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236 C H A P T E R 6. F U R T H E R TECHNIQUES IN DECISION A N A L Y S I S E x a m p l e 6.7 As in Example 6.4, suppose Joe has an investment opportunity that entails a .6 probability of gaining $4000 and a .4 probability of losing $2500. Again let dl be the decision alternative to take the investment opportunity and d2 be the decision alternative to reject it. Suppose Joe's risk preferences can be modeled using In(x). First, let's model the problem instance when Joe has a total wealth of $10,000. We then have that E U ( d l ) = .4In(10,000 + 4 0 0 0 ) + .6 In(10,000- 2500) = 9.1723 EU(d2) -- In 10,000 = 9.2103. So the decision is to reject the investment opportunity and do nothing. Next let's model the problem instance when Joe has a total wealth of $100,000. We then have that EU(dl) - . 4 I n ( 1 0 0 , 0 0 0 + 4 0 0 0 ) + .6 In(100,000- 2500) = 11.5134 EU(d2) = In 100,000 = 11.5129. So now the decision is to take the investment opportunity. Modeling risk attitudes is discussed much more in [Clemen, 1996]. 6.2 Analyzing Risk Directly Some decision makers may not be comfortable assessing personal utility func- tions and making decisions based on such functions. Rather, they may want to directly analyze the risk inherent in a decision alternative. One way to do this is to use the variance as a measure of spread from the expected value. Another way is to develop risk profiles. We discuss each technique in turn. 6.2.1 U s i n g the Variance to M e a s u r e Risk We start with an example. E x a m p l e 6.8 Suppose Patricia is going to make the decision modeled by the decision tree in Figure 6.4. If Patricia simply maximizes expected value, it is left as an exercise to show E(dl) E(d2) = = $1220 $1200. So dl is the decision alternative that maximizes expected value. However, the expected values by themselves tell us nothing of the risk involved in the alternatives. Let's also compute the variance of each decision alternative. If we choose alternative dl, then P(2000) P(1000) P(0) = = - .8· .1 .Sx.3+.1-.34.