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Games on concept lattices: Shapley value and core

Author

Listed:
  • Ulrich Faigle

    (Universität zu Köln = University of Cologne)

  • Michel Grabisch

    (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, CES - Centre d'économie de la Sorbonne - UP1 - Université Paris 1 Panthéon-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

  • Andres Jiménez-Losada

    (ETSI - Escuela Técnica Superior de Ingenieros de Sevilla)

  • Manuel Ordóñez

    (ETSI - Escuela Técnica Superior de Ingenieros de Sevilla)

Abstract

We introduce cooperative TU-games on concept lattices, where a concept is a pair (S, S ′) with S being a subset of players or objects, and S ′ a subset of attributes. Any such game induces a game on the set of players/objects, which appears to be a TU-game whose collection of feasible coalitions is a lattice closed under intersection, and a game on the set of attributes. We propose a Shapley value for each type of game, axiomatize it, and investigate the geometrical properties of the core (non-emptiness, boundedness, pointedness, extremal rays). In particular, we derive the equivalence of the intent and extent core for the class of distributive concepts.

Suggested Citation

  • Ulrich Faigle & Michel Grabisch & Andres Jiménez-Losada & Manuel Ordóñez, 2016. "Games on concept lattices: Shapley value and core," PSE-Ecole d'économie de Paris (Postprint) hal-01379699, HAL.
    • Handle: RePEc:hal:pseptp:hal-01379699
    DOI: 10.1016/j.dam.2015.08.004
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    References listed on IDEAS

    as
    1. Monjardet, Bernard, 2003. "The presence of lattice theory in discrete problems of mathematical social sciences. Why," Mathematical Social Sciences, Elsevier, vol. 46(2), pages 103-144, October.
    2. 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.
    3. Derks, Jean J M & Gilles, Robert P, 1995. "Hierarchical Organization Structures and Constraints on Coalition Formation," International Journal of Game Theory, Springer;Game Theory Society, vol. 24(2), pages 147-163.
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    8. Bilbao, J. M. & Lebron, E. & Jimenez, N., 1999. "The core of games on convex geometries," European Journal of Operational Research, Elsevier, vol. 119(2), pages 365-372, December.
    9. Ulrich Faigle & Michel Grabisch, 2011. "A Discrete Choquet Integral for Ordered Systems," Post-Print halshs-00563926, HAL.
    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.
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    Cited by:

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    3. Ren, Hongbo & Wu, Qiong & Li, Qifen & Yang, Yongwen, 2020. "Optimal design and management of distributed energy network considering both efficiency and fairness," Energy, Elsevier, vol. 213(C).
    4. Surajit Borkotokey & Loyimee Gogoi & Dhrubajit Choudhury & Rajnish Kumar, 2022. "Solidarity induced by group contributions: the MI $$^k$$ k -value for transferable utility games," Operational Research, Springer, vol. 22(2), pages 1267-1290, April.
    5. Wu, Qiong & Ren, Hongbo & Gao, Weijun & Ren, Jianxing & Lao, Changshi, 2017. "Profit allocation analysis among the distributed energy network participants based on Game-theory," Energy, Elsevier, vol. 118(C), pages 783-794.

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    More about this item

    Keywords

    cooperative game; restricted cooperation; concept lattice; core; Shapley value;
    All these keywords.

    JEL classification:

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

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