  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 M\"obius 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.
