MS&E 252 Decision Analysis I Handout 29 11/18/2016 Homework Assignment #7 Due on Thursday December 1st 11:59 pm Homework Submission Logistics: You can access the MS&E 252 homework submission site at https://mse252.stanford.edu. Click on \"Assignments\" on the left-hand menu. You may submit homework answers multiple times. Only the last submission will be graded. You are required to turn in answers only to the probabilistic questions. Assigned Reading: 1) Foundations of Decision Analysis, read Chapter 14. 2) Microrisks for Medical Decision Analysis Definitions: From the class lectures, explain the following concepts: Decision Diagram Informational arrow Influence arrow Relevance arrow Functional arrow Page 1 of 11 HW #7 MS&E 252 Decision Analysis I Handout 29 11/18/2016 Probabilistic Questions: (Submit answers on https://mse252.stanford.edu) (All decision makers follow the Five Rules of Actional Thought and prefer more money to less unless stated otherwise. Also, do not worry about time preference) Use the following decision diagram in answering Questions 1 and 2. A B C D E F G H I 1) How many of the following statements must be true? I. Uncertainty A has been observed at the time of Decision H. II. Uncertainties A and F are irrelevant given D and &. III. Uncertainties D and F are irrelevant given B and &. a) b) c) d) 0 1 2 3 2) How many of the following statements must be true? I. The Value of Clairvoyance for Uncertainty F at the time of Decision H is zero. II. The Value of Clairvoyance for Uncertainty A at the time of Decision H is zero. III. The Value of Clairvoyance for Uncertainty B at the time of Decision H is zero. a) b) c) d) 0 1 2 3 Page 2 of 11 HW #7 MS&E 252 Decision Analysis I Handout 29 11/18/2016 3) Sam follow the Delta Property and has a risk tolerance of $50,000. He is looking to purchase insurance for his new Hyundai Elantra which he values at $15,000. The company he goes to, Affordable Insurance, offers a \"name your own coverage\" alternative where for a payment of $0.01X/year Sam receives a payout of $X less his deductible of $1,500. if he gets into an accident. The most coverage he can purchase is $15,000 (i.e. $X$15,000). Sam assigns a probability of 0.01 to getting in an accident this year (regardless of whether or not he buys insurance), and that any accident results in his car being totaled (a cost of $15,000) to him. Let $X* be the value of $X such that Sam would be indifferent between purchasing $X* worth of coverage and not purchasing insurance at all. Which interval contains $X*? a) b) c) d) $0 $X* $3,000 $3,000 < $X* $6,000 $6,000 < $X* $10,000 $10,000 < $X* $15,000 4) (Continued from Question 3) Suppose now that X is fixed at $15000 (Sam can no longer choose his level of coverage) but Sam has the alternative of reducing his deductible to $Y by paying an additional charge of $(1500-Y)/50. Let $Y* be the value of $Y such that Sam is indifferent between purchasing insurance with a deductible equal to $Y* and not purchasing insurance at all. Which interval contains $Y*? a) b) c) d) $0 $Y* $250 $250 < $Y $500 $500 < $Y $1,000 $1,000 < $Y* $1,500 Page 3 of 11 HW #7 MS&E 252 Decision Analysis I Handout 29 11/18/2016 5) Philip is a graduate student working on his own startup idea to rent Rolls-Royces to faculty of elite universities. Philip is in talks with investors and is thinking of his company's future valuation as the following deal: Value Measure 0.2 a1 $5.0M 0.5 a2 $0.3M 0.3 a3 -$0.8M Currently, Philip's personal wealth is $1M and he is indifferent between staying independent or selling his business at a price matching his PISP for the deal above. He will sell for sure for any offer above and will pocket all the proceedings. Wenhao, who graduated two years ago, runs a successful business of renting electric scooters to hipsters in global cities. His company's valuation is estimated at $200M and he is the sole decision maker for mergers and acqusitions. Aishwarya, who is about to graduate, thinks Philip's company is a good acquisition target for Wenhao's business. Moreover, she thinks that Wenhao agrees to Philip's assessment of his company's prospects. At the same time, due to inefficiencies in the due diligence department at Wenhao's company, she thinks she can help secure the deal as an independent deal broker. Aishwarya learns that Philip follows the u-curve () = where x is his personal wealth plus his company's valuation. Similarly, Wenhao has the u-curve for his company valuation in the form of () = , where x is the valuation (which includes the original estimate and whatever gain/loss the company might face in a deal). Let v denote the maximum amount of money (in millions of dollars) Aishwarya thinks she can ever potentially make as net profit from brokering between Philip and Wenhao. Which one of the following ranges contains v? a) b) c) d) v $.03M $0.3M < v 0.5M $0.5M < v $0.8M $0.8M < v Page 4 of 11 HW #7 MS&E 252 Decision Analysis I Handout 29 11/18/2016 6) Chris is facing a decision with four alternatives, namely A, B, C and D, and only one uncertainty, Uncertainty X. Denote the two degrees of Uncertainty X as X1 and X2. The sensitivity plot of the certain equivalents for the four alternatives with respect to {X=X1 | &} is shown below. Chris follows the Five Rules of Actional Thought and prefers more money to less. Please do not assume anything other than the information given. Consider the following statements. I. Alternative D should not be chosen regardless of {X = X1 | &}. II. The graphs in the sensitivity plot above will always be in form of