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Unformatted text preview: Kata Bognar [email protected] Economics 142 Probabilistic Microeconomics UCLA Fall 2008 Homework Assignment 2. Solutions  1. (Value of Information) General Sternwood is blackmailed. The threat can be either real or just a bluff. The General does not know this for sure but he believes that the threat is real with probability 1 / 4 . The General has a total wealth of 250,000 dollars and the blackmailer is asking for 90,000 dollars. If he decides not to pay and the threat was real then he loses all his wealth. If he decides not to pay and the threat was a bluff then he has no loss at all. The General’s vNM utility is u ( x ) = √ x. Should General Sternwood pay the blackmailer? [2] Answer: To answer this question, you have to understand the consequences of each of the General’s actions, i.e. what the related acts are. Then you have to figure which of the two acts is preferred by the General by calculating and comparing their expected utilities. There are two actions available for the General, pay or not pay. The payoff relevant events for him are whether the threat is real or fake. Given his action and a state an outcome is defined, i.e. the amount of money he left with. • If the General pays the blackmailer then he has 250 , 000 90 , 000 = 160 , 000 dollars left for sure. The expected utility of this act is V (pay) = 1 p 160 , 000 = 400 . • If he does not pay and the threat was real, an event which has a probability 1 / 4, he ends up with no money. If he does not pay and the threat was fake, an event which has a probability 3 / 4, he ends up with with 250 , 000 dollars.The expected utility from this act is V (not pay) = 1 / 4 √ 0 + 3 / 4 p 250 , 000 = 1 / 4 * 0 + 3 / 4 * 500 = 375 . Therefore the General is better off if he pays (400 > 375) . The General figures the answer to the previous question himself but he is quite unhappy with it. Therefore he hires Philip Marlowe to find out some information about the case. Marlowe is a pretty good detective and if the blackmailing is a bluff he finds that out. However, he fails to recognize 1 in 3 real threats. (Clearly, there are no movies about those cases, however General Sternwood knows this.) What is the probability of a bluff given that Marlowe claims that the threat is real? [4] How likely it is that the threat is real given that Marlowe says it is a bluff? Answer: Denote by R the event that the threat is real, and by B the event that the threat is fake. Also, denote by r the event that Marlowe says that the threat is real, and by b the event that Marlowe says that the threat is fake. In the problem, the following probabilities are given: 1 • Prior probabilities for real or fake threat such as: P ( R ) = 1 / 4 P ( B ) = 3 / 4 ....
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 Fall '09
 Bognar
 Microeconomics, Conditional Probability, Philip Marlowe, Mrs Rutledge

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