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Unformatted text preview: I we obtain that p (6 z Z p ( z ) v ( z )) M ( 1 6 z Z p ( z ) v ( z )) m . And by the lemma p % q iff 6 z Z p ( z ) v ( z ) 6 z Z q ( z ) v ( z ) . The Uniqueness of vNM Utilities The vNM utilities are unique up to positive afne transformation (namely, multiplication by a positive number and adding any scalar) and are not invariant to arbitrary monotonic transformation. Con-sider a preference relation % dened over L ( Z ) and dene v ( z ) as in the proof above. Of course, dening w ( z ) = v ( z ) + for all z (for some > 0 and some ), the utility function W ( p ) = 6 z Z p ( z ) w ( z ) represents % ....
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- Fall '10