IDEAS home Printed from https://ideas.repec.org/p/hal/cesptp/halshs-01821712.html
   My bibliography  Save this paper

The cone of supermodular games on finite distributive lattices

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)

  • Tomáš Kroupa

    (UTIA / CAS - Institute of Information Theory and Automation of the Czech Academy of Sciences - CAS - Czech Academy of Sciences [Prague])

Abstract

In this article, we study supermodular functions on finite distributive lattices. Relaxing the assumption that the domain is a powerset of a finite set, we focus on geometrical properties of the polyhedral cone of such functions. Specifically, we generalize the criterion for extremal rays and study the face lattice of the supermodular cone. An explicit description of facets by the corresponding tight linear inequalities is provided.

Suggested Citation

  • Michel Grabisch & Tomáš Kroupa, 2018. "The cone of supermodular games on finite distributive lattices," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01821712, HAL.
    • Handle: RePEc:hal:cesptp:halshs-01821712
    Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-01821712
    as

    Download full text from publisher

    File URL: https://shs.hal.science/halshs-01821712/document
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    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. 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.
    4. Michel Grabisch, 2016. "Set Functions, Games and Capacities in Decision Making," Theory and Decision Library C, Springer, number 978-3-319-30690-2, March.
    5. Jeroen Kuipers & Dries Vermeulen & Mark Voorneveld, 2010. "A generalization of the Shapley–Ichiishi result," International Journal of Game Theory, Springer;Game Theory Society, vol. 39(4), pages 585-602, October.
    6. van den Brink, Rene & Gilles, Robert P., 1996. "Axiomatizations of the Conjunctive Permission Value for Games with Permission Structures," Games and Economic Behavior, Elsevier, vol. 12(1), pages 113-126, January.
    7. Schmeidler, David, 1989. "Subjective Probability and Expected Utility without Additivity," Econometrica, Econometric Society, vol. 57(3), pages 571-587, May.
    8. 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.
    9. 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.
    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. 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.
    2. Martin Cerny & Michel Grabisch, 2023. "Player-centered incomplete cooperative games," Documents de travail du Centre d'Economie de la Sorbonne 23006, 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, 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.
    2. 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.
    3. Michel Grabisch, 2011. "Ensuring the boundedness of the core of games with restricted cooperation," Annals of Operations Research, Springer, vol. 191(1), pages 137-154, November.
    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. Ulrich Faigle & Michel Grabisch, 2019. "Least Square Approximations and Linear Values of Cooperative Game," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-02381231, HAL.
    7. Herings, P. Jean-Jacques & van der Laan, Gerard & Talman, Dolf, 2007. "The socially stable core in structured transferable utility games," Games and Economic Behavior, Elsevier, vol. 59(1), pages 85-104, April.
    8. 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.
    9. 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.
    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. Mayag, Brice & Bouyssou, Denis, 2020. "Necessary and possible interaction between criteria in a 2-additive Choquet integral model," European Journal of Operational Research, Elsevier, vol. 283(1), pages 308-320.
    12. Michel Grabisch & Peter Sudhölter, 2016. "Characterizations of solutions for games with precedence constraints," PSE-Ecole d'économie de Paris (Postprint) hal-01297600, HAL.
    13. René van den Brink & Gerard van der Laan & Valeri Vasil'ev, 2007. "Distributing Dividends in Games with Ordered Players," Tinbergen Institute Discussion Papers 06-114/1, Tinbergen Institute.
    14. René van den Brink & Gerard van der Laan & Valeri Vasil'ev, 0000. "The Restricted Core for Totally Positive Games with Ordered Players," Tinbergen Institute Discussion Papers 09-038/1, Tinbergen Institute.
    15. Grabisch, Michel & Sudhölter, Peter, 2018. "On a class of vertices of the core," Games and Economic Behavior, Elsevier, vol. 108(C), pages 541-557.
    16. Suzuki, T. & Talman, A.J.J., 2011. "Solution Concepts for Cooperative Games with Circular Communication Structure," Discussion Paper 2011-100, Tilburg University, Center for Economic Research.
    17. Takao Asano & Hiroyuki Kojima, 2022. "Choquet Integrals and Belief Functions," KIER Working Papers 1077, Kyoto University, Institute of Economic Research.
    18. Mustapha Ridaoui & Michel Grabisch & Christophe Labreuche, 2018. "An axiomatisation of the Banzhaf value and interaction index for multichoice games," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-02381119, HAL.
    19. Bottero, M. & Ferretti, V. & Figueira, J.R. & Greco, S. & Roy, B., 2018. "On the Choquet multiple criteria preference aggregation model: Theoretical and practical insights from a real-world application," European Journal of Operational Research, Elsevier, vol. 271(1), pages 120-140.
    20. Cinfrignini, Andrea & Petturiti, Davide & Vantaggi, Barbara, 2023. "Dynamic bid–ask pricing under Dempster-Shafer uncertainty," Journal of Mathematical Economics, Elsevier, vol. 107(C).

    More about this item

    Keywords

    supermodular/submodular function; core; coalitional game; polyhedral cone; fonction sur/sous-modulaire; coeur; jeux coalitionnel; cône polyhédral;
    All these keywords.

    JEL classification:

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

    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-01821712. 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.