Bases and transforms of set functions

Abstract : The chapter studies the vector space of set functions on a finite set X, which can be alternatively seen as pseudo-Boolean functions, and including as a special cases games. We present several bases (unanimity games, Walsh and parity functions) and make an emphasis on the Fourier transform. Then we establish the basic duality between bases and invertible linear transform (e.g., the Möbius transform, the Fourier transform and interaction transforms). We apply it to solve the well-known inverse problem in cooperative game theory (find all games with same Shapley value), and to find various equivalent expressions of the Choquet integral.
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Submitted on : Wednesday, December 7, 2016 - 5:27:30 PM
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Michel Grabisch. Bases and transforms of set functions. 2016. ⟨halshs-01411966⟩

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