Rubinstein2005-page110

Rubinstein2005-page110 - p % q iff U ( p ) ≥ U ( q ) . We...

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October 21, 2005 12:18 master Sheet number 108 Page number 92 92 Lecture Eight Extreme preference for certainty : This preference relation is con- tinuous (as the function max { p 1 , ... , p K } which represents it is continuous in probabilities). It does not satisfy I since, for example, although [ z 1 ]∼[ z 2 ] ,1 / 2 [ z 1 ]⊕ 1 / 2 [ z 1 1 / 2 [ z 1 ]⊕ 1 / 2 [ z 2 ] . Lexicographic preferences : Such a preference relation satisFes I but not C (an exercise!). The worst case : Such a preference relation does not satisfy C .In the two-prize case where v ( z 1 )> v ( z 2 ) , [ z 1 1 / 2 [ z 1 ]⊕ 1 / 2 [ z 2 ] . Viewed as points in R 2 + , we can rewrite this as ( 1, 0 ) Â ( 1 / 2, 1 / 2 ) . Any neighborhood of ( 1, 0 ) contains lotteries that are not strict- ly preferred to ( 1 / 2, 1 / 2 ) and thus C is not satisFed. Such a pref- erence relation also does not satisfy I ( [ z 1 ]Â[ z 2 ] but 1 / 2 [ z 1 ]⊕ 1 / 2 [ z 2 ]∼[ z 2 ] .) Utility Representation By Debreu’s theorem we know that for any relation % deFned on the space of lotteries that satisFes C , there is a utility representation U : L ( Z ) →< , continuous in the probabilities, such that
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Unformatted text preview: p % q iff U ( p ) ≥ U ( q ) . We will use the above axioms to isolate a family of preference relations and to derive a more structured utility function. Theorem (vNM): Let % be a preference relation over L ( Z ) satisfying I and C . There are numbers ( v ( z )) z ∈ Z such that p % q iff U ( p ) = 6 z ∈ Z p ( z ) v ( z ) ≥ U ( q ) = 6 z ∈ Z q ( z ) v ( z ). Note the distinction between U ( p ) (the utility number of the lot-tery p ) and v ( z ) (called the Bernoulli numbers or the vNM utilities). The function v is a utility function representing the preferences on Z and is the building block for the construction of U ( p ) , a utility function representing the preferences on L ( Z ) . We will also often say that v is a vNM utility function representing the preferences % over L ( Z ) ....
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This note was uploaded on 12/29/2011 for the course ECO 443 taught by Professor Aswa during the Fall '10 term at SUNY Stony Brook.

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