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Axiomatic structure of \(k\)-additive capacities. (English) Zbl 1070.28009

Summary: We deal with the problem of axiomatizing the preference relations modeled through the 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.

MSC:

28E10 Fuzzy measure theory
28A12 Contents, measures, outer measures, capacities
91D10 Models of societies, social and urban evolution
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