  In this paper, we establish a mathematical duality between utility transforms
and probability distortions. These transforms play a central role in decision
under risk by forming the foundation for the classic theories of expected
utility, dual utility, and rank-dependent utility. Our main results establish
that probability distortions are characterized by commutation with utility
transforms, and utility transforms are characterized by commutation with
probability distortions. These results require no additional conditions, and
hence each class can be axiomatized with only one property. Moreover, under
monotonicity, rank-dependent utility transforms can be characterized by set
commutation with either utility transforms or probability distortions.
