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Bases and Transforms of Set Functions

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On Logical, Algebraic, and Probabilistic Aspects of Fuzzy Set Theory

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 336))

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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Authors and Affiliations

  1. Paris School of Economics, University of Paris I, 106-112, Bd de L’Hôpital, 75013, Paris, France

    Michel Grabisch

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  1. Michel Grabisch
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Correspondence to Michel Grabisch .

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Editors and Affiliations

  1. Dept of Knowledge-Based Math. Systems, Johannes Kepler University Linz, Linz, Austria

    Susanne Saminger-Platz

  2. Dept. of Mathematics & Descriptive Geo., Slovak University of Technology, BRATISLAVA, Slovakia

    Radko Mesiar

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Grabisch, M. (2016). Bases and Transforms of Set Functions. In: Saminger-Platz, S., Mesiar, R. (eds) On Logical, Algebraic, and Probabilistic Aspects of Fuzzy Set Theory. Studies in Fuzziness and Soft Computing, vol 336. Springer, Cham. https://doi.org/10.1007/978-3-319-28808-6_13

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  • DOI: https://doi.org/10.1007/978-3-319-28808-6_13

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-28807-9

  • Online ISBN: 978-3-319-28808-6

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