Rubinstein2005-page111

# Rubinstein2005-page111 - I ∗ we obtain that p ∼(6 z ∈...

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October 21, 2005 12:18 master Sheet number 109 Page number 93 Expected Utility 93 For the proof of the theorem, we need the following lemma: Lemma: Let % be a preference over L ( Z ) satisfying Axiom I . Let x , y Z such that [ x ]Â[ y ] and 1 α>β 0. Then α x ( 1 α) y Â β x ( 1 β) y . Proof: If either α = 1or β = 0, the claim is implied by I . Otherwise, by I , α x ( 1 α) y Â[ y ] . Using I again we get: α x ( 1 α) y Â (β/α)(α x ( 1 α) y ) ( 1 β/α) [ y ]= β x ( 1 β) y . Proof of the theorem: Let M and m be the best and worst certain lotteries in L ( Z ) . Consider ±rst the case that M m . It follows from I that p m for any p and thus p q for all p , q L ( Z ) . Choosing v ( z ) = 0 for all z we have 6 z Z p ( z ) v ( z ) = 0 for all p L ( Z ) . Thus, a constant utility function represents % . Now consider that M Â m .By C and the lemma, there is a single number v ( z ) ∈[ 0, 1 ] such that v ( z ) M ( 1 v ( z )) m ∼[ z ] . (For exam- ple, v ( M ) = 1 and v ( m ) = 0). By
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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 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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