When Belief Functions and Lower Probabilities are Indistinguishable

This paper reports on a geometrical investigation of de Finetti's Dutch Book method as an operational foundation for a wide range of generalisations of probability measures, including lower probabilities, necessity measures and belief functions. Our main result identifies a number of non-limiting circumstances under which de Finetti's coherence fails to lift from less to more general models. In particular our result shows that rich enough sets of events exist such that the coherence criteria for belief functions and lower probability collapse.