game are hierarchically ordered in the sense that there are players that need permission from other players before they are allowed to cooperate. The corresponding restricted game takes account of the limited cooperation possibilities by assigning to every coalition the worth of its largest feasible subset. In this paper we consider the class of non-negative additive games with an acyclic permission structure. This class generalizes the so-called peer group games being non-negative additive games on a permission tree. We provide a polynomial time algorithm for computing the nucleolus of every restricted game corresponding to some disjunctive non-negative additive game with an acyclic permission structure. We discuss an application to market situations where sellers can sell objects to buyers through a directed network of intermediaries.

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Bibliographic Info

Paper provided by Tinbergen Institute in its series Tinbergen Institute Discussion Papers with number 08-104/1.

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Date of creation: 31 Oct 2008
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Handle: RePEc:dgr:uvatin:20080104

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Web page: http://www.tinbergen.nl

Related research

Keywords: TU-game; additive game; acyclic permission structure; disjunctive approach; peer group game; nucleolus; algorithm; complexity;

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  1. Arin, J. & Feltkamp, V., 1994. "The nucleolus and kernel of veto-rich transferable utility games," Discussion Paper 1994-40, Tilburg University, Center for Economic Research.
  2. van den Brink, Rene & Gilles, Robert P., 1996. "Axiomatizations of the Conjunctive Permission Value for Games with Permission Structures," Games and Economic Behavior, Elsevier, vol. 12(1), pages 113-126, January.
  3. Brânzei, R. & Solymosi, T. & Tijs, S.H., 2003. "Strongly Essential Coalitions and the Nucleolus of Peer Group Games," Discussion Paper 2003-19, Tilburg University, Center for Economic Research.
  4. René van den Brink & Ilya Katsev & Gerard van der Laan, 2008. "Computation of the Nucleolus for a Class of Disjunctive Games with a Permission Structure," Tinbergen Institute Discussion Papers 08-060/1, Tinbergen Institute.
  5. Gilles, R.P. & Owen, G. & Brink, J.R. van den, 1991. "Games with permission structures: The conjunctive approach," Discussion Paper 1991-14, Tilburg University, Center for Economic Research.
  6. Muto, S. & Potters, J.A.M. & Tijs, S.H., 1989. "Information market games," Open Access publications from Tilburg University urn:nbn:nl:ui:12-154244, Tilburg University.
  7. RenÊ van den Brink, 1997. "An Axiomatization of the Disjunctive Permission Value for Games with a Permission Structure," International Journal of Game Theory, Springer, vol. 26(1), pages 27-43.
  8. Muto, Shigeo & Potters, Jos & Tijs, Stef, 1989. "Information Market Games," International Journal of Game Theory, Springer, vol. 18(2), pages 209-26.
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