On the Number and Characterization of the Extreme Points of the Core of Necessity Measures on Finite Spaces

This paper develops a combinatorial description of the extreme points of the core of a necessity measure on a finite space. We use the ingredients of Dempster-Shafer theory to characterize a necessity measure and the extreme points of its core in terms of the Möbius inverse, as well as an interpretation of the elements of the core as obtained through a transfer of probability mass from non-elementary events to singletons. With this understanding we derive an exact formula for the number of extreme points of the core of a necessity measure and obtain a constructive combinatorial insight into how the extreme points are obtained in terms of mass transfers. Our result sharpens the bounds for the number of extreme points given in [15] or [14, 13]. Furthermore, we determine the number of edges of the core of a necessity measure and additionally show how our results could be used to enumerate the extreme points of the core of arbitrary belief functions in a not too inefficient way.