Random variables coupled by the dependence structure of comonotonicity posses the property of increasing or decreasing simultaneously. Moreover, comonotone random variables share a number of interesting properties. One of them is that given two marginal probability distributions, there exists a joint comonotone model with the given marginals, such a joint model can be easily constructed, and it unique. This paper explores comonotonicity within the context of uncertainty modelled using imprecise probability models. Specifically, we assume that the uncertainty about the marginal models is described in terms of coherent lower probabilities and we seek a comonotone lower probability with the given marginals, referred to as the comonotone extension. In this setting, we analyse whether a comonotone extension can be built (existence), how to build it (construction) and if it unique (uniqueness). The potential lack of uniqueness leads us to examine the same questions in relation to the most conservative comonotone extension, known as the comonotone natural extension.

A common approach used in robust statistics to robustify a probabilistic model is to distort a probability measure or to create a neighbourhood around it with a given radius and with respect to an appropriate distorting function. This approach establishes a clear connection with lower probabilities, also referred to as non-additive measures or capacities, which serve as tools to model uncertainty in a probability measure and are formally equivalent to normalised coalitional games. In this contribution, we take this idea a step further by analysing the problem of directly distorting a lower probability. For this purpose, we introduce in first place a formal definition of a generic distortion procedure and examine some desirable properties such a procedure may satisfy. Afterwards, we focus particularly on the distortion procedure based on the total variation distance and investigate the properties it satisfies. Finally, we demonstrate that the distortion of lower probabilities has a clear interpretation from the perspective of coalitional games, showing that the distortion based on the total variation distance aligns with a procedure commonly known as strong- δ -core, used to relax the constraints imposed by coalitions in order to ensure the non-emptiness of the core.

We consider the problem of aggregating belief models elicited by experts when these are expressed by means of coherent lower previsions. These constitute a framework general enough so as to include as particular cases not only probability measures but also the majority of models from imprecise probability theory. Although the aggregation problem has already been tackled in the literature, our contribution provides a unified view by gathering a number of rationality criteria and aggregation rules studied in different papers. Specifically, we consider six aggregation rules and twenty rationality criteria. We exhaustively analyse the relationships between the rules, the properties satisfied by each rule and the characterisations of the rules in terms of the criteria.

The Total Variation is a common distance between probability distributions that measures the maximum difference in probability among all events. When comparing non-additive measures, that can be represented by closed and convex sets of probability measures, the Total Variation distance can be extended in multiple ways. This paper explores three specific approaches in detail. The first approach considers the minimum Total Variation distance between the probability measures dominating the non-additive measures. The second approach replaces the minimum with the maximum of the Total Variation distances. The third approach modifies the first one by using the supremum distance instead of the Total Variation. We analyse the main properties of each approach and, in particular, apply them to distort non-additive measures. Finally, we demonstrate that the distortion of non-additive measures with the proposed extensions are closely connected to the strong and weak cores in coalition game theory.

We investigate the role of some game solutions, such the Shapley and the Banzhaf values, as probability transformations. The first one coincides with the pignistic transformation proposed in the Transferable Belief Model; the second one is not efficient in general, leading us to consider its normalized version. We study a number of particular models of lower probabilities: minitive measures, coherent lower probabilities, as well as the lower probabilities induced by comparative or distortion models. For them, we provide some alternative expressions of the Shapley and Banzhaf values and study under which conditions they belong to the core of the lower probability.

Under an epistemic interpretation, an upper probability can be regarded as equivalent to the set of probability measures it dominates, sometimes referred to as its core. In this paper, we study the properties of the number of extreme points of the core of a possibility measure, and investigate in detail those associated with (uni- and bi-)variate p-boxes, that model the imprecise information about a cumulative distribution function.

A p-box is a simple generalization of a distribution function, useful to study a random number in the presence of imprecision. We propose an extension of p-boxes to cover imprecise evaluations of pairs of random numbers and term them bivariate p-boxes. We analyze their rather weak consistency properties, since they are at best (but generally not) equivalent to 2-coherence. We therefore focus on the relevant subclass of coherent p-boxes, corresponding to coherent lower probabilities on special domains. Several properties of coherent p-boxes are investigated and compared with those of (one-dimensional) p-boxes or of bivariate distribution functions.

The comparison of Atanassov intuitionistic fuzzy sets (AIF-sets) is a topic that has been widely studied due to its several applications in image segmentation or decision making, among other fields. Divergences for AIF-sets (AIF-divergences) were introduced as an adequate measure of comparison for AIF-sets. This study investigates a family of AIF-divergences that satisfies a local property. Such a property allows us to compute the divergence between AIF-sets pointwise. A characterization of those AIF-divergences satisfying the local property is provided. Several interesting properties of local divergence are also discussed. Some applications of these AIF-divergences in pattern recognition and decision making illustrate their utility.

Characterization of dissimilarity/divergence between intuitionistic fuzzy sets (IFSs) is important as it has applications in different areas including image segmentation and decision making. This study deals with the problem of comparison of intuitionistic fuzzy sets. An axiomatic definition of divergence measures for IFSs is presented, which are particular cases of dissimilarities between IFSs. The relationships among IF-divergences, IF-dissimilarities, and IF-distances are studied. Finally, we propose a very general framework for comparison of IFSs, where depending on the conditions imposed on a particular function, we can realize measures of distance, dissimilarity, and divergence for IFSs. Some methods for building divergence measures for IFSs are also introduced, as well as some examples of IF-divergences. In particular, we have proved some results that can be used to generate measures of divergence for fuzzy sets as well as for intuitionistic fuzzy sets.

Classically, the comparison of random variables have been done by means of a crisp order, which is known as stochastic dominance. In the last years, the classical stochastic dominance have been extended to a graded version by means of a probabilistic relation. In this work we propose different ways of measuring the gradual order among random variables by using fuzzy relations instead of probabilistic relations. The connection between the cycle-transitivity of the probabilistic relation and the T-transitivity of the associated fuzzy weak preference relation is characterized in the particular case of the minimum t-norm.

In some cases the fitness value of a knowledge base is not completely determined, but just bounded in an interval. In this case the fitness value is modelled by a random variable. Thus the comparison of random variables allows to compare the fitness values when they are not completely determined. In this contribution we consider a quite new proposal in stochastic comparison: statistical preference. We recall the advantages of this method with respect to (the classical) stochastic dominance. We also order by statistical preference two fitness values modelled by beta distributions with some special parameters.

The concept of (classical) complete preorder can be characterized in several ways. In previous works we have studied whether complete fuzzy preorders can be characterized by the same properties as in the crisp case. We have proven that this is not usually the case. We have studied five possible characterizations and we have proven that only one still characterizes a fuzzy preorder. In this work we study those properties for additive fuzzy preference structures without incomparability. Despite they do not characterize complete fuzzy preorders, they can be related among them. In this contribution we show their connection when the preference structure does not admit incomparable alternatives.