straight lines even if Chris' u-curve changes. III. If we plot the sensitivity plot of the Certain Equivalents of the four alternatives with respect to {X = X1 | &} (change the y-axis to Certain Equivalents), the four graphs on that plot will always remain straight lines. How many of the statements above must be true? a) b) c) d) 0 1 2 3 Page 5 of 11 HW #7 MS&E 252 Decision Analysis I Handout 29 11/18/2016 7) (Continued from Question 6) Chris currently believes that {X = X1 | &} = 0.4. A symmetric binary test on X with accuracy p is available (sensitivity = specificity = p). Suppose that p is always greater than (or equal to) 0.5. Let p* denote the minimum accuracy for this test to be material to Chris. Which of the following ranges contains p*? a) b) c) d) 0.5 p* 0.7 0.7 < p* 0.8 0.8 < p* 0.9 0.9 < p* 8) Nirmit played Mockemon Go all summer and is very impressed with geolocation games. However, he is also worried about this genre being a fad. Recently Nirmit was approached by Pedantic Labs, the leading developer of geolocation games, with a job offer. Nirmit also has an offer from Pear Computer on hand and needs to decide between the two. He is interested in maximizing his earnings over the next ten years, after which he plans to quit the industry and start an organic strawberry farm. After checking his equity package in the offer, Nirmit has estimated his futurediscounted earnings prospects in dollars as follows: Branch Join Pedantic Labs, geolocation games are a hit Join Pedantic Labs, geolocation games are a fad Prospect $3M $1M Currently, Nirmit assigns a probability of 0.6 to geolocation games being a long-term hit. Nirmit follows the Delta Property and thinks he has a risk tolerance of $4M. Let denote the value of the prospect of working for Pear Computer such that Nirmit is indifferent between joining Pedantic Labs and Pear Computer. Which of the following ranges contains ? a) b) c) d) $1.5M $1.5M < $2.5M $2.5M < $3.5M $3.5M < Page 6 of 11 HW #7 MS&E 252 Decision Analysis I Handout 29 11/18/2016 9) (Continued from Question 8) Nirmit decides that he values the prospect of joining Pear Computer at $2.6M. However, while he is certain he follows the Delta Property in the range of the prospects of this decision situation, he is not sure of his exact risk tolerance anymore. Let denote the feasible risk tolerance for Nirmit such that he is just indifferent between his two alternatives. Which of the following ranges contains ? a) b) c) d) $0.8M $0.8M < $0.9M $0.9M < cannot be established. 10) (Continued from Questions 8 and 9) Continue to assume that Nirmit values the prospect of joining Pear Computer at $2.6M. Unfortunately, after doing the class project in MS&E 252, Nirmit has discovered that he is actually not a Delta Person at all. 1 His best guess is now a u-curve of a form () = 1 2 where x is Nirmit's total wealth in millions of dollars. Nirmit's current wealth is just $0.2M. Let VoC denote Nirmit's Value of Clairvoyance on geolocation games being a long-term hit. In what range does VoC lie? a) b) c) d) VoC = $0 $0 < VoC $0.3M $0.3M < VoC $0.5M $0.5M < VoC Page 7 of 11 HW #7 MS&E 252 Decision Analysis I Handout 29 11/18/2016 Quantitative Problems 1) An aircraft company is considering using a new lightweight fastener, called the Superlight, to rivet certain sheet metal parts together. These particular items are not critical to flight safety, and so the decision of whether to use them is a financial one. The weight savings realized by using Superlights instead of the Basic fastener has a value of $100K to the aircraft manufacturer. The Superlight fastener has never been used before on a production aircraft, however, and engineers are wary of its durability. From what little they know of the design and initial lab tests, they have high uncertainty about the number of Superlight fasteners that would need replacement during routine maintenance. They translate this to costs of $300K with probability of 0.3, $120K with probability 0.5, and $75K with probability 0.2. (Label these possible cases High, Moderate, and Low repair cost, for future reference.) These costs include both parts and labor. The alternative to the Superlight is the Basic fastener, which, although heavier than the Superlight, has been used extensively on past aircraft. Stress engineers feel that the Basic has a \"tried and true\" performance record: they conclude from past statistics that the repair cost associated with Basic fastener failure is $125K. Within these dollar ranges, assume that the aircraft company is risk-neutral. a. What is the certain equivalent for the best alternative? b. What is the value of clairvoyance on the repair costs for the Superlights? c. What probability of high repair costs would make the aircraft company indifferent between the two alternatives? When you vary the probability, keep the ratio of the other two probabilities constant. d. What probability of high repair costs leads to the highest value of clairvoyance on repair costs? Again, when you vary the probability, keep the ratio of the other two probabilities constant. The Test Engineering group could run a batch of Superlights through a test program to determine their strength and durability characteristics under simulated flight conditions. Given what the stress group has told him regarding the Superlight, the Test group manager claims that his test program would be of value. He says that for Page 8 of 11 HW #7 MS&E 252 Decision Analysis I Handout 29 11/18/2016 Superlights with High repair cost, the test batch will pass his test with a likelihood of only 30%. If the Superlights have Moderate repair cost, they will pass the test with a probability of 80%, and if they would be Low in repair cost a 90% probability of passing is estimated. The Test group manager cautions that a lot of time and hardware would be needed to conduct the test. e. What is the value of the Test Engineering group's test program? f. Draw the decision diagram for the aircraft company's decision in part a. g. Draw the decision diagram for the aircraft company's decision to buy clairvoyance. h. Draw the decision diagram for the aircraft company's decision to use the Test Engineering group's test program. 