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.