  We consider a family of conditional nonlinear expectations defined on the
space of bounded random variables and indexed by the class of all the
sub-sigma-algebras of a given underlying sigma-algebra. We show that if this
family satisfies a natural consistency property, then it collapses to a
conditional certainty equivalent defined in terms of a state-dependent utility
function. This result is obtained by embedding our problem in a decision
theoretical framework and providing a new characterization of the Sure-Thing
Principle. In particular we prove that this principle characterizes those
preference relations which admit consistent backward conditional projections.
We build our analysis on state-dependent preferences for a general state space
as in Wakker and Zank (1999) and show that their numerical representation
admits a continuous version of the state-dependent utility. In this way, we
also answer positively to a conjecture posed in the aforementioned paper.
