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
Version
  • 1