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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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- Fall '10