eco331-08-StochasticDominance - Northwestern University...

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Northwestern University Marciano Siniscalchi Fall 2009 Econ 331-0 STOCHASTIC DOMINANCE 1. Introduction Our analysis so far builds upon expected-utility theory. An unpleasant implication of this ap- proach is that, without knowledge of an individual’s utility function, there is little we can say about her preferences. True, if we know that she is risk-averse, we may be able to determine whether she prefers a certain amount of money to a random variable describing uncertain or risky terminal wealth; we are able to construct the efficient frontier if we consider normally distributed returns; etc. But note that, in both cases, we are restricting attention to specific comparisons—e.g. comparing certain sums of money with random variables, comparing normally distributed portfolios, etc. However, for general random variables X,Y , we cannot say whether X ± Y , Y ± X , or X Y . It is clear that we cannot completely remedy this situation—otherwise, utility theory would be useless! However, we can make some progress on the sole basis of economically meaningful assumptions such as “more money is better” and risk aversion. This leads to the notion of stochastic dominance . 2. First-Order Stochastic Dominance With this as backdrop, consider two random variables X and Y . The following seems a plausible definition of the assertion that X is more likely to yield better outcomes than Y : Definition 1. For any two random variables X,Y , say that X first-order stochastically dom- inates Y (or X FOSD Y ) iff, for all m R , Pr[ X > m ] Pr[ Y > m ] . The important point is that the above inequality is required to hold for all real numbers m . In a way, X is “unequivocally better” than Y . Note also that FOSD may be seen as establishing a transitive, but partial (as opposed to complete) order on random variables. On the other hand, EU yields a complete order. For instance, suppose
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This note was uploaded on 11/17/2009 for the course ECONOMICS 331 taught by Professor Marciano during the Spring '09 term at Northwestern.

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eco331-08-StochasticDominance - Northwestern University...

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