Rubinstein2005-page31 - f U b(since f is strictly...

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October 21, 2005 12:18 master Sheet number 29 Page number 13 Utility 13 because we fnd it more convenient to talk about the maximization oF a numerical Function than oF a preFerence relation. Note that when defning a preFerence relation using a utility Func- tion, the Function has an intuitive meaning that carries with it addi- tional inFormation. In contrast, when the utility Function is Formed in order to represent an existing preFerence relation, the utility Func- tion has no meaning other than that oF representing a preFerence relation. Absolute numbers are meaningless in the latter case; only relative order has meaning. Indeed, iF a preFerence relation has a utility representation, then it has an infnite number oF such repre- sentations, as the Following simple claim shows: Claim: IF U represents % , then For any strictly increasing Function f :<→< , the Function V ( x ) = f ( U ( x )) represents % as well. Proof: a % b iFF U ( a ) U ( b ) (since U represents % ) iFF f ( U ( a ))
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Unformatted text preview: f ( U ( b )) (since f is strictly increasing) iFF V ( a ) ≥ V ( b ) . Existence of a Utility Representation IF any preFerence relation could be represented by a utility Function, then it would “grant a license” to use utility Functions rather than preFerence relations with no loss oF generality. Utility theory inves-tigates the possibility oF using a numerical Function to represent a preFerence relation and the possibility oF numerical representations carrying additional meanings (such as, a is preFerred to b more than c is preFerred to d ). We will now examine the basic question oF “utility theory”: Under what assumptions do utility representations exist? Our frst observation is quite trivial. When the set X is fnite, there is always a utility representation. The detailed prooF is pre-sented here mainly to get into the habit oF analytical precision. We...
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