Inverse stochastic orders and generalized Gini functionals

We investigate the class of stochastic orders induced by Generalized Gini Functionals (GGF) [Yaari (1987) dual functionals] and identify the maximal classes of functionals associated with these orders. Our results are inspired by Marshall (1991) and are dual to those obtained for additive representations in Müller (1997) and in Castagnoli and Maccheroni (1998). The closure of the convex hull generated by a given set of probability distortion functions (F) [or by a set of rank-dependent weighting functions (V)] identifies the maximal class of functionals associated with the stochastic orders that are consistent with F [or V]. Rank-dependent weighting functions obtained as convex combinations of indicator functions identify GGFs that can be considered the "basis" of relevant stochastic orders in decision theory and inequality measurement. As hinted by Wang and Young (1998) and Zoli (1999, 2002) the stochastic orders obtained are related to the class of inverse stochastic dominance (ISD) conditions introduced in Muliere and Scarsini (1989). Making use of our results we review some stochastic dominance conditions that can be applied in decision theory, inequality, welfare and poverty measurement. These conditions are associated with orders implied by first order ISD and implying second order ISD, as well as with orders implied by the latter.