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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 infinite 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 definition of the values U ( x 1 ) , . . . , U ( x n 1 ) so that x k x l iff U ( x k ) U ( x l ) . If 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 defined 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 . For some k and l we have x
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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 < 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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