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CORNELL ECON 2040 - ps6

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Networks: Fall 2013 Homework 6David Easley and´Eva Tardos Due at 11:15am, Friday, November 8, 2013As noted on the course home page, homework solutions must be submitted by upload to course’sBlackboard site. The file you upload must be in PDF format. It is fine to write the homeworkin another format such as Word; from Word, you can save the file out as PDF for uploading.You can choose type ”pdf” when you save the file, or print it and choose ”Adobe pdf” as yourprinter. (Changing the file extension from doc, or docx to pdf does not change the format, onlymakes the file unreadable)Blackboard will stop accepting homework uploads after the posted due date. We cannot ac-cept late homework except for University-approved excuses (which include illness, a familyemergency, or travel as part of a University sports team or other University activity).Reading: The questions below are primarily based on the material in Chapter 16 of the bookon Information Cascade.(1) (8 points) Consider the balls and bins experiment discussed in class. Each urn has threeballs: one has two red balls and one blue ball; and, the other has two blue balls and one redball. In class we used Bayes’ Rule to compute the probability that the bin is majority blue if thefirst randomly drawn ball is blue. We use B to denote that a blue ball was drawn, and m − Bthat the bin is majority blue. So we know that P r(m − B) = 1/2, and P r(B|m − B) = 2/3 astwo of the three balls are blue if we are drawing from the majority blue bin. So,P r(m − B|B) =P r(B|m − B)P r(m − B)P r(B)=2/3 · 1/21/2= 2/3using the fact that P r(B) = P r(B|m − B)P r(m − B) + P r(B|m − R )P r(m − R) = 2/3 · 1/2 +1/3 · 1/2 = 1/2.(a) Use Bayes’ Rule to compute the probability the bin is majority blue, if the first tworandomly drawn balls are blue. (Recall that we return the ball to the urn after every experi-ment.)(b) Use Bayes’ Rule to compute the probability the bin is majority blue, if the sequence ofthree balls randomly drawn from the bin in this experiment is Blue, Blue, Red.(2) )(10 points) Bayes’ Rule can arise in a wide range of everyday situations. For example,consider the following question that can be answered using Bayes’ Rule.A patient is being tested for a rare disease. What we mean by saying that the disease israre is that only one out of every 1,000 people who get tested for the disease actually have the1disease. The test for this disease is quite accurate. In fact, if a patient has the disease thetest result is positive with probability 0.99; for someone who does not have the disease the testresult is negative with probability 0.99.(a) Use Bayes’ Rule to compute the probability that a randomly selected patient who hasa positive test result actually has the disease.(b) Use Bayes’ Rule to compute the probability that a randomly selected patient who hasa negative test result actually has the disease. Provide a brief explanation of why the resultsin parts (a) and (b) are so different.(c) Another patient has a family history of this disease and it is known that for patientswith this family history one out of every 100 people have the disease. Suppose that this patienttoo gets a positive result on the test for the disease (and that the accuracy rates given abovehold for this population of patients too). What is the probability that this patient has thedisease?(3) (10 points) Let’s consider the model of information cascades. Assume that the proba-bility that the state is Good (G) is p =12, and hence the probability that the state is Bad (B)is also 1 − p =12. Let’s say that the probability of a High signal given a Good state is differentfrom the running example in class; for this question it’s q =23. The probability of a Low signalgiven a Bad state is also q =23. For this problem, unlike the model used in Chapter 16, thepayoffs are random and have the property that the average payoff is positive if the technologyis Good, and negative if the technology is Bad.As in the model used in Chapter 16, each person receives a private signal about the technol-ogy and observes the actions of all those who chose previously. In this problem assume thereis also a wiki where people who accepted the product discuss their experience with the prod-uct. As a result each person now can know the payoffs received by each of those who movedpreviously: it’s 0 for the players who didn’t accept the product, and the value is discussed onthe wiki for those who did accept the product.(a) Suppose that the new technology is actually Bad. How does this new information aboutpayoffs (the payoffs received by each of those who moved previously) affect the potential foran incorrect information cascade of choices to adopt the new technology to form and persist?[You do not need to write a proof. A brief argument is sufficient.](b) Suppose that the new technology is actually Good. How does this new informationabout payoffs (the payoffs received by each of those who moved previously) affect the potentialfor an incorrect information cascade of choices to reject the new technology to form and persist?[You do not need to write a proof. A brief argument is sufficient.](4) (12 points) An interesting issue in analyzing information cascades is to consider thepossibility that some people have better information than others. Let’s consider the model2from Sections 16.5 and 16.6, but change things slightly as follows. First, we’ll choose specificvalues for the quantities p, q, vgand vbused to specify the model: the prior probability of aGood state is p =12, and the probability of a High signal given a Good state (or a Low signalgiven a Bad state) is q =34. We’ll have vg= 1 and vb= −1.Next, there will be only two people making decisions, with one person deciding after theother. Person 1 receives one signal as usual, but Person 2 receives three independent signals,each drawn independently with the probabilities given by the model.The two people will decide in order whether to accept or reject, basing their decision asusual on choosing the alternative that provides the higher expected payoff, given all the signalsthey’ve received together with what they’ve observed about earlier decisions.(a) Suppose that Person 1 decides first, and Person 2 decides second (after observing Person1’s decision but not his signal). How would you expect Person 1 and Person 2 to behave inmaking their respective decisions? Give an explanation for your


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