Axiomatic structure of k-additive capacities

TL;DR

This paper develops an axiomatic framework for preference relations modeled by the Choquet integral with respect to k-additive capacities, covering from probability measures to general capacities, with a focus on 2-additive cases.

Contribution

It provides a step-by-step axiomatization of k-additive capacities, emphasizing 2-additive capacities, and extends previous social welfare axiomatizations.

Findings

Axiomatization for symmetric 2-additive capacities related to Gini index.

Extension of axioms to general k-additive capacities.

Complete framework connecting preference relations and 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 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.