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.

我们建立了由设定值风险度量引起的各种偏好关系的数值表示。由于此类首选项的一般性不完整，因此我们必须处理多用途表示形式。我们寻找既简约（表示功能的族由一组易处理的参数索引）又表现良好的表示（相对于替代项的基础空间的结构，表示功能满足良好的正则性）。我们的结果的关键是集合值风险度量的一般双重表示，它统一了文献中现有的双重表示，并突出了它们与标量风险度量的双重性结果之间的联系。