Weakly consistent extensions of lower previsions

Several consistency notions are available for a lower prevision assessed on a set of gambles (bounded random variables), ranging from the well known coherence to convexity and to the recently introduced 2-coherence and 2-convexity. In all these instances, a procedure with remarkable features, called (coherent, convex, 2-coherent or 2-convex) natural extension, is available to extend , preserving its consistency properties, to an arbitrary superset of gambles. We analyse the 2-coherent and 2-convex natural extensions, and respectively, showing that they may coincide with the other extensions in certain, special but rather common, cases of ‘full’ conditional lower prevision or probability assessments. This does generally not happen if is a(n unconditional) lower probability on the powerset of a given partition and is extended to the gambles defined on the same partition. In this framework we determine alternative formulae for and . We also show that may be nearly vacuous in some sense, while the Choquet integral extension is 2-coherent if is, and bounds from above the 2-coherent natural extension. Relationships between the finiteness of the various natural extensions and conditions of avoiding sure loss or weaker are also pointed out.