Generalising the pari-mutuel model

We introduce two models for imprecise probabilities which generalise the Pari-Mutuel Model while retaining its simple structure. Their consistency properties are investigated, as well as their capability of formalising an assessor\u2019s different attitudes. It turns out that one model is always coherent, while the other is (occasionally coherent but) generally only 2-coherent, and may elicit a conflicting attitude towards risk

Inference with Nearly-Linear uncertainty models

Several simplified uncertainty models are derived from a given probability of which they are a perturbation. Among these, we introduced in previous work Nearly-Linear (NL) models. They perform a linear affine transformation of with barriers, obtaining a couple of conjugate lower/upper probabilities, and generalise several well known neighbourhood models. We classified NL models, partitioning them into three subfamilies, and established their basic consistency properties in [5]. In this paper we investigate how to extend NL models that avoid sure loss by means of their natural extension, a basic, although operationally not always simple, inferential procedure in Imprecise Probability Theory. We obtain formulae for computing directly the natural extension in a number of cases, supplying a risk measurement interpretation for one of them. The results in the paper also broaden our knowledge of NL models: we characterise when they avoid sure loss, express some of them as linear (or even convex) combinations of simpler models, and explore relationships with interval probabilities