Parametric representation of preferences

A preference is invariant with respect to a transformation if its ranking of acts is unaffected by a reshuffing of the states under. We show that any invariant preference must be parametric: there is a unique sufficient set of parameters such that the preference ranks acts according to their expected utility given the parameters. This property holds for all non-trivial preferences, provided only that they are reflexive, transitive, monotone, continuous and mixture linear.