Arrovian impossibilities in aggregating preferences over non-resolute outcomes

Let A be a set of alternatives whose power set is (A) under bar. Elements of (A) under bar are interpreted as non-resolute outcomes. We consider the aggregation of preference profiles over (A) under bar into a (social) preference over (A) under bar. In case we allow individuals to have any complete and transitive preference over (A) under bar, Arrow's impossibility theorem naturally applies. However, the Arrovian impossibility prevails, even when the set of admissible preferences over A is severely restricted. In fact, we identify a mild regularity condition which ensures the dictatoriality of a domain. Regularity is compatible with almost all standard extension axioms of the literature. Thus, we interpret our results as the strong prevalence of Arrow's impossibility theorem in aggregating preferences over non-resolute outcomes.