Bases and Transforms of Set Functions

Michel Grabisch (1, 2)

1 CES - Centre d'économie de la Sorbonne
2 PSE - Paris School of Economics

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.

Mots clés: set function; capacity; basis; transform; Moebius transform; Choquet integral

Domaines: Mathématique discrète [cs.DM]; Economies et finances

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⟩