Bases and linear transforms of TU-games and cooperation systems

Ulrich Faigle (1) , Michel Grabisch (2, 3)

1 Universität zu Köln = University of Cologne
2 PSE - Paris School of Economics
3 CES - Centre d'économie de la Sorbonne

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 .

Mots clés: Cooperation system; cooperative game; basis; Fourier analysis; inverse problem; potential; transform

Domaines: Mathématique discrète [cs.DM]; Economies et finances

Ulrich Faigle, Michel Grabisch. Bases and linear transforms of TU-games and cooperation systems. International Journal of Game Theory, 2016, 45 (4), pp.875-892. ⟨10.1007/s00182-015-0490-x⟩. ⟨hal-01404509⟩