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/|
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"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.
- 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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- Hart, Sergiu & Mas-Colell, Andreu, 1989. "Potential, Value, and Consistency," Econometrica, Econometric Society, vol. 57(3), pages 589-614, May.
- 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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