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On the vertices of the k-additive core

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Author Info

  • Michel Grabisch

    ()
    (CES - Centre d'économie de la Sorbonne - CNRS : UMR8174 - Université Paris I - Panthéon-Sorbonne)

  • Pedro Miranda

    ()
    (Universidad Complutense de Madrid - Universidad Complutense de Madrid)

Abstract

The core of a game v on N, which is the set of additive games φ dominating v such that φ(N)=v(N), is a central notion in cooperative game theory, decision making and in combinatorics, where it is related to submodular functions, matroids and the greedy algorithm. In many cases however, the core is empty, and alternative solutions have to be found. We define the k-additive core by replacing additive games by k-additive games in the definition of the core, where k-additive games are those games whose Möbius transform vanishes for subsets of more than k elements. For a sufficiently high value of k, the k-additive core is nonempty, and is a convex closed polyhedron. Our aim is to establish results similar to the classical results of Shapley and Ichiishi on the core of convex games (corresponds to Edmonds' theorem for the greedy algorithm), which characterize the vertices of the core.

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Bibliographic Info

Paper provided by HAL in its series Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) with number hal-00321625.

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Date of creation: Nov 2008
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Publication status: Published, Discrete Mathematics, 2008, 308, 22, 5204-5217
Handle: RePEc:hal:cesptp:hal-00321625

Note: View the original document on HAL open archive server: http://hal.archives-ouvertes.fr/hal-00321625
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Web page: http://hal.archives-ouvertes.fr/

Related research

Keywords: Cooperative games; Core; k-additive games; Vertices;

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Cited by:
  1. Stéphane Gonzalez & Michel Grabisch, 2014. "Autonomous coalitions," Documents de travail du Centre d'Economie de la Sorbonne, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne 14044, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
  2. Stéphane Gonzalez & Michel Grabisch, 2012. "Preserving coalitional rationality for non-balanced games," Documents de travail du Centre d'Economie de la Sorbonne, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne 12022, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
  3. Grabisch, Michel & Li, Tong, 2011. "On the set of imputations induced by the k-additive core," European Journal of Operational Research, Elsevier, Elsevier, vol. 214(3), pages 697-702, November.
  4. Stéphane Gonzalez & Michel Grabisch, 2013. "Multicoalitional solutions," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers), HAL halshs-00881108, HAL.
  5. repec:hal:journl:halshs-00718358 is not listed on IDEAS
  6. Stéphane Gonzalez & Michel Grabisch, 2014. "Autonomous coalitions," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers), HAL halshs-01021328, HAL.

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