A theory of desirable things

The theory of imprecise probabilities [1, 2] is often thought of as a theory of partially specified probabilities, which involves manipulating sets of probabilities and their lower and upper expectations. Its mathematical underpinnings, however, are provided by an underlying theory of sets of desirable gambles [2, 3, 4, 5, 6]: sets of gambles--rewards with an uncertain payoff--that a subject finds desirable, in the sense that she prefers those gambles to the status quo--to the trivial gamble with zero payoff. Rewards are typically taken to be expressed in units of some linear utility scale, and this them implies that positive linear combinations of desirable gambles are desirable themselves. Sets of desirable gambles that satisfy this condition (as well as some other, less essential conditions) are called coherent. Due to the geometric nature of the coherence conditions, inference with desirable gambles is typically simple and intuitive, a feature that is particularly handy, also when it comes to designing proofs. Most crucially, however, well known imprecise probability models such as credal sets (closed convex sets of probabilites), lower and upper expectations (or previsions), partial preference oderings, belief functions and lower and upper probabilities, all correspond to special cases of coherent sets of desirable gambles [4], which explains the importance of the latter as a basis for impreciseprobabilistic reasoning.