DeterminingGammaWithaSingleLotteryQuestion.pdf - Page 1 Determining γ(and Thereby ρ for a Delta Person Using a Single Certain Equivalent Question By

DeterminingGammaWithaSingleLotteryQuestion.pdf - Page 1...

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Page 1 October 31, 2019 Determining γ (and Thereby ρ ) for a Delta Person Using a Single Certain Equivalent Question By Dale M. Nesbitt Draft 1 How is it possible to get an assessment of the γ and therefore the ρ parameter for a delta person by asking a single certain equivalent question? This development, absolutely essential to this course, tells you precisely how and why. It isn’t as well articulated in the book as I would like, and the risk odds discussion there confuses matters. This discussion gives the answer in general and completely clarifies the risk odds approach in the book. Questions related to assessment of the risk aversion coefficient for a delta person with a single question have come up in my office hours for the past two years, so I put together a definitive characterization. 1 Derivation of Indifference Probability from Any Given u-Curve This section harks back to Lecture 6 and the rules of actional thought. Consider the deal for a more preferred prize A and a less preferred prize C. If we have any valid u-curve u(.), the expected utility is indicated in the following figure p ( ) u A ( ) u C 1 p u = We know that the u-curve u(.) can be subjected to an affine transformation without affecting the answer in any way. In particular, the affine transformed u-curve v(.) will give the same answer as the original u curve u(.). The affine transformed u-curve is ( ) ( ) v x a bu x b 0 = + > We want to set the constants a and b>0 of the affine transformation so that v(C)=0 and v(A)=1, just like Kim did in the party problem where the best price had u value 1 and the worst prize had u value 0. In particular, we want to set a and b so that ( ) ( ) ( ) ( ) v A a bu A 1 v C a bu C 0 = + = = + = This is exactly what we did in Lecture 6 with the best and worst prizes in the original development by von Neumann. We manipulate the second of these two equations
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Page 2 October 31, 2019 ( ) ( ) ( ) v C a bu C 0 a bu C = + = ⇒ =− Substitute this into the transformed u-curve v to write it ( ) ( ) ( ) ( ) ( ) v x bu C bu x b u x u C = + = Our intention to have v(A)=1 implies that ( ) ( ) ( ) ( ) ( ) v A 1 b u A u C 1 b u A u C = = = If we substitute this into the foregoing v curve, we can write the affine transformed utility function v in final form ( ) ( ) ( ) ( ) ( ) u z u C v z z u A u C = When z=A, we get v(A)=0. When z=C, we get v(C)=1. This works for ANY u curve, not just an exponential u-curve.
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