Rubinstein2005-page115

Rubinstein2005-page115 - one of the proposed alternatives...

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October 21, 2005 12:18 master Sheet number 113 Page number 97 Problem Set 8 Problem 1. ( Standard ) Consider the following preference relations that were described in the text: “The size of the support” and “Comparing the most likely prize.” a. Check carefully whether they satisfy axioms I and C . b. These preference relations are not immune to a certain “framing prob- lem.” Explain. Problem 2. ( Standard. Based on Markowitz 1959. ) One way to construct preferences over lotteries with monetary prizes is by evaluating each lottery L on the basis of two numbers, Ex ( L ) , the expectation of L and var ( L ) , L ’s variance. Such a procedure may or may not be consistent with vNM assumptions. a. Show that u ( L ) = Ex ( L ) ( 1 / 4 ) var ( L ) induces a preference relation that is not consistent with the vNM assumptions. (For example, consider the mixtures of each of the lotteries [ 1 ] and 0 . 5 [ 0 ]⊕ 0 . 5 [ 4 ] with the lottery 0 . 5 [ 0 ]⊕ 0 . 5 [ 2 ] .) b. Show that u ( L ) = Ex ( L ) ( Ex ( L )) 2 var ( L ) is consistent with vNM as- sumptions. Problem 3. ( More difFcult. Based on Yaari 1987. ) In this problem you will encounter the functional of Quiggin and Yaari,
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Unformatted text preview: one of the proposed alternatives to expected utility theory. Consider a world with the prizes z , z 1 ,, z K . A decision maker attaches a number v ( z k ) to each z k such that v ( z ) = < v ( z 1 ) < v ( z 2 ) < . . . < v ( z K ) and eval-uates each lottery L by the number U ( L ) = 6 K k = 1 f ( G L ( z k )) [ v ( z k ) v ( z k 1 ) ] , where f : [ 0, 1 ] [ 0, 1 ] is a continuous increasing function and G L ( z k ) = 6 j k L ( z j ) . ( L ( z ) is the probability that the lottery L yields z and G L is the anti-distribution of L .) a. Verify that for f ( x ) = x , U ( L ) is the standard expected v-utility of L . b. Show that the induced preference relation satises the continuity ax-iom but may not satisfy the independence axiom. c. What are the difculties with a functional form of the type 6 z f ( p ( z )) u ( z ) ?...
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