  We establish a variety of numerical representations of preference relations
induced by set-valued risk measures. Because of the general incompleteness of
such preferences, we have to deal with multi-utility representations. We look
for representations that are both parsimonious (the family of representing
functionals is indexed by a tractable set of parameters) and well behaved (the
representing functionals satisfy nice regularity properties with respect to the
structure of the underlying space of alternatives). The key to our results is a
general dual representation of set-valued risk measures that unifies the
existing dual representations in the literature and highlights their link with
duality results for scalar risk measures.
