We are pleased to announce that our paper
The Total Variation Distance for Comparing Non-Additive Measures
has been accepted for publication in the International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems.
This paper, written by David Nieto-Barba, Enrique Miranda and Ignacio Montes, explores the comparison of non-additive probabilities, with the Total Variation distance between probability measures as baseline. The first proposal extends this distance by taking the minimum Total Variation distance among the pairs of probability measures dominating the respective non-additive measures, while the second one replaces the minimum by a maximum. In our last approach, we consider the minimum of the supremum distances instead.
The preliminary Section 2 is followed by the analysis of each of these approaches in the respective Sections 3, 4 and 5. More concretely, for each proposal we investigate the distance-like properties fulfilled, computational simplifications in terms of the boundary or the extreme points of the set of dominating probabilities, alternative expressions in terms of the values of the non-additive measures and their application to the distortion of non additive measures.
In doing so, we demonstrate that neither of the extensions considered is a distance and, while the computation of the first a third approaches reduces to the boundary of the sets of dominating probabilities, the third one may be simply computed in terms of the extreme points. Moreover, we find sufficent conditions allowing us to express the minimum and maximum Total Variation distances and subsequent distortions directly in terms of the values of the non-additive measures.
To conclude, the Section 7 proves the connection of the distortions by the minimum distances proposals with the strong and weak cores in coalition game theory.