Axiomatic structure of k-additive capacities

Pedro Miranda (1) , Michel Grabisch (2, 3) , Pedro Gil (4)

1 UCM - Universidad Complutense de Madrid = Complutense University of Madrid [Madrid]
2 DECISION
3 CERMSEM - CEntre de Recherche en Mathématiques, Statistique et Économie Mathématique
4 Universidad de Oviedo [Oviedo]

In this paper we deal with the problem of axiomatizing the preference relations modelled through Choquet integral with respect to a $k$-additive capacity, i.e. whose Möbius transform vanishes for subsets of more than $k$ elements. Thus, $k$-additive capacities range from probability measures ($k=1$) to general capacities ($k=n$). The axiomatization is done in several steps, starting from symmetric 2-additive capacities, a case related to the Gini index, and finishing with general $k$-additive capacities. We put an emphasis on 2-additive capacities. Our axiomatization is done in the framework of social welfare, and complete previous results of Weymark, Gilboa and Ben Porath, and Gajdos.

Mots clés: Axiomatic; Capacities; k-Additivity

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

Pedro Miranda, Michel Grabisch, Pedro Gil. Axiomatic structure of k-additive capacities. Mathematical Social Sciences, 2005, 49 (2), pp.153-178. ⟨10.1016/j.mathsocsci.2004.06.001⟩. ⟨hal-00188165⟩