Bases and linear transforms of TU-games and cooperation systems

Ulrich Faigle () and Michel Grabisch

International Journal of Game Theory, 2016, vol. 45, issue 4, No 6, 875-892

Abstract: 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 $${\Phi }$$ Φ and a game v, find all games $$v'$$ v ′ such that $${\Phi (v)=\Phi (v')}$$ 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.

Keywords: Cooperation system; Cooperative game; Basis; Fourier analysis; Inverse problem; Potential; Transform (search for similar items in EconPapers)
JEL-codes: C71 (search for similar items in EconPapers)
Date: 2016
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Citations: View citations in EconPapers (13)

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Working Paper: Bases and linear transforms of TU-games and cooperation systems (2016)
Working Paper: Bases and linear transforms of TU-games and cooperation systems (2016)
Working Paper: Bases and linear transforms of TU-games and cooperation systems (2016)
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DOI: 10.1007/s00182-015-0490-x

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