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Lecture12-2010

# Lecture12-2010 - Von Neumann-Morgenstern Proof I Lecture...

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Von Neumann-Morgenstern - Proof I: Lecture XII Charles B. Moss September 16, 2010 I. A:A If u v then α < β implies (1 - α ) u + α v (1 - β ) u + β v (1) 1. The direction of the assertion is that if u v and α < β , then the preference ordering must follow. 2. To demonstrate this we start with axiom 3:B:a given 0 α 1 u v u α u + (1 - α ) v u (1 - β ) + β v (2) 3. Intuitively, this axiom states that if u is the inferior bundle, then any bundle constructed with any combination of v must be pre- ferred to u . 4. bf Axiom 3:B:b reverses this axiom by saying that if u is the pre- ferred bundle then it must also be preferred to a bundle containing any amount of v . u v u α u + (1 - α ) v (3) 5. We start from the first equation, replace with and by replacing the first in the right-hand side with preceding equation yields (1 - β ) u + β v ((1 - β ) u + β v ) + (1 - γ ) v (4) 1

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AEB 6182 Agricultural Risk Analysis and Decision Making Professor Charles B. Moss Lecture XII Fall 2010 6. Next, we reverse the order of the combining in the first set of
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