2-coherent and 2-convex Conditional Lower Previsions

TL;DR

This paper introduces and analyzes relaxations of coherent and convex conditional previsions, specifically 2-coherent and 2-convex types, exploring their properties, extensions, and applications in risk measurement and decision theory.

Contribution

It defines and studies 2-coherent and 2-convex conditional previsions, establishing their properties, natural extensions, and relevance to risk measures like Value-at-Risk.

Findings

2-coherent and 2-convex previsions satisfy the Generalized Bayes Rule.

They always have a 2-convex or 2-coherent natural extension.

Value-at-Risk is centered 2-convex.

Abstract

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 Generalized 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 $n\geq 3$ 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 generalizations 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.