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Bases and Linear Transforms of Cooperation systems

  • Ulrich Faigle


    (Universität zu Köln - Mathematisches Institut)

  • Michel Grabisch


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

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.

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Paper provided by HAL in its series Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) with number halshs-00971393.

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Date of creation: May 2015
Date of revision:
Publication status: Published in Documents de travail du Centre d'Economie de la Sorbonne 2014.10R - ISSN : 1955-611X. 2015
Handle: RePEc:hal:cesptp:halshs-00971393
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  1. Monderer, Dov & Shapley, Lloyd S., 1996. "Potential Games," Games and Economic Behavior, Elsevier, vol. 14(1), pages 124-143, May.
  2. 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.
  3. Grabisch, M. & Roubens, M., 1998. "An Axiomatic Approach to the Concept of Interaction Among Players in Cooperative Games," Liege - Groupe d'Etude des Mathematiques du Management et de l'Economie 9818, UNIVERSITE DE LIEGE, Faculte d'economie, de gestion et de sciences sociales, Groupe d'Etude des Mathematiques du Management et de l'Economie.
  4. Hart, Sergiu & Mas-Colell, Andreu, 1989. "Potential, Value, and Consistency," Econometrica, Econometric Society, vol. 57(3), pages 589-614, May.
  5. Guillermo Owen, 1972. "Multilinear Extensions of Games," Management Science, INFORMS, vol. 18(5-Part-2), pages 64-79, January.
  6. 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.
  7. 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.
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