Notes on desirability and conditional lower previsions

We detail the relationship between sets of desirable gambles and conditional lower previsions. The former is one the most general models of uncertainty. The latter corresponds to Walley's celebrated theory of imprecise probability. We consider two avenues: when a collection of conditional lower previsions is derived from a set of desirable gambles, and its converse. In either case, we relate the properties of the derived model with those of the originating one. Our results constitute basic tools to move from one formalism to the other, and thus to take advantage of work done in the two fronts.