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Lecture13-2010 - Von Neumann-Morgenstern Proof II Lecture...

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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 < α 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. Specifically, 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 α u + (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 γ u 0 - γα 0 u 0 + γα 0 v 0 + v 0 - γ v 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 : 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 (1 - α 0 ) u 0 + α 0 v 0 w 0 (3) again substituting a linear combination of α 0 for the bundle w .
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