002 blume 1 ordinal

002 blume 1 ordinal - Ordinal Representations September 1...

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Unformatted text preview: Ordinal Representations September 1, 2006 Note: The marked exercises are not (yet) homework assignments. They are additional things I thought it would amuse you to think about. 1 What is an Ordinal Representation ? We are given a preference order ≻ on X . Definition 1. A utility representation of the preference order ≻ is a function U : X → R such that x ≻ y if and only if u ( x ) > u ( y ) . What do we mean by an ordinal representation? First, a representa- tion is a numerical scaling — a thermometer to measure preference. Thus if x is better than y , x gets a higher utility number than y , just as if New York City is hotter than Boston, NY gets a higher temperature number. But with utility, only the ordinal ranking matters. Temperature is not an ordinal scale. New York is only slightly hotter than Boston, while Miami is much hotter than Cleveland. T (Miami) − T (Cleveland) > T (New York) − T (Boston) > The temperature difference between New York and Boston is smaller than the temperature difference between Miami and Cleveland. But to say that u ( x ) − u ( y ) > u ( a ) − u ( b ) > 1 does not mean that the incremental satisfaction from x over y is more than the incremental satisfaction from a over b . We express this as follows: Definition 2. A utility representation for ≻ is ordinal . If U is a utility representation for ≻ and f : R → R is a strictly increasing function, f ◦ U is also a utility representation for ≻ . 2 Why do we want an ordinal representation? Summary: An ordering is just a list of pairs, which is hard to grasp. A utility function is a convenient way of summarizing properties of the order. For instance, with expected utility preferences of the form U ( p ) = ∑ a u ( a ) p a , risk aversion — not preferring a gamble to its ex- pected value — is equivalent to the concavity of u . The curvature of u measures how risk-averse the decision-maker is. Optimization: We want to find optimal elements of orders on feasible sets. Sometimes these are more easily computed with utility functions. For instance, if U is C 1 and B is of the form { x : F ( x ) ≤ } , then optima can be found with the calculus. So why not start with utilities? • Preferences, after all, are the primitive concept, and we don’t know that utility representations exist for all kinds of preferences we’d want to talk about. • Some characteristic properties of classes of preferences are better un- derstood by expressing them in terms of orderings. • Preferences are the primitive concept, and some properties of utility functions are not readily interpreted in terms of the preference order. 2 3 When do ordinal representations exist? There are really two questions to ask: • Does every preference order have a representation? More generally, what binary relations have numerical representations?...
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002 blume 1 ordinal - Ordinal Representations September 1...

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