A von Neumann-Morgenstern utility function U is said to be invariant with respect to a family of transformations G provided for every member g of G, the function U and Uog  represent the same preference. According to the classical result of Pfanzagl [P] a continuous utility function is invariant with respect to the shift transformations if and only if it is either a linear or an exponential function. Recently, Abbas [A] has proved that a utility function invariant with respect to a single shift value may depend on an arbitrary periodic function. Therefore the following problem arises naturally: given a nonempty set T of shifts determine all utility functions invariant with respect to every shift from T. In the present talk we give a complete answer to this question. As a consequence of our results we obtain the forms of utility functions invariant with respect to the families of commuting transformations. In this way we generalize the results from [A].

[A] A.E. Abbas, Invariant utility functions and certain equivalent transformations, Decision Analysis 4 (2007), 17--31.

[P] J. Pfanzagl, A general theory of measurement. Applications to utility, Naval Res. Logist. Quart. 6 (1959), 283--294.

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