Extreme points of the credal sets generated by comparative probabilities

• We consider comparative probability models on the singletons of a finite space.
• We study the set of probability measures compatible with such a model.
• We characterize its extreme points by means of a graphical representation.
• We investigate the properties of the lower probability induced by this set.
• We provide tight bounds on the number of extreme points.

When using convex probability sets (or, equivalently, lower previsions) as uncertainty models, identifying extreme points can help simplifying various computations or the use of some algorithms. In general, sets induced by specific models such as possibility distributions, linear vacuous mixtures or 2-monotone measures may have extreme points easier to compute than generic convex sets. In this paper, we study extreme points of another specific model: comparative probability orderings between the singletons of a finite space. We characterize these extreme points by means of a graphical representation of the comparative model, and use them to study the properties of the lower probability induced by this set. By doing so, we show that 2-monotone capacities are not informative enough to handle this type of comparisons without a loss of information. In addition, we connect comparative probabilities with other uncertainty models, such as imprecise probability masses.