Monetary utility functions are -- except for the expected value -- not of von Neumann-Morgenstern type.  In case the utility function has convex level sets in the set of probability measures on the real line, we can give some characterisation that comes close to the vN-M form.  For coherent utility functions this was solved by Ziegel.  The general concave case is not yet fully understood but with some extra weak compactness property and using some results of Stephan Weber we can characterise this class of utility functions.  Having convex level sets can be seen as a weakened form of the independence axiom in the vN-M theorem.

This is joint work with Bignozzi, Bellini and Ziegel.