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Ordinal Representations



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 representation 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. T (Boston) − T (Miami)> T (Boston) − T (New York)> 0 The temperature difference between New York and Boston is smaller than the temperature difference between Miami and Boston. But to say that u(x) − u(y)> u(a) − u(b)> 0 1 does not mean that x is more preferred to y than a is to b. We express this as follows: Definition 2. A utility representation for ≻ is ordinal iff for any strictly increasing function f: R → R, f ◦ U is also a utility representation for ≻. Proposition 1. A utility representation for a preference order ≻ is ordinal. Proof. If f is strictly increasing and U is a utility representation for ≻, then x ≻ y iff U(x)> U(y) iff f ( U(x) )> f ( U(y) ). 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)pa, risk aversion — not preferring a gamble to its expected 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) ≤ 0}, then optima can be found with the calculus. So why not start with utilities? • We don’t know that utility representations exist. • Some characteristic properties of classes of preferences are better understood by expressing them in terms of orderings

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Year: 2002
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