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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, IUF - Institut universitaire de France - M.E.N.E.S.R. - Ministère de l'Education nationale, de l’Enseignement supérieur et de la Recherche)

  • Tomáš Kroupa

    (University of West Bohemia [Plzeň ])

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 extremality 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, 2019. "The cone of supermodular games on finite distributive lattices," PSE-Ecole d'économie de Paris (Postprint) halshs-02381019, HAL.
    • Handle: RePEc:hal:pseptp:halshs-02381019
    DOI: 10.1016/j.dam.2019.01.024
    Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-02381019v2
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    References listed on IDEAS

    as
    1. 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.
    2. repec:hal:pseose:hal-00803233 is not listed on IDEAS
    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, 2013. "The core of games on ordered structures and graphs," Annals of Operations Research, Springer, vol. 204(1), pages 33-64, April.
    5. 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.
    6. 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.
    7. Michel Grabisch, 2016. "Set Functions, Games and Capacities in Decision Making," Theory and Decision Library C, Springer, number 978-3-319-30690-2, December.
    8. Schmeidler, David, 1989. "Subjective Probability and Expected Utility without Additivity," Econometrica, Econometric Society, vol. 57(3), pages 571-587, May.
    9. repec:hal:pseose:hal-01372858 is not listed on IDEAS
    10. 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.
    11. 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.
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    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.

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    More about this item

    Keywords

    supermodular function; finite distributive lattice; coalitional game; polyhedral cone;
    All these keywords.

    JEL classification:

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

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