2-Coherent and 2-convex conditional lower previsions

In this paper we explore relaxations of (Williams) coherent and convex conditional previsions that form the families of n-coherent and n-convex conditional previsions, at the varying of n. We investigate which such previsions are the most general one may reasonably consider, suggesting (centered) 2-convex or, if positive homogeneity and conjugacy is needed, 2-coherent lower previsions. Basic properties of these previsions are studied. In particular, we prove that they satisfy the Generalised Bayes Rule and always have a 2-convex or, respectively, 2-coherent natural extension. The role of these extensions is analogous to that of the natural extension for coherent lower previsions. On the contrary, n-convex and n-coherent previsions with n≥3 either are convex or coherent themselves or have no extension of the same type on large enough sets. Among the uncertainty concepts that can be modelled by 2-convexity, we discuss generalisations of capacities and niveloids to a conditional framework and show that the well-known risk measure Value-at-Risk only guarantees to be centered 2-convex. In the final part, we determine the rationality requirements of 2-convexity and 2-coherence from a desirability perspective, emphasising how they weaken those of (Williams) coherence.