2) Recall Kim's party decision. What is the minimum \"accuracy\" that a rain detector would need for Kim to be willing to pay $10 for its use in helping her make her party decision? By \"accuracy\" we mean the probability that the detector indicates sunshine when it will actually be sunny, which equals the probability that the detector indicates rain when it will actually be rainy. 3) Konstantin plans to graduate at the end of the year and enter the working world. He is currently evaluating several job prospects, and his choice has come down to one of two. He will either work for Clairvoyant Consultants, Inc. (CCI) or for the Wizard of Wall Street and Company (WWS & Co.). If he goes to work for CCI, Konstantin will start out as a driver for their famous Field Action Minivan. He has been promised a starting salary of $25,000/year. CCI has a firm rule that everyone must start out as a driver in the Bay Area for three weeks before getting a permanent assignment. The position of driver has long been known to carry a great deal of prestige at CCI, and Konstantin figures that, starting as a driver, he has a 90% chance of being promoted to some better position after only three weeks on the job. Unfortunately, he must sign a one-year contract with CCI before he finds out whether or not he will Page 9 of 11 HW #7 MS&E 252 Decision Analysis I Handout 29 11/18/2016 receive the promotion, and since CCI is in desperate need of drivers, there is still that 10% chance that they will keep him there for the rest of this year. Despite this, Konstantin suspects that there is a 30% chance of being promoted to Associate Clairvoyant. Given that he is promoted, he suspects there is a fifty-fifty chance that he will be sent to either CCI's San Francisco office or their San Jose office. The San Francisco office pays its Associate Clairvoyants $42,000/year while the San Jose office pays its Associate Clairvoyants on a very strange commission schedule. If they bring in a new client, San Jose's Associate Clairvoyants are paid $53,000/year, but if they do not bring in a new client, they are only paid $32,000/year. Konstantin figures there is a 40% chance that, as an Associate Clairvoyant, he could bring in a new client. Having listened closely to the rumor mills, Konstantin discovers that CCI is desperately in need of people to start up a new office in Osaka, Japan. Being fluent in Japanese, he suspects there is a 60% chance that CCI will choose to send him to Japan rather than keeping him in the Bay Area. But once he gets there, he doesn't know whether he will be an Associate Clairvoyant or a Senior Clairvoyant. After consulting with Blair's probability wheel, Konstantin determines that, if he is sent to Osaka, there is a 85% chance that he'll be made a Senior Clairvoyant, and only a 15% chance that he will be an Associate Clairvoyant. Japanese Senior Clairvoyants are paid $95,000/year but Associate Clairvoyants are only paid $35,000/year. The Wizard of Wall Street & Company has offered Konstantin $60,000/year to work in New York City. However, there is no chance of promotion once he is there. Konstantin decides that he would be equally happy working in any of these cities. His only concern is for his salary level at the end of the first year. He doesn't care about anything after the first yearhe plans to join a Tibetan monastery and will relinquish all worldly possessions at that time. After pondering his predicament for several hours in Nuts and Mud, Konstantin comes up with the following information: He wishes to follow the delta property over all the prospects in his problem. He considers a deal for $100,000 or nothing with equal chances to be worth about $38,000. He considers that same deal for $100,000 or nothing, with a 75% probability of getting $100,000 to be worth about $64,000. a. Draw a decision diagram for Konstantin's problem. b. Find Konstantin's best decision. Page 10 of 11 HW #7 MS&E 252 Decision Analysis I Handout 29 11/18/2016 c. How much would WWS need to offer Konstantin for him to be indifferent between the two offers? d. Konstantin decides that he wasn't quite sure of his risk preference. What risk tolerance would make him indifferent between the two offers? e. Konstantin meets secretly with his friend the clairvoyant, who will reveal the outcome of any uncertainty Konstantin chooses. Draw Konstantin's new decision diagram. How much should Konstantin pay for his friend's information? f. Konstantin's friend has changed his mind; she will not allow Konstantin to choose which uncertainty she reveals. Konstantin is uncertain which of his uncertainties his friend will reveal, and assigns equal probabilities to each of his four uncertainties (Driver/Bay Area/Japan, San Jose/San Francisco, Finding a new client in San Francisco, Being promoted to Senior Clairvoyant in Japan). Draw his new decision diagram, and determine his PIBP for his friend's services. Page 11 of 11 HW #7