IDEAS home Printed from https://ideas.repec.org/p/hal/cesptp/halshs-00544134.html
   My bibliography  Save this paper

Ensuring the boundedness of the core of games with restricted cooperation

Author

Listed:
  • 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," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00544134, HAL.
    • Handle: RePEc:hal:cesptp:halshs-00544134
    Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-00544134
    as

    Download full text from publisher

    File URL: https://shs.hal.science/halshs-00544134/document
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    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.
    4. Marina Núñez & Carles Rafels, 1998. "On extreme points of the core and reduced games," Annals of Operations Research, Springer, vol. 84(0), pages 121-133, December.
    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. Lange, Fabien & Grabisch, Michel, 2009. "Values on regular games under Kirchhoff's laws," Mathematical Social Sciences, Elsevier, vol. 58(3), pages 322-340, November.
    7. Christophe Labreuche & Michel Grabisch, 2008. "A value for bi-cooperative games," Post-Print halshs-00308738, HAL.
    8. 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.
    9. 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.
    10. 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.
    11. 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.
    12. 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.
    13. 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.
    14. Jeroen Suijs & Peter Borm & Herbert Hamers & Marieke Quant & Maurice Koster, 2005. "Communication and Cooperation in Public Network Situations," Annals of Operations Research, Springer, vol. 137(1), pages 117-140, July.
    15. René Brink & Gerard Laan & Valeri Vasil’ev, 2007. "Component efficient solutions in line-graph games with applications," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 33(2), pages 349-364, November.
    16. Michel Grabisch & Lijue Xie, 2008. "The core of games on distributive lattices: how to share benefits in a hierarchy," Post-Print halshs-00344802, HAL.
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Michel Grabisch, 2013. "The core of games on ordered structures and graphs," Annals of Operations Research, Springer, vol. 204(1), pages 33-64, April.
    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 & 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.
    4. 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.
    5. 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.
    6. Michel Grabisch & Peter Sudhölter, 2016. "Characterizations of solutions for games with precedence constraints," PSE-Ecole d'économie de Paris (Postprint) hal-01297600, HAL.
    7. 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.
    8. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Michel Grabisch, 2013. "The core of games on ordered structures and graphs," Annals of Operations Research, Springer, vol. 204(1), pages 33-64, April.
    2. 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.
    3. 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.
    4. Michel Grabisch & Lijue Xie, 2008. "The core of games on distributive lattices: how to share benefits in a hierarchy," Post-Print halshs-00344802, HAL.
    5. Bilbao, J.M. & Jiménez, N. & López, J.J., 2010. "The selectope for bicooperative games," European Journal of Operational Research, Elsevier, vol. 204(3), pages 522-532, August.
    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. 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.
    8. 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.
    9. Rene van den Brink & Ilya Katsev & Gerard van der Laan, 2023. "Properties of Solutions for Games on Union-Closed Systems," Mathematics, MDPI, vol. 11(4), pages 1-16, February.
    10. Michel Grabisch, 2015. "Fuzzy Measures and Integrals: Recent Developments," Post-Print hal-01302377, HAL.
    11. Sébastien Courtin & Zéphirin Nganmeni & Bertrand Tchantcho, 2016. "The Shapley–Shubik power index for dichotomous multi-type games," Theory and Decision, Springer, vol. 81(3), pages 413-426, September.
    12. Michel Grabisch & Tomáš Kroupa, 2018. "The core of supermodular games on finite distributive lattices," Documents de travail du Centre d'Economie de la Sorbonne 18010, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    13. Michel Grabisch & Peter Sudhölter, 2016. "Characterizations of solutions for games with precedence constraints," PSE-Ecole d'économie de Paris (Postprint) hal-01297600, HAL.
    14. René Brink, 2017. "Games with a permission structure - A survey on generalizations and applications," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(1), pages 1-33, April.
    15. 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.
    16. Branzei, R. & Tijs, S. & Zarzuelo, J., 2009. "Convex multi-choice games: Characterizations and monotonic allocation schemes," European Journal of Operational Research, Elsevier, vol. 198(2), pages 571-575, October.
    17. Encarnacion Algaba & René van den Brink & Chris Dietz, 2015. "Power Measures and Solutions for Games under Precedence Constraints," Tinbergen Institute Discussion Papers 15-007/II, Tinbergen Institute.
    18. Grabisch, Michel & Sudhölter, Peter, 2018. "On a class of vertices of the core," Games and Economic Behavior, Elsevier, vol. 108(C), pages 541-557.
    19. Sylvain Béal & Sylvain Ferrières & Philippe Solal, 2022. "The priority value for cooperative games with a priority structure," International Journal of Game Theory, Springer;Game Theory Society, vol. 51(2), pages 431-450, June.
    20. Freixas, Josep & Zwicker, William S., 2009. "Anonymous yes-no voting with abstention and multiple levels of approval," Games and Economic Behavior, Elsevier, vol. 67(2), pages 428-444, November.

    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

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:hal:cesptp:halshs-00544134. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: CCSD (email available below). General contact details of provider: https://hal.archives-ouvertes.fr/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.