Extreme points of credal sets generated by 2-alternating capacities

Abstract

The characterization of the extreme points constitutes a crucial issue in the investigation of convex sets of probabilities, not only from a purely theoretical point of view, but also as a tool in the management of imprecise information. In this respect, different authors have found an interesting relation between the extreme points of the class of probability measures dominated by a second order alternating Choquet capacity and the permutations of the elements in the referential. However, they have all restricted their work to the case of a finite referential space. In an infinite setting, some technical complications arise and they have to be carefully treated. In this paper, we extend the mentioned result to the more general case of separable metric spaces. Furthermore, we derive some interesting topological properties about the convex sets of probabilities here investigated. Finally, a closer look to the case of possibility measures is given: for them, we prove that the number of extreme points can be reduced even in the finite case.