Utility Functions Invariant with Respect to Some Classes of Shifts

A utility function U is said to be invariant with respect to a family of transformations Γ provided, for every member γ of Γ , U and U ◦ γ represent the same preference relation over lotteries. An invariance with respect to a wide class of transformations can be reduced to an invariance with respect to the shift transformations. We give a complete answer to the following question: given a nonempty set T of shifts determine all utility functions invariant with respect to the shift transformations by every element of T . As a consequence of our results we obtain the forms of utility functions invariant with respect to the families of commuting transformations.