Preferences on Intervals: a general framework

Preferences are usually considered as binary relations applied on a set of objects, let’s say A. Preference modelling is concerned by two basic problems (see [31]). The first can be summarised as follows. Consider a decision maker replying to a set of preference queries concerning a the elements of the set A: “do you prefer a to b?”, “do you prefer b to c?” etc.. Given such replies the problem is to check whether exists (and under which conditions) one or more real valued functions which, when applied to A, will return (faithfully) the preference statements of the decision maker. As an example consider a decision maker claiming that, given three candidates a, b and c, he is indifferent between a and b as well as between b and c, although he clearly prefers a to c. There are several different numerical representations which could account for such preferences. For instance we could associate to a the interval [5, 10], to b the interval [3, 6] and to c the interval [1, 4]. Under the rule that x is preferred to y iff the interval associated to x is completely to the right (in the sense of the reals) of the one associated to y and indifferent otherwise, the above numerical representation faithfully represents the decision makers preference statements. The second problem goes the opposite way. We have a numerical representation for all elements of the set A and we would like to construct preference relations for a given decision maker. As an example consider three objects a, b and c whose cost is 10, 12 and 20 respectively. For a certain decision