Characterizing the Set of Coherent Lower Previsions with a Finite Number of Constraints or Vertices

TL;DR

This paper reformulates the coherence criterion for finite lower previsions using a finite set of constraints, enabling the computation of extreme coherent lower previsions as vertices of a convex polytope.

Contribution

It introduces a finite constraint reformulation for coherence, allowing practical computation of extreme lower previsions in finite settings.

Findings

Finite constraints characterize coherence for finite lower previsions.

Vertices of the convex polytope are the extreme coherent lower previsions.

The method enables explicit computation of these extreme points.

Abstract

The standard coherence criterion for lower previsions is expressed using an infinite number of linear constraints. For lower previsions that are essentially defined on some finite set of gambles on a finite possibility space, we present a reformulation of this criterion that only uses a finite number of constraints. Any such lower prevision is coherent if it lies within the convex polytope defined by these constraints. The vertices of this polytope are the extreme coherent lower previsions for the given set of gambles. Our reformulation makes it possible to compute them. We show how this is done and illustrate the procedure and its results.