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Ensuring the boundedness of the core of games with restricted cooperation

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  • Michel Grabisch

    (CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique, PSE - Paris School of Economics - UP1 - Université Paris 1 Panthéon-Sorbonne - ENS-PSL - École normale supérieure - Paris - PSL - Université Paris Sciences et Lettres - EHESS - École des hautes études en sciences sociales - ENPC - École des Ponts ParisTech - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement)

Abstract

The core of a cooperative game on a set of players N is one of the most popular concept of solution. When cooperation is restricted (feasible coalitions form a subcollection F of 2N), the core may become unbounded, which makes it usage questionable in practice. Our proposal is to make the core bounded by turning some of the inequalities defining the core into equalities (additional efficiency constraints). We address the following mathematical problem : can we find a minimal set of inequalities in the core such that, if turned into equalities, the core becomes bounded ? The new core obtained is called the restricted core. We completely solve the question when F is a distributive lattice, introducing also the notion of restricted Weber set. We show that the case of regular set systems amounts more or less to the case of distributive lattices. We also study the case of weakly union-closed systems and give some results for the general case.

Suggested Citation

  • Michel Grabisch, 2010. "Ensuring the boundedness of the core of games with restricted cooperation," Post-Print halshs-00544134, HAL.
    • Handle: RePEc:hal:journl:halshs-00544134
    Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-00544134
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    1. Michel Grabisch & Lijue Xie, 2007. "A new approach to the core and Weber set of multichoice games," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 66(3), pages 491-512, December.
    2. Michel Grabisch, 2013. "The core of games on ordered structures and graphs," Annals of Operations Research, Springer, vol. 204(1), pages 33-64, April.
    3. Hans Reijnierse & Jean Derks, 1998. "Note On the core of a collection of coalitions," International Journal of Game Theory, Springer;Game Theory Society, vol. 27(3), pages 451-459.
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    5. Edward M. Bolger, 2000. "A consistent value for games with n players and r alternatives," International Journal of Game Theory, Springer;Game Theory Society, vol. 29(1), pages 93-99.
    6. Honda, Aoi & Grabisch, Michel, 2008. "An axiomatization of entropy of capacities on set systems," European Journal of Operational Research, Elsevier, vol. 190(2), pages 526-538, October.
    7. Michel Grabisch & Lijue Xie, 2008. "The core of games on distributive lattices: how to share benefits in a hierarchy," Post-Print halshs-00344802, HAL.
    8. Lange, Fabien & Grabisch, Michel, 2009. "Values on regular games under Kirchhoff's laws," Mathematical Social Sciences, Elsevier, vol. 58(3), pages 322-340, November.
    9. Christophe Labreuche & Michel Grabisch, 2008. "A value for bi-cooperative games," Post-Print halshs-00308738, HAL.
    10. Faigle, U. & Grabisch, M. & Heyne, M., 2010. "Monge extensions of cooperation and communication structures," European Journal of Operational Research, Elsevier, vol. 206(1), pages 104-110, October.
    11. Michel Grabisch & Lijue Xie, 2011. "The restricted core of games on distributive lattices: how to share benefits in a hierarchy," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 73(2), pages 189-208, April.
    12. Michel Grabisch & Fabien Lange, 2007. "Games on lattices, multichoice games and the shapley value: a new approach," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 65(1), pages 153-167, February.
    13. E. Algaba & J. M. Bilbao & P. Borm & J. J. López, 2001. "The Myerson value for union stable structures," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 54(3), pages 359-371, December.
    14. J. Bilbao & J. Fernández & N. Jiménez & J. López, 2008. "The Shapley value for bicooperative games," Annals of Operations Research, Springer, vol. 158(1), pages 99-115, February.
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    17. R. Amer & F. Carreras & A. Magaña, 1998. "Extension of values to games withmultiple alternatives," Annals of Operations Research, Springer, vol. 84(0), pages 63-78, December.
    18. Jesús Mario Bilbao & Julio R. Fernández & Nieves Jiménez & Jorge Jesús López, 2004. "The Shapley value for bicooperative games," Economic Working Papers at Centro de Estudios Andaluces E2004/56, Centro de Estudios Andaluces.
    19. Derks, J J M, 1992. "A Short Proof of the Inclusion of the Core in the Weber Set," International Journal of Game Theory, Springer;Game Theory Society, vol. 21(2), pages 149-150.
    20. Faigle, U & Kern, W, 1992. "The Shapley Value for Cooperative Games under Precedence Constraints," International Journal of Game Theory, Springer;Game Theory Society, vol. 21(3), pages 249-266.
    21. Hsiao Chih-Ru & Raghavan T. E. S., 1993. "Shapley Value for Multichoice Cooperative Games, I," Games and Economic Behavior, Elsevier, vol. 5(2), pages 240-256, April.
    22. Ichiishi, Tatsuro, 1981. "Super-modularity: Applications to convex games and to the greedy algorithm for LP," Journal of Economic Theory, Elsevier, vol. 25(2), pages 283-286, October.
    23. J.M. Bilbao & E. Lebrón & N. Jiménez, 1998. "Probabilistic values on convex geometries," Annals of Operations Research, Springer, vol. 84(0), pages 79-95, December.
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    1. repec:hal:pseose:hal-00803233 is not listed on IDEAS
    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. Michel Grabisch & Peter Sudhölter, 2012. "The bounded core for games with precedence constraints," Annals of Operations Research, Springer, vol. 201(1), pages 251-264, December.
    4. Grabisch, Michel & Sudhölter, Peter, 2014. "On the restricted cores and the bounded core of games on distributive lattices," European Journal of Operational Research, Elsevier, vol. 235(3), pages 709-717.
    5. Michel Grabisch, 2013. "The core of games on ordered structures and graphs," Annals of Operations Research, Springer, vol. 204(1), pages 33-64, April.
    6. Michel Grabisch & Peter Sudhölter, 2014. "The positive core for games with precedence constraints," Documents de travail du Centre d'Economie de la Sorbonne 14036, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    7. 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.
    8. Michel Grabisch & Peter Sudhölter, 2016. "Characterizations of solutions for games with precedence constraints," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) hal-01297600, HAL.
    9. repec:hal:pseose:hal-01297600 is not listed on IDEAS
    10. P. García-Segador & P. Miranda, 2020. "Order cones: a tool for deriving k-dimensional faces of cones of subfamilies of monotone games," Annals of Operations Research, Springer, vol. 295(1), pages 117-137, December.
    11. repec:hal:pseose:hal-01372858 is not listed on IDEAS
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    More about this item

    Keywords

    Cooperative game; core; restricted cooperation; bounded core; Weber set; Jeu coopératif; coeur; coopération restreinte; coeur borné; ensemble de Weber;
    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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