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Unformatted text preview: Chapter 1 Stochastic Dominance The concept of stochastic dominance is designed to capture the technical properties of statistical distributions for lotteries that enable broad rankings of those lotteries (with only limited information about the utility function of a particular consumer). Practically speaking, it is a way of comparing di/erent lotteries or distributions of outcomes. Let L 1 be a lottery with cumulative distribution F ( x ) and L 2 be a lottery with cumulative distribution G ( x ) . One approach to comparing these lotteries (and thus examining stochastic dominance) is to ask the following two questions: 1) When can we say that everyone will prefer L 1 to L 2 ? 2) When can we say that anyone who is risk averse will prefer L 1 to L 2 ? The answer to the &rst question is de&ned as the property of First-Order Stochastic Dom- inance (FOSD) , while the answer to the second question is the property of Second-Order Stochastic Dominance (SOSD) . A second approach to stochastic dominance asks two related questions: 1a) Can we write L 1 = L 2 + ¡something good¢? If we can do so, then everyone should prefer L 1 to L 2 for the right de&nition of ¡something good.¢ 2a) Can we write L 2 = L 1 + ¡risk¢? If we can do so, then every risk averse person should prefer L 1 to L 2 (and every risk loving person should prefer L 2 to L 1 ) for the right de&nition of ¡risk.¢ This section explains the de&nitions of ¡something good¢ and ¡risk,¢ and then shows how the two approaches to stochastic dominance are equivalent for these de&nitions. There is also a separate set of technical conditions that can be used to check for FOSD and SOSD, but they are just simpli&ed versions of the conditions for (1a) and (2a). 1 .2 .3 .1 .4 State L 1 L 2 s 1 $80 $10 s 2 $30 $50 s 3 $60 $70 s 4 $50 $30 A &nal important general point is that FOSD and SOSD require only weak preference for L 1 vs. L 2 , corresponding to weak conditions on utility functions (e.g. weak rather than strict concavity for risk aversion). 1.1 First-Order Stochastic Dominance (FOSD) We want to &nd conditions where we can write L 1 = L 2 + ¡something good,¢ and we want to &nd the appropriate de&nition (so that everyone will prefer L 1 = L 2 ) of ¡something good.¢ We will impose only the most minimal restriction on the utility function, specifying that u ( x ) is non- decreasing. This means that more wealth is at least as good as less wealth. For our de&nition, it must be that every person at least weakly prefers L 1 to L 2 . No matter how strange the utility function, if it nondecreasing, it must be true that L 1 & L 2 . In line with this restriction on u ( x ) , if we can match up the outcomes in L 1 and L 2 so that the outcomes in L 1 are at least as good as the outcomes in L 2 (in pairwise fashion) and L 1 is sometimes strictly better than L 2 , then everyone will prefer L 1 to L 2 ....
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- Fall '08