Slides16-2010 - Problems with Expected Utility Charles B....

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Problems with Expected Utility Charles B. Moss Outline Allais’ Paradox Common Consequence Efect Problems with Expected Utility: Lecture XVI Charles B. Moss September 28, 2010 Charles B. Moss Problems with Expected Utility
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Problems with Expected Utility Charles B. Moss Outline Allais’ Paradox Common Consequence Efect 1 Allais’ Paradox 2 Common Consequence Efect Charles B. Moss Problems with Expected Utility
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Problems with Expected Utility Charles B. Moss Outline Allais’ Paradox Common Consequence Efect Allais’ Paradox Machina starts by describing how the expected utility formulation is linear in probabilities. We start by assuming that there are three possible outcomes for a random variable { x 1 , x 2 , x 3 } such that x 1 < x 2 < x 3 with probabilities P =( p 1 , p 2 , p 3 ). Given this formulation 3 X i =1 p i =1 p 2 p 1 p 2 (1) Which implies U ( x )= p 1 U ( x 1 )+ p 2 U ( x 2 p 3 U ( x 3 ) = p 1 U ( x 1 )+(1 p 1 p 3 ) U ( x 2 p 3 U ( x 3 ) (2) Charles B. Moss Problems with Expected Utility
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Problems with Expected Utility Charles B. Moss Outline Allais’ Paradox Common Consequence Efect Continued Holding the payofs in Equation 3.52 constant and diferentiating with respect to the levels oF probability yields dU ( x )= U ( x 1 ) dp 1 +( dp 1 dp 3 ) U ( x 2 )+ U ( x 3 ) dp 3 =0 [ U ( x 1 ) U ( x 2 )] dp 1 = [ U ( x 2 U ( x 3 )] dp 3 [ U ( x 2 ) U ( x 1 )] [ U ( x 3 ) U ( x 2 )] = dp 3 dp 1 (3) which given that x 1 < x 2 < x 3 and iF U 0 ( x ) > 0 implies that any increase in p 3 relative to p 1 that holds the total probability constant (i.e., the probabilities still sum to one) yields more expected utility.
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This note was uploaded on 07/15/2011 for the course AEB 6182 taught by Professor Weldon during the Fall '08 term at University of Florida.

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Slides16-2010 - Problems with Expected Utility Charles B....

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