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Bases and transforms of set functions

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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

Suggested Citation

  • Michel Grabisch, 2016. "Bases and transforms of set functions," Documents de travail du Centre d'Economie de la Sorbonne 16078, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    • Handle: RePEc:mse:cesdoc:16078
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    References listed on IDEAS

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    1. Michel Grabisch & Christophe Labreuche, 2002. "The symmetric and asymmetric Choquet integrals on finite spaces for decision making," Statistical Papers, Springer, vol. 43(1), pages 37-52, January.
    2. Michel Grabisch & Christophe Labreuche, 2010. "A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid," Annals of Operations Research, Springer, vol. 175(1), pages 247-286, March.
    3. Ulrich Faigle & Michel Grabisch, 2014. "Linear Transforms, Values and Least Square Approximation for Cooperation Systems," Documents de travail du Centre d'Economie de la Sorbonne 14010, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    4. Chateauneuf, Alain & Jaffray, Jean-Yves, 1989. "Some characterizations of lower probabilities and other monotone capacities through the use of Mobius inversion," Mathematical Social Sciences, Elsevier, vol. 17(3), pages 263-283, June.
    5. Norman L. Kleinberg & Jeffrey H. Weiss, 1985. "Equivalent N -Person Games and the Null Space of the Shapley Value," Mathematics of Operations Research, INFORMS, vol. 10(2), pages 233-243, May.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    basis; set functions; TU games; Fourier transform; Möbius transform; interaction Shapley value; Choquet integral;
    All these keywords.

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games

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