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Rubinstein2005-page112

# Rubinstein2005-page112 - 12:18 94 master Sheet number 110...

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October 21, 2005 12:18 master Sheet number 110 Page number 94 94 Lecture Eight Furthermore, assume that W ( p ) = z p ( z ) w ( z ) represents the pref- erences as well. We will show that w must be a positive affine transformation of v . To see this, let α > 0 and β satisfy w ( M ) = α v ( M ) + β and w ( m ) = α v ( m ) + β (the existence of α > 0 and β is guaranteed by v ( M ) > v ( m ) and w ( M ) > w ( m ) ). For any z Z we have [ z ] ∼ v ( z ) M ( 1 v ( z )) m , so it must be that w ( z ) = v ( z ) w ( M ) + ( 1 v ( z )) w ( m ) = v ( z ) [ α v ( M ) + β ] + ( 1 v ( z )) [ α v ( m ) + β ] = α v ( z ) + β. The Dutch Book Argument There are those who consider expected utility maximization to be a normative principle. One of the arguments made to support this view is the following Dutch book argument. Assume that L 1 L 2 but that α L ( 1 α) L 2 α L ( 1 α) L 1 . We can perform the fol- lowing trick on the decision maker: 1. Take α L ( 1 α) L 1 (we can describe this as a contingency with random event E , which we both agree has probability 1 α ). 2. Take instead
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