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Bases and Linear Transforms of Cooperation systems

Author

Listed:
  • Ulrich Faigle

    (Universität zu Köln = University of Cologne)

  • 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

We study linear properties of TU-games, revisiting well-known issues like interaction transforms, the inverse Shapley value problem and potentials. We embed TU-games into the model of cooperation systems and influence patterns, which allows us to introduce linear operators on games in a natural way. We focus on transforms, which are linear invertible maps, relate them to bases and investigate many examples (Möbius transform, interaction transform, Walsh transform and Fourier analysis etc.). In particular, we present a simple solution to the inverse problem in its general form: Given a linear value Φ and a game v, find all games v' such that Φ(v) = Φ(v' ). Generalizing Hart and Mas-Colell's concept of a potential, we introduce general potentials and show that every linear value is induced by an appropriate potential.

Suggested Citation

  • Ulrich Faigle & Michel Grabisch, 2015. "Bases and Linear Transforms of Cooperation systems," Post-Print halshs-00971393, HAL.
    • Handle: RePEc:hal:journl:halshs-00971393
    Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-00971393v2
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    References listed on IDEAS

    as
    1. Luis Ruiz & Federico Valenciano & José Zarzuelo, 1998. "Some new results on least square values for TU games," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 6(1), pages 139-158, June.
    2. Marichal, Jean-Luc & Mathonet, Pierre, 2011. "Weighted Banzhaf power and interaction indexes through weighted approximations of games," European Journal of Operational Research, Elsevier, vol. 211(2), pages 352-358, June.
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    4. Irinel Dragan, 2006. "The least square values and the shapley value for cooperative TU games," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 14(1), pages 61-73, June.
    5. Hart, Sergiu & Mas-Colell, Andreu, 1989. "Potential, Value, and Consistency," Econometrica, Econometric Society, vol. 57(3), pages 589-614, May.
    6. Marc Roubens & Michel Grabisch, 1999. "An axiomatic approach to the concept of interaction among players in cooperative games," International Journal of Game Theory, Springer;Game Theory Society, vol. 28(4), pages 547-565.
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    8. Ruiz, Luis M. & Valenciano, Federico & Zarzuelo, Jose M., 1998. "The Family of Least Square Values for Transferable Utility Games," Games and Economic Behavior, Elsevier, vol. 24(1-2), pages 109-130, July.
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    More about this item

    Keywords

    cooperation system; cooperative game; basis; Fourier analysis; inverse problem; potential; transform;
    All these keywords.

    JEL classification:

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

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