Bases and Linear Transforms of Cooperation systems
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.
|Date of creation:||May 2015|
|Publication status:||Published in Documents de travail du Centre d'Economie de la Sorbonne 2014.10R - ISSN : 1955-611X. 2015|
|Note:||View the original document on HAL open archive server: https://halshs.archives-ouvertes.fr/halshs-00971393v2|
|Contact details of provider:|| Web page: https://hal.archives-ouvertes.fr/|
References listed on IDEAS
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- 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.
- 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.
- Hart, Sergiu & Mas-Colell, Andreu, 1989. "Potential, Value, and Consistency," Econometrica, Econometric Society, vol. 57(3), pages 589-614, May.
- 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.
- 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.
- Guillermo Owen, 1972. "Multilinear Extensions of Games," Management Science, INFORMS, vol. 18(5-Part-2), pages 64-79, January.
- 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.
- Monderer, Dov & Shapley, Lloyd S., 1996. "Potential Games," Games and Economic Behavior, Elsevier, vol. 14(1), pages 124-143, May. Full references (including those not matched with items on IDEAS)
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