Utility function Suppose ˆ L q 1 L1 q2 L 2 qm L m and for each j 1 j m let Lj p

Utility function suppose ˆ l q 1 l1 q2 l 2 qm l m

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Utility function. Suppose ˆ L = [ q 1 ( L 1 ) , q 2 ( L 2 ) ,..., q m ( L m )] and for each j 1 j m let L j = [ p j 1 ( x 1 ) , p j 2 ( x 2 ) ,..., p j k ( x k )] . Define for each i = 1 ,..., k , r i = q 1 p 1 i + q 2 p 2 i + ... + q m p m i . u ( ˆ L ) = r 1 θ 1 + r 2 θ 2 + ... + r k θ k . For every simple lottery L = [ p 1 ( x 1 ) ,..., p k ( x k )] , u ( L ) = k j = 1 p j θ j Outcome x j is same as the lottery L = [ 1 ( x j )] which is same as ˆ L = [ 1 ( L )] . So outcome of ˆ L is x j with probability 1. So we have r i = 1 if i = j , 0 if i j . We deduce that u ( x j ) = θ j . Sunil Simon Utility theory
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A characterization theorem To show: The function u is linear. Need to show that for each simple lottery L = [ p 1 ( x 1 ) ,... , p k ( x k )] , u ( L ) = k j = 1 p j u ( x j ) We have, 1 u ( L ) = k j = 1 p j θ j , 2 u ( x j ) = θ j . (1) implies that u ( L ) = k j = 1 p j θ j and (2) implies that k j = 1 p j u ( x j ) = k j = 1 p j θ j . Sunil Simon Utility theory
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A characterization theorem To show: The function u is a utility function. Need to show that for any pair of compound lotteries ˆ L and ˆ L , ˆ L ˆ L iff u ( ˆ L ) u ( ˆ L ) . Claim 2. ˆ L [ u ( ˆ L )( x k ) , ( 1 - u ( ˆ L ))( x 1 )] for every ˆ L . Monotonicity. Suppose x y , [ α ( x ) , ( 1 - α )( y )] [ β ( x ) , ( 1 - β )( y )] iff α β . Assuming Claim 2, the result follows from monotonicity of . Sunil Simon Utility theory
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A characterization theorem Claim 2. ˆ L [ u ( ˆ L )( x k ) , ( 1 - u ( ˆ L )( x 1 )] for every ˆ L . Proof. Let ˆ L = [ q 1 ( L 1 ) , q 2 ( L 2 ) ,.. ., q m ( L m )] and for each j 1 j m let L j = [ p j 1 ( x 1 ) , p j 2 ( x 2 ) ,..., p j k ( x k )] . Let r i = q 1 p 1 i + q 2 p 2 i + ... + q m p m i (the probability that the outcome is x i in ˆ L ). By simplification axiom, ˆ L [ r 1 ( x 1 ) ,..., r k ( x k )] . Let M i = [ θ i ( x k ) , ( 1 - θ i )( x 1 )] for every 1 i k . By definition x i M i . Thus k application of the independence axiom ˆ L [ r 1 ( M 1 ) ,..., r k ( M k )] .
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