Bases and Transforms of Set Functions
Chapitre d'ouvrage: 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.
Auteur(s)
Michel Grabisch
Éditeur(s) scientifique(s)
- S. Saminger-Platz and R. Mesiar
Titre de l’ouvrage
- On Logical, Algebraic and Probabilistic Aspects of Fuzzy Set Theory
Date de publication
- 2016
Mots-clés
- Set function
- Capacity
- Basis
- Transform
- Moebius transform
- Choquet integral
URL de la notice HAL
Version
- 1