On the Existence of Subjective Upper and Lower Probabilities

Consistency between preference under uncertainty modeled by Choquet expected utility theory and preference under imprecise risk (a particular situation of uncertainty characterized by the existence of objective upper and lower probabilities) modeled by linear utility theory is shown to imply that the Choquet capacity must coincide on the imprecisely probabilized events with a fixed linear combination of their upper and lower probabilities. This result is used to derive conditions for the existence of subjective upper and lower probabilities on all the events.