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K-balanced games and capacities

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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 game. Based on the concept of k-additivity, we define to 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.

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  • Pedro Miranda & Michel Grabisch, 2008. "K-balanced games and capacities," Documents de travail du Centre d'Economie de la Sorbonne b08079, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
  • Handle: RePEc:mse:cesdoc:b08079
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    File URL: ftp://mse.univ-paris1.fr/pub/mse/CES2008/B08079.pdf
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    1. repec:hal:journl:hal-00321625 is not listed on IDEAS
    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.
    3. 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.
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    Cited by:

    1. 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.
    2. 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.
    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. Pedro Miranda & Michel Grabisch, 2012. "An algorithm for finding the vertices of the k-additive monotone core," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-00806905, HAL.
    5. 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.
    6. 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.
    7. repec:hal:journl:hal-00625339 is not listed on IDEAS

    More about this item

    Keywords

    Cooperative games; k-additivity; balanced games; capacities; core.;

    JEL classification:

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

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