Rubinstein2005-page33

Rubinstein2005-page33 - y = x l . The above applied to n =...

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October 21, 2005 12:18 master Sheet number 31 Page number 15 Utility 15 countable (recall that X is countable and infnite iF there is a one-to- one Function From the natural numbers to X , namely, it is possible to speciFy an enumeration oF all its members { x n } n = 1,2, ... ). Claim: IF X is countable, then any preFerence relation on X has a utility representation with a range ( 1, 1 ) . Proof: Let { x n } be an enumeration oF all elements in X . We will construct the utility Function inductively. Set U ( x 1 ) = 0. Assume that you have completed the defnition oF the values U ( x 1 ) , ... , U ( x n 1 ) so that x k % x l iFF U ( x k ) U ( x l ) .I F x n is indiFFerent to x k For some k < n , then assign U ( x n ) = U ( x k ) . IF not, by transitivity, all num- bers in the set { U ( x k ) | x k x n }∪{− 1 } are below all numbers in the set { U ( x k ) | x n x k }∪{ 1 } . Choose U ( x n ) to be between the two sets. This guarantees that For any k < n we have x n % x k iFF U ( x n ) U ( x k ) . Thus, the Function we defned on { x 1 , ... , x n } represents the preFer- ence on those elements. To complete the prooF that U represents % , take any two elements, x and y X . ±or some k and l we have x = x k and
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Unformatted text preview: y = x l . The above applied to n = max { k , l } yields x k % x l iFF U ( x k ) U ( x l ) . Lexicographic Preferences Lexicographic preFerences are the outcome oF applying the Follow-ing procedure For determining the ranking oF any two elements in a set X . The individual has in mind a sequence oF criteria that could be used to compare pairs oF elements in X . The criteria are applied in a fxed order until a criterion is reached that succeeds in distinguish-ing between the two elements, in that it determines the preFerred alternative. ormally, let ( % k ) k = 1, ... , K be a K-tuple oF orderings over the set X . The lexicographic ordering induced by those orderings is defned by x % L y iF (1) there is k such that For all k &lt; k we have x k y and x k y or (2) x k y For all k . VeriFy that % L is a preFerence relation....
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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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