Additivity with multiple priors

The functional defined as the 'min' of integrals with respect to probabilities in a given non-empty closed and convex class appears prominently in recent work on uncertainty in economics. In general, such a functional violates the additivity of the expectations operator. We characterize the types of functions over which additivity of this functional is preserved. This happens exactly when 'integrating' functions which are positive affine transformations of each other (or when one is constant). We show that this result is quite general by restricting the types of classes of probabilities considered. Finally, we prove that with a very peculiar exception, all the results hold more generally for functionals which are linear combinations of the 'min' and the 'max' functional.