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Rubinstein2005-page30 - X is the set of all conceivable...

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October 21, 2005 12:18 master Sheet number 28 Page number 12 LECTURE 2 Utility The Concept of Utility Representation Think of examples of preferences. In the case of a small number of alternatives, we often describe a preference relation as a list arranged from best to worst. In some cases, the alternatives are grouped into a small number of categories and we describe the preferences on X by specifying the preferences on the set of categories. But, in my experience, most of the examples that come to mind are similar to: “I prefer the taller basketball player,” “I prefer the more expensive present,” “I prefer a teacher who gives higher grades,” “I prefer the person who weighs less.” Common to all these examples is that they can naturally be spec- ified by a statement of the form “ x y if V ( x ) V ( y ) ” (or V ( x ) V ( y ) ), where V : X is a function that attaches a real number to each element in the set of alternatives X . For example, the prefer- ences stated by “I prefer the taller basketball player” can be expressed
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Unformatted text preview: X is the set of all conceivable basketball players, and V ( x ) is the height of player x . Note that the statement x % y if V ( x ) ≥ V ( y ) always deFnes a pref-erence relation since the relation ≥ on < satisFes completeness and transitivity. Even when the description of a preference relation does not in-volve a numerical evaluation, we are interested in an equivalent nu-merical representation. We say that the function U : X → < represents the preference % if for all x and y ∈ X , x % y if and only if U ( x ) ≥ U ( y ) . If the function U represents the preference relation % , we refer to it as a utility function and we say that % has a utility representation . It is possible to avoid the notion of a utility representation and to “do economics” with the notion of preferences. Nevertheless, we usually use utility functions rather than preferences as a means of de-scribing an economic agent’s attitude toward alternatives, probably...
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