Lower Desirability Functions: A Convenient Imprecise Hierarchical Uncertainty Model

I introduce and study a fairly general imprecise secondorder uncertainty model, in terms of lower desirability. A modeller’s lower desirability for a gamble is defined as her lower probability for the event that a given subject will find the gamble (at least marginally) desirable. For lower desirability assessments, rationality criteria are introduced that go back to the criteria of avoiding sure loss and coherence in the theory of (first-order) imprecise probabilities. I also introduce a notion of natural extension that allows the least committal coherent extension of lower desirability assessments to larger domains, as well as to a first-order model, which can be used in statistical reasoning and decision making. The main result of the paper is what I call Precision–Imprecision Equivalence: as far as certain behavioural implications of this model are concerned, it does not matter whether the subject’s underlying first-order model is assumed to be precise or imprecise.