Bases and Transforms of Set Functions

Abstract : The paper 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 dual-ity 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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Chapitre d'ouvrage
S. Saminger-Platz and R. Mesiar. On Logical, Algebraic and Probabilistic Aspects of Fuzzy Set Theory, 2016, 〈10.1007/978-3-319-28808-6_13〉
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Michel Grabisch. Bases and Transforms of Set Functions. S. Saminger-Platz and R. Mesiar. On Logical, Algebraic and Probabilistic Aspects of Fuzzy Set Theory, 2016, 〈10.1007/978-3-319-28808-6_13〉. 〈hal-01302376〉

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