Weakly consistent extensions of lower previsions

Several consistency notions are available for a lower prevision P_ assessed on a set D 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 P_, preserving its consistency properties, to an arbitrary superset of gambles. We analyse the 2-coherent and 2-convex natural extensions, E_2 and E_2c 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 P_ 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 E_2 and E_2c. We also show that E_2c may be nearly vacuous in some sense, while the Choquet integral extension is 2-coherent if P_ 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.