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Exam 16: Decision Analysis

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Question 1

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Consider the following decision tree,where the probabilities associated with the branches emanating from each event node are shown in parentheses. The dollar amount given next to each branch is the cash flow generated along that branch,where these intermediate cash flows add up to the total net cash flow shown to the right of each terminal branch.(The unknown amount for the top branch is represented by the variable x.)The decision maker has a utility function u(y)= where y is the total net cash flow after a terminal branch.The resulting utilities for the various terminal branches are shown to the right of the decision tree.Use these utilities to analyze the decision tree.Then determine the value of x for which the decision maker is indifferent between decision alternatives A₁ and A₂.

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The decision tree below shows the result...

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An individual makes decisions according to Bayes' decision rule.For her current problem,she has constructed the following payoff table,and she now wishes to maximize the expected payoff. The value of x currently is 50,but there is an opportunity to increase x by spending some money now.What is the maximum amount that should be spent to increase x to 75?

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For a₁: E[p(a₁,\(\theta\))] = 0.4(2x)+ 0.2(50)+0.4(10)= 14 + 0.8x For a₂: E[p(a₂,\(\theta\))] = 0.4(25)+ 0.2(40)+0.4(90)= 54 For a₃: E[p(a₃,\(\theta\))] = 0.4(35)+ 0.2(3x)+0.4(30)= 26 + 0.6x At x = 50,the optimal decision alternative is a_3,with expected payoff 56.At x = 75,the optimal decision alternative is a_1,with expected payoff 74.The minimum amount that should be spent to increase x to 75 is (74 - 56)= 18.

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You are given the opportunity to guess whether a coin is fair or two-headed,where the prior probabilities are 0.5 for each of these possibilities.If you are correct,you win $5; otherwise,you lose $5.You are also given the option of seeing a demonstration flip of the coin before making your guess.You wish to use Bayes' decision rule to maximize expected profit.(a)Develop a decision analysis formulation of this problem by identifying the decision alternatives,states of nature,and payoff table.(b)What is the optimal decision alternative,given that you decline the option of seeing a demonstration flip? (c)Find EVPI.d)Calculate the posterior distribution if the demonstration flip is a tail.Do the same if the flip is a head. (e)Determine your optimal policy.(f)Now suppose that you must pay to see the demonstration flip.What is the most that you should be willing to pay? (g)Draw the complete decision tree for this problem and label all the branches.Write the cash flow incurred at each branch or terminal node underneath that branch or node.Write the probability associated with each branch emanating from an event node above that branch.At each decision node,insert a 1 or 2 inside the node to indicate whether the optimal decision there is to choose the upper branch (indicated by a 1)or to choose the lower branch (indicated by a 2).Next to each decision node and event node,write the expected payoff at that node if the optimal policy is being followed.

Meredith Delgado owns a small firm that has developed software for organizing and playing music on a computer.Her software contains a number of unique features that she has patented so her company's future has looked bright.However,there now has been an ominous development.It appears that a number of her patented features were copied in similar software developed by MusicMan Software,a huge software company with annual sales revenue in excess of $1 billion.Meredith is distressed.MusicMan Software has stolen her ideas and that company's marketing power is likely to enable it to capture the market and drive Meredith out of business.In response,Meredith has sued MusicMan Software for patent infringement.With attorney fees and other expenses,the cost of going to trial (win or lose)is expected to be $1 million.She feels that she has a 60% chance of winning the case,in which case she would receive $5 million in damages.If she loses the case,she gets nothing.Moreover,if she loses the case,there is a 50% chance that the judge would also order Meredith to pay for court expenses and lawyer fees for MusicMan (an additional $1 million cost).Music Man Software has offered Meredith $1.5 million to settle this case out of court.(a)Construct and use a decision tree to determine whether Meredith should go to court or accept the settlement offer,assuming she wants to maximize her expected payoff.(b)To implement the equivalent lottery method to determine appropriate utility values for all the possible payoffs in this problem,what questions would need to be asked of Meredith? (c)Suppose that Meredith's attitude toward risk is such that she would be indifferent between doing nothing and a gamble where she would win $1 million with 50% probability and lose $500 thousand with 50% probability.Use the exponential utility function to re-solve the decision tree from part a.

The General Ford Motors Corporation (GFMC)is planning the introduction of a brand new SUV-the Vector.There are two options for production.One is to build the Vector at the company's existing plant in Indiana,sharing production time with its line of minivans that are currently being produced there.If sales of the Vector are just moderate,this will work out well as there is sufficient capacity to produce both types of vehicles at the same plant.However,if sales of the Vector are strong,this option would require the operation of a third shift,which would lead to significantly higher costs.A second option is to open a new plant in Georgia.This plant would have sufficient capacity to meet even the largest projections for sales of the Vector.However,if sales are only moderate,the plant would be underutilized and therefore less efficient.This is a new design,so sales are hard to predict.However,GFMC predicts that there would be about a 60% chance of strong sales (annual sales of 100,000),and a 40% chance of moderate sales (annual sales of 50,000).The average revenue per Vector sold is $30,000.Production costs per vehicle for the two production options depend upon sales,as indicated in the table below. The amortized annual cost of plant construction and other associated fixed costs for the Georgia plant would total $400 million per year (regardless of sales volume).The fixed costs for adding Vector production to the plant in Indiana would total $200 million per year (regardless of sales volume).(a)Construct a decision tree to determine which production option maximizes the expected annual profit,considering fixed costs,production costs,and sales revenues.(b)Due to the uncertainty in expected sales for the Vector,GFMC is considering conducting a marketing survey to determine customer attitudes toward the Vector and better predict the likelihood of strong sales.The marketing survey would give one of two results-a positive attitude or a negative attitude toward the design.GFMC has used this marketing survey for other vehicles.For vehicles that eventually had strong sales,the marketing survey indicated positive attitudes toward the design 70% of the time and negative attitudes 30% of the time.For vehicles that eventually had moderate sales,the marketing survey indicated positive attitudes toward the design 20% of the time and negative attitudes 80% of the time.Assuming GFMC conducts such a survey,construct a decision tree to determine how the company should proceed and what the expected annual profit would be (ignoring the cost of the survey).(c)What is the expected value of experimentation (EVE)in part b? What does this say about how large the cost of the marketing survey can be before it would no longer be worthwhile to conduct the survey?