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 af±ne transformation (namely, multiplication by a positive number and adding any scalar) and are not invariant to arbitrary monotonic transformation. Con-sider a preference relation % de±ned over L ( Z ) and de±ne v ( z ) as in the proof above. Of course, de±ning w ( z ) = α v ( z ) + β for all z (for some α > 0 and some β ), the utility function W ( p ) = 6 z ∈ Z p ( z ) w ( z ) represents % ....
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- Fall '10
- Utility, following lemma, worst certain lotteries, vNM Utilities, arbitrary monotonic transformation