1. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.1 Table of Contents Chapter 12 (Decision Analysis) Decision Analysis Examples12.2–12.3 A Case Study: The Goferbroke Company Problem (Section 12.1)12.4–12.8 Decision Criteria (Section 12.2)12.9–12.13 Decision Trees (Section 12.3)12.14–12.19 Sensitivity Analysis with Decision Trees (Section 12.4)12.20–12.24 Checking Whether to Obtain More Information (Section 12.5)12.25–12.27 Using New Information to Update the Probabilities (Section 12.6)12.28–12.35 Decision Tree to Analyze a Sequence of Decisions (Section 12.7)12.36–12.39 Sensitivity Analysis with a Sequence of Decisions (Section 12.8)12.40–12.47 Using Utilities to Better Reflect the Values of Payoffs (Section 12.9)12.48–12.64 Introduction to Decision Analysis (UW Lecture)12.65–12.80 These slides are based upon a lecture from the MBA core course in Management Science at the University of Washington (as taught by one of the authors). Sequential Decisions and the Value of Information (UW Lecture)12.81–12.91 These slides are based upon a lecture from the MBA core course in Management Science at the University of Washington (as taught by one of the authors). Risk Attitude and Utility Functions (UW Lecture)12.92–12.102 These slides are based upon a lecture from the MBA core course in Management Science at the University of Washington (as taught by one of the authors).

2. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.2 Decision Analysis Managers often must make decisions in environments that are fraught with uncertainty. Some Examples –A manufacturer introducing a new product into the marketplace What will be the reaction of potential customers? How much should be produced? Should the product be test-marketed? How much advertising is needed? –A financial firm investing in securities Which are the market sectors and individual securities with the best prospects? Where is the economy headed? How about interest rates? How should these factors affect the investment decisions?

3. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.3 Decision Analysis Managers often must make decisions in environments that are fraught with uncertainty. Some Examples –A government contractor bidding on a new contract. What will be the actual costs of the project? Which other companies might be bidding? What are their likely bids? –An agricultural firm selecting the mix of crops and livestock for the season. What will be the weather conditions? Where are prices headed? What will costs be? –An oil company deciding whether to drill for oil in a particular location. How likely is there to be oil in that location? How much? How deep will they need to drill? Should geologists investigate the site further before drilling?

4. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.4 The Goferbroke Company Problem The Goferbroke Company develops oil wells in unproven territory. A consulting geologist has reported that there is a one-in-four chance of oil on a particular tract of land. Drilling for oil on this tract would require an investment of about $100,000. If the tract contains oil, it is estimated that the net revenue generated would be approximately $800,000. Another oil company has offered to purchase the tract of land for $90,000. Question: Should Goferbroke drill for oil or sell the tract?

5. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.5 Prospective Profits Profit Status of LandOilDry Alternative Drill for oil$700,000–$100,000 Sell the land90,000 Chance of status1 in 43 in 4

6. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.6 Decision Analysis Terminology The decision maker is the individual or group responsible for making the decision. The alternatives are the options for the decision to be made. The outcome is affected by random factors outside the control of the decision maker. These random factors determine the situation that will be found when the decision is executed. Each of these possible situations is referred to as a possible state of nature. The decision maker generally will have some information about the relative likelihood of the possible states of nature. These are referred to as the prior probabilities. Each combination of a decision alternative and a state of nature results in some outcome. The payoff is a quantitative measure of the value to the decision maker of the outcome. It is often the monetary value.

7. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.7 Prior Probabilities State of NaturePrior Probability The tract of land contains oil0.25 The tract of land is dry (no oil)0.75

8. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.8 Payoff Table (Profit in $Thousands) State of Nature AlternativeOilDry Drill for oil700–100 Sell the land90 Prior probability0.250.75

9. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.9 The Maximax Criterion The maximax criterion is the decision criterion for the eternal optimist. It focuses only on the best that can happen. Procedure: –Identify the maximum payoff from any state of nature for each alternative. –Find the maximum of these maximum payoffs and choose this alternative. State of Nature AlternativeOilDryMaximum in Row Drill for oil700–100 700  Maximax Sell the land90

10. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.10 The Maximin Criterion The maximin criterion is the decision criterion for the total pessimist. It focuses only on the worst that can happen. Procedure: –Identify the minimum payoff from any state of nature for each alternative. –Find the maximum of these minimum payoffs and choose this alternative. State of Nature AlternativeOilDryMinimum in Row Drill for oil700–100 Sell the land90 90  Maximin

11. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.11 The Maximum Likelihood Criterion The maximum likelihood criterion focuses on the most likely state of nature. Procedure: –Identify the state of nature with the largest prior probability –Choose the decision alternative that has the largest payoff for this state of nature. State of Nature AlternativeOilDry Drill for oil700–100 Sell the land90 90  Step 2: Maximum Prior probability0.250.75  Step 1: Maximum

12. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.12 Bayes’ Decision Rule Bayes’ decision rule directly uses the prior probabilities. Procedure: –For each decision alternative, calculate the weighted average of its payoff by multiplying each payoff by the prior probability and summing these products. This is the expected payoff (EP). –Choose the decision alternative that has the largest expected payoff.

13. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.13 Bayes’ Decision Rule Features of Bayes’ Decision Rule –It accounts for all the states of nature and their probabilities. –The expected payoff can be interpreted as what the average payoff would become if the same situation were repeated many times. Therefore, on average, repeatedly applying Bayes’ decision rule to make decisions will lead to larger payoffs in the long run than any other criterion. Criticisms of Bayes’ Decision Rule –There usually is considerable uncertainty involved in assigning values to the prior probabilities. –Prior probabilities inherently are at least largely subjective in nature, whereas sound decision making should be based on objective data and procedures. –It ignores typical aversion to risk. By focusing on average outcomes, expected (monetary) payoffs ignore the effect that the amount of variability in the possible outcomes should have on decision making.

14. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.14 Decision Trees A decision tree can apply Bayes’ decision rule while displaying and analyzing the problem graphically. A decision tree consists of nodes and branches. –A decision node, represented by a square, indicates a decision to be made. The branches represent the possible decisions. –An event node, represented by a circle, indicates a random event. The branches represent the possible outcomes of the random event.

15. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.15 Decision Tree for Goferbroke

16. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.16 Using TreePlan TreePlan, an Excel add-in developed by Professor Michael Middleton, can be used to construct and analyze decision trees on a spreadsheet. 1.Choose Decision Tree under the Tools menu. 2.Click on New Tree, and it will draw a default tree with a single decision node and two branches, as shown below. 3.The labels in D2 and D7 (originally Decision 1 and Decision 2) can be replaced by more descriptive names (e.g., Drill and Sell).

17. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.17 Using TreePlan 4.To replace a node (such as the terminal node of the drill branch in F3) by a different type of node (e.g., an event node), click on the cell containing the node, choose Decision Tree again from the Tools menu, and select “Change to event node”.

18. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.18 Using TreePlan 5.Enter the correct probabilities in H1 and H6. 6.Enter the partial payoffs for each decision and event in D6, D14, H4, and H9.

19. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.19 TreePlan Results The numbers inside each decision node indicate which branch should be chosen (assuming the branches are numbered consecutively from top to bottom). The numbers to the right of each terminal node is the payoff if that node is reached. The number 100 in cells A10 and E6 is the expected payoff at those stages in the process.

20. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.20 Consolidate the Data and Results

21. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.21 Sensitivity Analysis: Prior Probability of Oil = 0.15

22. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.22 Sensitivity Analysis: Prior Probability of Oil = 0.35

23. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.23 Using Data Tables to Do Sensitivity Analysis

24. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.24 Data Table Results The Effect of Changing the Prior Probability of Oil

25. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.25 Checking Whether to Obtain More Information Might it be worthwhile to spend money for more information to obtain better estimates? A quick way to check is to pretend that it is possible to actually determine the true state of nature (“perfect information”). EP (with perfect information) = Expected payoff if the decision could be made after learning the true state of nature. EP (without perfect information) = Expected payoff from applying Bayes’ decision rule with the original prior probabilities. The expected value of perfect information is then EVPI = EP (with perfect information) – EP (without perfect information).

26. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.26 Expected Payoff with Perfect Information

27. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.27 Expected Payoff with Perfect Information

28. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.28 Using New Information to Update the Probabilities The prior probabilities of the possible states of nature often are quite subjective in nature. They may only be rough estimates. It is frequently possible to do additional testing or surveying (at some expense) to improve these estimates. The improved estimates are called posterior probabilities.

29. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.29 Seismic Survey for Goferbroke Goferbroke can obtain improved estimates of the chance of oil by conducting a detailed seismic survey of the land, at a cost of $30,000. Possible findings from a seismic survey: –FSS: Favorable seismic soundings; oil is fairly likely. –USS: Unfavorable seismic soundings; oil is quite unlikely. P(finding | state) =Probability that the indicated finding will occur, given that the state of nature is the indicated one. P(finding | state) State of NatureFavorable (FSS)Unfavorable (USS) OilP(FSS | Oil) = 0.6P(USS | Oil) = 0.4 DryP(FSS | Dry) = 0.2P(USS | Dry) = 0.8

30. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.30 Calculating Joint Probabilities Each combination of a state of nature and a finding will have a joint probability determined by the following formula: P(state and finding) = P(state) P(finding | state) P(Oil and FSS) = P(Oil) P(FSS | Oil) = (0.25)(0.6) = 0.15. P(Oil and USS) = P(Oil) P(USS | Oil) = (0.25)(0.4) = 0.1. P(Dry and FSS) = P(Dry) P(FSS | Dry) = (0.75)(0.2) = 0.15. P(Dry and USS) = P(Dry) P(USS | Dry) = (0.75)(0.8) = 0.6.

31. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.31 Probabilities of Each Finding Given the joint probabilities of both a particular state of nature and a particular finding, the next step is to use these probabilities to find each probability of just a particular finding, without specifying the state of nature. P(finding) = P(Oil and finding) + P(Dry and finding) P(FSS) = 0.15 + 0.15 = 0.3. P(USS) = 0.1 + 0.6 = 0.7.

32. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.32 Calculating the Posterior Probabilities The posterior probabilities give the probability of a particular state of nature, given a particular finding from the seismic survey. P(state | finding) = P(state and finding) / P(finding) P(Oil | FSS) = 0.15 / 0.3 = 0.5. P(Oil | USS) = 0.1 / 0.7 = 0.14. P(Dry | FSS) = 0.15 / 0.3 = 0.5. P(Dry | USS) = 0.6 / 0.7 = 0.86.

33. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.33 Probability Tree Diagram

34. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.34 Posterior Probabilities P(state | finding) FindingOilDry Favorable (FSS)P(Oil | FSS) = 1/2P(Dry | FSS) = 1/2 Unfavorable (USS)P(Oil | USS) = 1/7P(Dry | USS) = 6/7

35. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.35 Template for Posterior Probabilities

36. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.36 Decision Tree for the Full Goferbroke Co. Problem

37. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.37 Decision Tree with Probabilities and Payoffs

38. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.38 The Final Decision Tree

39. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.39 TreePlan for the Full Goferbroke Co. Problem

40. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.40 Organizing the Spreadsheet for Sensitivity Analysis

41. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.41 The Plot Option of SensIt

42. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.42 SensIt Plot

43. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.43 Optimal Policy Let p = Prior probability of oil Ifp ≤ 0.168, then sell the land (no seismic survey). If 0.169 ≤ p ≤ 0.308, then do the survey; drill if favorable, sell if not. If p ≥ 0.309, then drill for oil (no seismic survey).

44. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.44 The Spider Option of SensIt

45. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.45 SensIt Spider Graph

46. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.46 The Tornado Option of SensIt

47. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.47 SensIt Tornado Diagram

48. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.48 Using Utilities to Better Reflect the Values of Payoffs Thus far, when applying Bayes’ decision rule, we have assumed that the expected payoff in monetary terms is the appropriate measure. In many situations, this is inappropriate. Suppose an individual is offered the following choice: –Accept a 50-50 chance of winning $100,000. –Receive $40,000 with certainty. Many would pick $40,000, even though the expected payoff on the 50-50 chance of winning $100,000 is $50,000. This is because of risk aversion. A utility function for money is a way of transforming monetary values to an appropriate scale that reflects a decision maker’s preferences (e.g., aversion to risk).

49. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.49 A Typical Utility Function for Money

50. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.50 Shape of Utility Functions

51. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.51 Utility Functions When a utility function for money is incorporated into a decision analysis approach, it must be constructed to fit the current preferences and values of the decision maker. Fundamental Property: Under the assumptions of utility theory, the decision maker’s utility function for money has the property that the decision maker is indifferent between two alternatives if the two alternatives have the same expected utility. When the decision maker’s utility function for money is used, Bayes’ decision rule replaces monetary payoffs by the corresponding utilities. The optimal decision (or series of decisions) is the one that maximizes the expected utility.

52. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.52 Illustration of Fundamental Property By the fundamental property, a decision maker with the utility function below- right will be indifferent between each of the three pairs of alternatives below-left. 25% chance of $100,000 $10,000 for sure Both have E(Utility) = 0.25. 50% chance of $100,000 $30,000 for sure Both have E(Utility) = 0.5. 75% chance of $100,000 $60,000 for sure Both have E(Utility) = 0.75.

53. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.53 The Lottery Procedure 1.We are given three possible monetary payoffs—M 1, M 2, M 3 (M 1 < M 2 < M 3 ). The utility is known for two of them, and we wish to find the utility for the third. 2.The decision maker is offered the following two alternatives: a)Obtain a payoff of M 3 with probability p. Obtain a payoff of M 1 with probability (1–p). b)Definitely obtain a payoff of M 2. 3.What value of p makes you indifferent between the two alternatives? 4.Using this value of p, write the fundamental property equation, E(utility for a) = E(utility for b) so p U(M 3 ) + (1–p) U(M 1 ) = U(M 2 ). 5.Solve this equation for the unknown utility.

54. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.54 Procedure for Constructing a Utility Function 1.List all the possible monetary payoffs for the problem, including 0. 2.Set U(0) = 0 and then arbitrarily choose a utility value for one other payoff. 3.Choose three of the payoffs where the utility is known for two of them. 4.Apply the lottery procedure to find the utility for the third payoff. 5.Repeat steps 3 and 4 for as many other payoffs with unknown utilities as desired. 6.Plot the utilities found on a graph of the utility U(M) versus the payoff M. Draw a smooth curve through these points to obtain the utility function.

55. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.55 Generating the Utility Function for Max Flyer The possible monetary payoffs in the Goferbroke Co. problem are –130, –100, 0, 60, 90, 670, and 700 (all in $thousands). Set U(0) = 0. Arbitrarily set U(–130) = –150.

56. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.56 Finding U(700) The known utilities are U(–130) = –150 and U(0) = 0. The unknown utility is U(700). Consider the following two alternatives: a)Obtain a payoff of 700 with probability p. Obtain a payoff of –130 with probability (1–p). b)Definitely obtain a payoff of 0. What value of p makes you indifferent between these two alternatives? Max chooses p = 0.2. By the fundamental property of utility functions, the expected utilities of the two alternatives must be equal, so pU(700) + (1–p)U(–130) = U(0) 0.2U(700) + 0.8(–150) = 0 0.2U(700) – 120 = 0 0.2U(700) = 120 U(700) = 600

57. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.57 Finding U(–100) The known utilities are U(–130) = –150 and U(0) = 0. The unknown utility is U(–100). Consider the following two alternatives: a)Obtain a payoff of 0 with probability p. Obtain a payoff of –130 with probability (1–p). b)Definitely obtain a payoff of –100. What value of p makes you indifferent between these two alternatives? Max chooses p = 0.3. By the fundamental property of utility functions, the expected utilities of the two alternatives must be equal, so pU(0) + (1–p)U(–130) = U(–100) 0.3(0) + 0.7(–150) = U(–100) U(–100) = –105

58. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.58 Finding U(90) The known utilities are U(700) = 600 and U(0) = 0. The unknown utility is U(90). Consider the following two alternatives: a)Obtain a payoff of 700 with probability p. Obtain a payoff of 0 with probability (1–p). b)Definitely obtain a payoff of 90. What value of p makes you indifferent between these two alternatives? Max chooses p = 0.15. By the fundamental property of utility functions, the expected utilities of the two alternatives must be equal, so pU(700) + (1–p)U(0) = U(90) 0.15(600) + 0.85(0) = U(90) U(90) = 90

59. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.59 Max’s Utility Function for Money

60. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.60 Utilities for the Goferbroke Co. Problem Monetary Payoff, MUtility, U(M) –130–150 –100–105 00 60 90 670580 700600

61. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.61 Decision Tree with Utilities

62. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.62 Exponential Utility Function The procedure for constructing U(M) requires making many difficult decisions about probabilities. An alternative approach assumes a certain form for the utility function and adjusts this form to fit the decision maker as closely as possible. A popular form is the exponential utility function U(M) = R (1 – e –M/R ) where R is the decision maker’s risk tolerance. An easy way to estimate R is to pick the value that makes you indifferent between the following two alternatives: a)A 50-50 gamble where you gain R dollars with probability 0.5 and lose R/2 dollars with probability 0.5. b)Neither gain nor lose anything.

63. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.63 Using TreePlan with an Exponential Utility Function Specify the value of R in a cell on the spreadsheet. Give the cell a range name of RT (TreePlan refers to this term as the risk tolerance). Click on the Option button in the TreePlan dialogue box and select the “Use Exponential Utility Function” option.

64. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.64 Decision Tree with an Exponential Utility Function

65. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.65 Decisions Under Certainty State of nature is certain (one state). Select decision that yields highest return (e.g., linear programming, integer programming). Examples: –Product mix –Diet problem –Distribution –Scheduling

66. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.66 Decisions Under Uncertainty (or Risk) State of nature is uncertain (several possible states) Examples –Drilling for oil Uncertainty: Oil found? How much? How deep? Selling Price? Decision: Drill or not? –Developing a new product Uncertainty: R&D Cost, demand, etc. Decisions: Design, quantity, produce or not? –Newsvendor problem Uncertainty: Demand Decision: Stocking levels –Producing a movie Uncertainty: Cost, gross, etc. Decisions: Develop? Arnold or Keanu?

67. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.67 Oil Drilling Problem Consider the problem faced by an oil company that is trying to decide whether to drill an exploratory oil well on a given site. Drilling costs $200,000. If oil is found, it is worth $800,000. If the well is dry, it is worth nothing. State of Nature DecisionWetDry Drill600–200 Do not drill00

68. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.68 Decision Criteria Which decision is best? “Optimist” “Pessimist” “Second–Guesser” “Joe Average” State of Nature DecisionWetDry Drill600–200 Do not drill00

69. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.69 Bayes’ Decision Rule Suppose that the oil company estimates that the probability that the site is “Wet” is 40%. State of Nature DecisionWetDry Drill600–200 Do not drill00 Prior Probability0.40.6 Expected value of payoff (Drill) = (0.4)(600) + (0.6)(–200) = 120 Expected value of payoff (Do not drill) = (0.4)(0) + (0.6)(0) = 0 Bayes’ Decision Rule: Choose the decision that maximizes the expected payoff (Drill).

70. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.70 Features of Bayes’ Decision Rule Accounts not only for the set of outcomes, but also their probabilities. Represents the average monetary outcome if the situation were repeated indefinitely. Can handle complicated situations involving multiple related risks.

71. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.71 Using a Decision Tree to Analyze Oil Drilling Problem Folding Back: At each event node (circle): calculate expected value (SUMPRODUCT of payoffs and probabilities for each branch). At each decision node (square): choose “best” branch (maximum value).

72. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.72 Using TreePlan to Analyze Oil Drilling Problem 1.Choose Decision Tree under the Tools menu. 2.Click on “New Tree” and it will draw a default tree with a single decision node and two branches, as shown below. 3.Label each branch. Replace “Decision 1” with “Drill” (cell D2). Replace “Decision 2” with “Do not drill” (cell D7). 4.To replace the terminal node of the drill branch with an event node, click on the terminal node (cell F3) and then choose Decision Tree under the Tools menu. Click on “Change to event node,” choose two branches, then click OK.

73. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.73 Using TreePlan to Analyze Oil Drilling Problem 5.Change the labels “Event 3” and “Event 4” to “Wet” and “Dry”, respectively. 6.Change the default probabilities (cells H1 and H6) from 0.5 and 0.5 to the correct values of 0.4 and 0.6. 7.Enter the partial payoffs under each branch: (-200) for “Drill” (D6), 0 for “Do not drill” (D14), 800 for “Wet” (H4), and 0 for “Dry” (H9). The terminal value cash flows are calculated automatically from the partial cash flows.

74. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.74 Final Decision Tree

75. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.75 Features of TreePlan Terminal values (payoff) are calculated automatically from the partial payoffs (K3 = D6+H4, K8 = D6+H9, K13 = D14). Foldback values are calculated automatically (I4 = K3, I9 = K8, E6 = H1*I4 + H6*I9, E14 = K13, A10 = Max(E6,E14)). Optimal decisions are indicated inside decision node squares (labeled by branch number from top to bottom, e.g., branch #1 = Drill, branch #2 = Do not drill). Changes in the tree can be made by clicking on a node and choosing Decision Tree under the Tools menu (change type of node, # of branches, etc.) Clicking “Options…” in the Decision Tree dialogue box allows the choice of Maximize Profit or Minimize Cost.

76. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.76 Making Sequential Decisions Consider a pharmaceutical company that is considering developing an anticlotting drug. They are considering two approaches –A biochemical approach (more likely to be successful) –A biogenetic approach (more radical) While the biogenetic approach is not nearly as likely to succeed, if would likely capture a much larger portion of the market if it did. R&D ChoiceInvestmentOutcomes Profit (excluding R&D)Probability Biochemical$10 millionLarge success Small success $90 million $50 million 0.7 0.3 Biogenetic$20 millionSuccess Failure $200 million $0 million 0.2 0.8

77. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.77 Biochemical vs. Biogenetic

78. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.78 Simultaneous Development

79. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.79 Biochemical First

80. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.80 Biogenetic First

81. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.81 Incorporating New Information Often, a preliminary study can be done to better determine the true state of nature. Examples: –Market surveys –Test marketing –Seismic testing (for oil) Question: What is the value of this information?

82. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.82 Oil Drilling Problem Consider again the problem faced by an oil company that is trying to decide whether to drill an exploratory oil well on a given site. Drilling costs $200,000. If oil is found, it is worth $800,000. If the well is dry, it is worth nothing. The prior probability that the site is wet is estimated at 40%. State of Nature DecisionWetDry Drill600–200 Do not drill00 Prior Probability0.40.6 Expected Payoff (Drill) = (0.4)(600) + (0.6)(–200) = 120 Expected Payoff (Do not drill) = (0.4)(0) + (0.6)(0) = 0

83. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.83 Expected Value of Perfect Information (EVPI) State of Nature DecisionWetDry Drill600–200 Do not drill00 Prior Probability0.40.6 Suppose they had a test that could predict ahead of time whether the side would be wet or dry. Expected Payoff = (0.4)(600) + (0.6)(0) = 240 Expected Value of Perfect Information (EVPI) = Expected Payoff (with perfect info) – Expected Payoff (without info) = 240 – 120 = 120

84. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.84 Using TreePlan to Calculate EVPI

85. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.85 Imperfect Information (Seismic Test) Suppose a seismic test is available that would better (but not perfectly) indicate whether or not the site was wet or dry. –Good result usually means the site is wet (but not always) –Bad results usually means the site is dry (but not always) Record of 100 Past Seismic Test Sites Seismic Result Actual State of Nature Wet (W)Dry (D)Total Good (G)302050 Bad (B)104050 Total4060100

86. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.86 Decision Tree with Seismic Test

87. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.87 Conditional Probabilities Need probabilities of each test result: –P(G) = 50 / 100 = 0.5 –P(B) = 50 / 100 = 0.5 Need conditional probabilities of each state of nature, given a test result: –P(W | G) = 30 / 50 = 0.6 –P(D | G) = 20 / 50 = 0.4 –P(W | B) = 10 / 50 = 0.2 –P(D | B) = 40 / 50 = 0.8 Seismic Result Actual State of Nature Wet (W)Dry (D)Total Good (G)302050 Bad (B)104050 Total4060100

88. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.88 Expected Value of Sample Information (EVSI) Expected Value of Sample Information = EVSI = 140 – 120 = 20.

89. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.89 Revising Probabilities Suppose they don’t have the “Record of Past 100 Seismic Test Sites”. Vendor of test certifies: –Wet sites test “good” three quarters of the time. –Dry sites test “bad” two thirds of the time P(G | W) = 3 / 4 P(B | W) = 1 / 4 P(B | D) = 2 / 3 P(G | D) = 1 / 3 Is this the information needed in the decision tree?

90. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.90 Revising Probabilities (Probability Tree Diagram)

91. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.91 Template for Posterior Probabilities Template available on textbook CD.

92. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.92 Risk Attitude Consider the following coin-toss gambles. How much would you sell each of these gambles for? Heads:You win $200 Tails: You lose $0 Heads:You win $300 Tails:You lose $100 Heads:You win $20,000 Tails:You lose $0 Heads:You win $30,000 Tails:You lose $10,000

93. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.93 Demand for Insurance House Value = $150,000 Insurance Premium = $500 Probability of fire destroying house (in one year) = 1 / 1,000 Question: Should you buy insurance?

94. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.94 Utilities and Risk Aversion PayoffUtility $600,0001.0 200,0000.75 00.50 –120,0000.25 –200,0000

95. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.95 Oil Drilling Problem (Risk Aversion) Risk Neutral:Risk Averse:

96. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.96 Creating a Utility Function (Equivalent Lottery Method) 1.Set U(Min) = 0. 2.Set U(Max) = 1. 3.To find U(x): Choose p such that you are indifferent between the following: a)A payment of x for sure. b)A payment of Max with probability p and a payment of Min with probability 1–p. 4.U(x) = p.

97. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.97 Equivalent Lottery Method Uncertain situation:–$200 in worst caseU(–$200) = 0 $1,800 in best caseU($1,800) = 1 U($800) = U($200) = U($400) = U($600) =

98. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.98 Utility Curve

99. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.99 Biochemical vs. Biogenetic First (Expected Payoff)

100. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.100 Biochemical vs. Biogenetic First (with Utilities)

101. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.101 Exponential Utility Function Choose R so that you are indifferent between the following: U(M) = R(1 – e –M / R )

102. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.102 Exponential Utility Function U(M) = R(1 – e –M / R )

103. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.103 Using an Exponential Utility Function with TreePlan To use an exponential utility function in TreePlan, enter the R value in a cell on the spreadsheet Give this cell the range name RT (TreePlan calls this value the risk tolerance). Choose “Use Exponential Utility Function” in the dialogue box shown below (available by clicking on “Options…” in the Decision Tree dialogue box).

104. McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2003 12.104 Biochemical vs. Biogenetic First (with Exponential Utility)