Rubinstein2005-page108

Rubinstein2005-page108 - Independence(I ±or any p q r ∈...

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October 21, 2005 12:18 master Sheet number 106 Page number 90 90 Lecture Eight Figure 8.1 The compound lottery K k = 1 α k p k . We think of K k = 1 α k p k as a compound lottery with the following two stages: Stage 1 : It is randomly determined which of the lotteries p 1 , ... , p K is realized; α k is the probability that p k is realized. Stage 2 : The prize Fnally received is randomly drawn from the lottery determined in stage 1. When we compare two compound lotteries, α p ( 1 α) r and α q ( 1 α) r , we tend to simplify the comparison and form our pref- erence on the basis of the comparison between p and q . This intu- ition is translated into the following axiom:
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Unformatted text preview: Independence (I): ±or any p , q , r ∈ L ( Z ) and any α ∈ ( 0, 1 ) , p % q iff α p ⊕ ( 1 − α) r % α q ⊕ ( 1 − α) r . The following property follows from I : I ∗ : ⊕ K k = 1 α k p k % ⊕ K k = 1 α k q k when p k = q k for all k but k ∗ iff p k ∗ % q k ∗ . To see it, ⊕ k = 1, ... , K α k p k = α k ∗ p k ∗ ⊕ ( 1 − α k ∗ )( ⊕ k 6= k ∗ [ α k /( 1 − α k ∗ ) ] p k ) % α k ∗ q k ∗ ⊕ ( 1 − α k ∗ )( ⊕ k 6= k ∗ [ α k /( 1 − α k ∗ ) ] q k ) = ⊕ K k = 1 α k q k iff p k ∗ % q k ∗ ....
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This note was uploaded on 12/29/2011 for the course ECO 443 taught by Professor Aswa during the Fall '10 term at SUNY Stony Brook.

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