Slides13-2010

Slides13-2010 - Outline Separating Classes Mapping to...

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Outline Separating Classes Mapping to Utility Space Von Neumann-Morgenstern - Proof II: Lecture XIII Charles B. Moss September 20, 2010 Charles B. Moss Von Neumann-Morgenstern

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Outline Separating Classes Mapping to Utility Space Separating Classes Class I Class II Limitations Mapping to Utility Space Charles B. Moss Von Neumann-Morgenstern
Outline Separating Classes Mapping to Utility Space Class I Class II Limitations Separating Classes I There must exist α 0 with 0 0 < 1 which separates the classes. I Thus, α 0 will be such that for α<α 0 the resulting bundle is in Class I, I And if α>α 0 then the resulting set is in Class II. I First consider α 0 in Class I. I Speci±cally, we start by trying to generate a new point such that 0 , but w w 0 . I In this case (1 α 0 ) u 0 + α 0 v 0 w 0 I 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 . Charles B. Moss Von Neumann-Morgenstern

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Outline Separating Classes Mapping to Utility Space Class I Class II Limitations I Continued I 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 + 0 v 0 + v 0 γ v 0 γ (1 α 0 ) u 0 +(1 γ (1 α 0 )) v 0 (2) I Hence α =1 γ (1 α 0 ) α 0 3 :0 <γ< 1 belongs to I. I This forms the contradiction, so that w cannot be preferred to w 0 if α<α 0 . Charles B. Moss Von Neumann-Morgenstern
Outline Separating Classes Mapping to Utility Space Class I Class II Limitations I Second, consider α 0 in Class II.

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Slides13-2010 - Outline Separating Classes Mapping to...

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