  If uncertainty is modelled by a probability measure, decisions are typically
made by choosing the option with the highest expected utility. If an imprecise
probability model is used instead, this decision rule can be generalised in
several ways. We here focus on two such generalisations that apply to sets of
probability measures: E-admissibility and maximality. Both of them can be
regarded as special instances of so-called choice functions, a very general
mathematical framework for decision making. For each of these two decision
rules, we provide a set of necessary and sufficient conditions on choice
functions that uniquely characterises this rule, thereby providing an axiomatic
foundation for imprecise decision making with sets of probabilities. A
representation theorem for Archimedean choice functions in terms of coherent
lower previsions lies at the basis of both results.
