Axiomatic structure of k-additive capacities

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

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