Lecture13-2010

# Lecture13-2010 - Von Neumann-Morgenstern - Proof II:...

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Von Neumann-Morgenstern - Proof II: Lecture XIII Charles B. Moss September 20, 2010 I. Separating Classes A. There must exist α 0 with 0 0 < 1 which separates the classes. 1. Thus, α 0 will be such that for α<α 0 the resulting bundle is in Class I, 2. And if α>α 0 then the resulting set is in Class II. B. First consider α 0 in Class I. 1. Speci±cally, we start by trying to generate a new point such that α>α 0 , but w w 0 . 2. In this case (1 α 0 ) u 0 + α 0 v 0 w 0 3. Using 3:B:e u w v αu +(1 α ) v w for some α γ ((1 α 0 ) u 0 + α 0 v 0 )+(1 γ ) v 0 w 0 (1) since w v 0 . 4. Therefore by the combining axiom γ ((1 α 0 ) u 0 + α 0 v 0 )+(1 γ ) v 0 γu 0 γα 0 u 0 + γα 0 v 0 + v 0 γv 0 γ (1 α 0 ) u 0 +(1 γ (1 α 0 )) v 0 (2) 5. Hence α =1 γ (1 α 0 ) α 0 3 :0 <γ< 1 belongs to I. 6. This forms the contradiction, so that w cannot be preferred to w 0 if α<α 0 . C. Second, consider α 0 in Class II. 1

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AEB 6182 Agricultural Risk Analysis and Decision Making Professor Charles B. Moss Lecture XII Fall 2010 1. Like the scenario above, we begin by generating the counter point that α<α 0 , but w ± w 0 . 2. Then
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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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Lecture13-2010 - Von Neumann-Morgenstern - Proof II:...

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