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Preserving coalitional rationality for non-balanced games

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  • Stéphane Gonzalez

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  • Michel Grabisch

    ()

Abstract

In cooperative games, the core is one of the most popular solution concept since it ensures coalitional rationality. For non-balanced games however, the core is empty, and other solution concepts have to be found. We propose the use of general solutions, that is, to distribute the total worth of the game among groups rather than among individuals. In particular, the $$k$$ k -additive core proposed by Grabisch and Miranda is a general solution preserving coalitional rationality which distributes among coalitions of size at most $$k$$ k , and is never empty for $$k\ge 2$$ k ≥ 2 . The extended core of Bejan and Gomez can also be viewed as a general solution, since it implies to give an amount to the grand coalition. The $$k$$ k -additive core being an unbounded set and therefore difficult to use in practice, we propose a subset of it called the minimal negotiation set. The idea is to select elements of the $$k$$ k -additive core mimimizing the total amount given to coalitions of size greater than 1. Thus the minimum negotiation set naturally reduces to the core for balanced games. We study this set, giving properties and axiomatizations, as well as its relation to the extended core of Bejan and Gomez. We give a method of computing the minimum bargaining set, and lastly indicate how to eventually get classical solutions from general ones. Copyright Springer-Verlag Berlin Heidelberg 2015

Suggested Citation

  • 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.
  • Handle: RePEc:spr:jogath:v:44:y:2015:i:3:p:733-760
    DOI: 10.1007/s00182-014-0451-9
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    References listed on IDEAS

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    1. repec:hal:journl:hal-00625339 is not listed on IDEAS
    2. Miranda, Pedro & Grabisch, Michel, 2010. "k-Balanced games and capacities," European Journal of Operational Research, Elsevier, vol. 200(2), pages 465-472, January.
    3. repec:hal:journl:hal-00321625 is not listed on IDEAS
    4. 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.
    5. Thomson, William, 2003. "Axiomatic and game-theoretic analysis of bankruptcy and taxation problems: a survey," Mathematical Social Sciences, Elsevier, vol. 45(3), pages 249-297, July.
    6. Aumann, Robert J. & Maschler, Michael, 1985. "Game theoretic analysis of a bankruptcy problem from the Talmud," Journal of Economic Theory, Elsevier, vol. 36(2), pages 195-213, August.
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    8. 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.
    9. 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.
    10. repec:spr:thdlic:978-3-540-72945-7 is not listed on IDEAS
    11. Jean Derks & Hans Haller & Hans Peters, 2000. "The selectope for cooperative games," International Journal of Game Theory, Springer;Game Theory Society, vol. 29(1), pages 23-38.
    12. Camelia Bejan & Juan Gómez, 2009. "Core extensions for non-balanced TU-games," International Journal of Game Theory, Springer;Game Theory Society, vol. 38(1), pages 3-16, March.
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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. Stéphane Gonzalez & Michel Grabisch, 2015. "Autonomous coalitions," Annals of Operations Research, Springer, vol. 235(1), pages 301-317, December.
    3. Gonzalez, Stéphane & Grabisch, Michel, 2016. "Multicoalitional solutions," Journal of Mathematical Economics, Elsevier, vol. 64(C), pages 1-10.
    4. Michel Grabisch, 2016. "Rejoinder 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 335-337, July.
    5. Stéphane Gonzalez & Aymeric Lardon, 2016. "Optimal Deterrence of Cooperation," Working Papers halshs-01333392, HAL.
    6. 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.
    7. Stéphane Gonzalez & Michel Grabisch, 2013. "Multicoalitional solutions," Post-Print halshs-00881108, HAL.
    8. Stéphane Gonzalez & Michel Grabisch, 2015. "Autonomous coalitions," Annals of Operations Research, Springer, vol. 235(1), pages 301-317, December.
    9. Gonzalez, Stéphane & Grabisch, Michel, 2016. "Multicoalitional solutions," Journal of Mathematical Economics, Elsevier, vol. 64(C), pages 1-10.

    More about this item

    Keywords

    Cooperative game; Core; Balancedness; General solution;

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games

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