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k-balanced games and capacities

Author

Listed:
  • Pedro Miranda

    (UCM - Universidad Complutense de Madrid = Complutense University of Madrid [Madrid])

  • Michel Grabisch

    (CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

Abstract

In this paper we present a generalization of the concept of balanced game for finite games. Balanced games are those having a nonempty core, and this core is usually considered as the solution of the game. Based on the concept of $k$-additivity, we define the so-called $k$-balanced games and the corresponding generalization of core, the $k$-additive core, whose elements are not directly imputations but $k$-additive games. We show that any game is $k$-balanced for a suitable choice of $k,$ so that the corresponding $k$-additive core is not empty. For the games in the $k$-additive core, we propose a sharing procedure to get an imputation and a representative value for the expectations of the players based on the pessimistic criterion. Moreover, we look for necessary and sufficient conditions for a game to be $k$-balanced. For the general case, it is shown that any game is either balanced or 2-balanced. Finally, we treat the special case of capacities.

Suggested Citation

  • Pedro Miranda & Michel Grabisch, 2010. "k-balanced games and capacities," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00445073, HAL.
  • Handle: RePEc:hal:cesptp:halshs-00445073
    DOI: 10.1016/j.ejor.2008.12.020
    Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-00445073
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    References listed on IDEAS

    as
    1. Chateauneuf, Alain & Jaffray, Jean-Yves, 1989. "Some characterizations of lower probabilities and other monotone capacities through the use of Mobius inversion," Mathematical Social Sciences, Elsevier, vol. 17(3), pages 263-283, June.
    2. Michel Grabisch & Pedro Miranda, 2008. "On the vertices of the k-additive core," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-00321625, HAL.
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    Citations

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    Cited by:

    1. Michel Grabisch, 2016. "Remarkable polyhedra related to set functions, games," Documents de travail du Centre d'Economie de la Sorbonne 16081, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    2. Pedro Miranda & Michel Grabisch, 2012. "An algorithm for finding the vertices of the k-additive monotone core," Post-Print hal-00806905, HAL.
    3. Grabisch, Michel & Li, Tong, 2011. "On the set of imputations induced by the k-additive core," European Journal of Operational Research, Elsevier, vol. 214(3), pages 697-702, November.
    4. Stéphane Gonzalez & Michel Grabisch, 2015. "Preserving coalitional rationality for non-balanced games," International Journal of Game Theory, Springer;Game Theory Society, vol. 44(3), pages 733-760, August.
    5. Michel Grabisch, 2016. "Remarkable polyhedra related to set functions, games and capacities," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(2), pages 301-326, July.
    6. repec:hal:pseose:hal-01372858 is not listed on IDEAS
    7. van den Brink, René & Chun, Youngsub & Funaki, Yukihiko & Zou, Zhengxing, 2023. "Balanced externalities and the proportional allocation of nonseparable contributions," European Journal of Operational Research, Elsevier, vol. 307(2), pages 975-983.
    8. Hans Peters, 2016. "Comments on: Remarkable polyhedra related to set functions, games and capacities," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(2), pages 330-332, July.
    9. repec:hal:pseose:halshs-01235625 is not listed on IDEAS

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    More about this item

    Keywords

    Cooperative Games; k-additivity; balanced games; capacities; core;
    All these keywords.

    JEL classification:

    • C7 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory
    • D7 - Microeconomics - - Analysis of Collective Decision-Making

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