Decreasing aversion under ambiguity

Under which condition does the set of desirable uncertain prospects expand when wealth increases? We show that the decreasing concavity (DC) of the utility function u is necessary and sufficient in the α-maxmin expected utility model. In the smooth ambiguity aversion model with the ambiguity valuation function ϕ, the DC of u and of ϕ∘u is necessary and sufficient. An alternative classical definition of decreasing aversion is based on the hypothesis that the investment in a risky asset is increasing in wealth. We show that this hypothesis does not hold in general under ambiguity aversion, and that one needs to constrain the structure of ambiguity to obtain unambiguous results.