1. Decision Analysis Anderson, Sweeney and Williams Chapter 4 Read: Sections 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, and appendix 4.1

2. Chapter 4: Decision Analysis Problem Formulation Decision Making without Probabilities Decision Making with Probabilities Risk Analysis and Sensitivity Analysis Decision Analysis with Sample Information Computing Posterior Probabilities

3. Decision Analysis A quantitative technique to use in deciding what course of action is likely to be "best" (most profitable, least costly, least risky,...)

4. Model Formulation A decision problem is characterized by decision alternatives, states of nature, and resulting payoffs. The decision alternatives are the different possible strategies the decision maker can employ. (denoted d i )

5. Model Formulation The states of nature refer to future events, not under the control of the decision maker, which may occur. States of nature should be defined so that they are mutually exclusive and collectively exhaustive. (denoted s j )

6. Model Formulation The consequence resulting from a specific combination of a decision alternative and a state of nature is a payoff. (denoted v ij ) Payoffs can be expressed in terms of profit, cost, time, distance or any other appropriate measure.

7. Model Formulation A table showing payoffs for all combinations of decision alternatives and states of nature is a payoff table. A decision tree is a chronological representation of the decision problem.

8. Example: Optimal Drilling Optimal Drilling wishes to decide what to do with a certain piece of property to which it owns mineral rights. Optimal will either Drill a gas well Unconditionally lease the mineral rights, or Conditionally lease the mineral rights.

9. Example: Optimal Drilling An unconditional lease means that Optimal will lease the property rights to a third party for a fixed fee ($5,000,000). A conditional lease means that Optimal will lease the property rights to a third party for a portion of the proceeds from any gas recovered on the property. ($2.25 per mcf, but only if at least 3,500,000 mcf are recovered) If Optimal drills, the estimated cost for a producing well is $6,000,000. The estimated cost for a “dry hole” is $5,000,000. Revenues are $4.50 per mcf of gas.

10. Example: Optimal Drilling Optimal’s forecasters have determined that there are four likely outcomes. 1. About 5,000,000 mcf of recoverable gas are on the property. 2. About 3,500,000 mcf of recoverable gas are on the property. 3. About 2,500,000 mcf of recoverable gas are on the property. 4. No recoverable gas is on the property.

11. What are the decision alternatives? What are the states of nature? Confirm that the following payoff table is correct  Example: Optimal Drilling

12. Payoff Table MCF of gas  Decision ↓ 5 million (s1) 3.5 million (s2) 2.5 million (s3) 0 (s4) Drill (d1) $16.5 million $9.75 million $5.25 million $-5 million Uncond. Lease (d2) $.5 million Cond. Lease (d3) $11.25 million $7.875 million $0

13. Decision Making without Probabilities Three commonly used criteria for decision making when probability information regarding the likelihood of the states of nature is unavailable are: the optimistic approach the conservative approach the minimax regret approach.

14. Optimistic Approach The optimistic approach would be used by an optimistic decision maker. The decision with the largest possible payoff is chosen.

15. Payoff Table MCF of gas  Decision ↓ 5 million (s1) 3.5 million (s2) 2.5 million (s3) 0 (s4) Drill (d1) $16.5 million $9.75 million $5.25 million $-5 million Uncond. Lease (d2) $.5 million Cond. Lease (d3) $11.25 million $7.875 million $0

16. Conservative Approach The conservative approach would be used by a conservative decision maker. For each decision the minimum payoff is listed and then the decision corresponding to the maximum of these minimum payoffs is selected. (Hence, the minimum possible payoff is maximized.)

17. Payoff Table MCF of gas  Decision ↓ 5 million (s1) 3.5 million (s2) 2.5 million (s3) 0 (s4) Drill (d1) $16.5 million $9.75 million $5.25 million $-5 million Uncond. Lease (d2) $.5 million Cond. Lease (d3) $11.25 million $7.875 million $0

18. Minimax Regret Approach The minimax regret approach requires the construction of a regret table or an opportunity loss table. This is done by calculating, for each state of nature, the difference between each payoff and the largest payoff for that state of nature.

19. Minimax Regret Approach Then, using this regret table, the maximum regret for each possible decision is listed. The decision chosen is the one corresponding to the minimum of the maximum regrets.

20. Payoff Table MCF of gas  Decision ↓ 5 million (s1) 3.5 million (s2) 2.5 million (s3) 0 (s4) Drill (d1) $16.5 million $9.75 million $5.25 million $-5 million Uncond. Lease (d2) $.5 million Cond. Lease (d3) $11.25 million $7.875 million $0

21. Regret Table MCF of gas  Decision ↓ 500,000 (s1) 200,000 (s2) 50,000 (s3) 0 (s4) Drill (d1) Uncond. Lease (d2) Cond. Lease (d3)

22. Homework status Before the next class, you should complete the following homework problems in chapter 4: 1 and 2

23. Question Which of these three decision strategies is Mark Cuban, owner of the Dallas Mavericks, employing in the following quote?

25. Decision Making with Probabilities Expected Value Approach If probabilistic information regarding the states of nature is available, one may use the expected value (EV) approach.

26. Decision Making with Probabilities Expected Value Approach (EV) Here the expected return for each decision is calculated by computing a weighted average of the possible payoffs under each state of nature. The decision yielding the best expected return is chosen. This is sometimes called the Expected Monetary Value (EMV) approach.

27. The expected value (EV) of decision alternative d i is defined as: where: N = the number of states of nature P(s j ) = the probability of state of nature s j V ij = the payoff corresponding to decision alternative d i and state of nature s j Expected Value of a Decision Alternative

28. Expected Value Approach Suppose we have studied historical data about the amount of recoverable gas on 100 similar properties. The number of times each outcome occurred is shown below: S1: 10 S2: 15 S3: 35 S4: 40

29. Expected Value Approach We can use these frequencies to estimate the probability of each state occurring as follows: P(S1) =.10 P(S2) =.15 P(S3) =.35 P(S4) =.40 Calculate the expected value for each decision.

30. Payoff Table MCF of gas  Probability Decision ↓ 5 million (s1).1 3.5 million (s2).15 2.5 million (s3).35 0 (s4).40 Drill (d1) $16.5 million $9.75 million $5.25 million $-5 million Uncond. Lease (d2) $.5 million Cond. Lease (d3) $11.25 million $7.875 million $0

31. Expected Value Approach For decision 1 (Drill), the expected value of the payoff is: EV(d1) =.1(16,500,000) +.15(9,750,000) +.35(5,250,000) +.40(-5,000,000) = 1,650,000+1,462,500+1,837,500 – 2,000,000 = 2,950,000

32. Expected Value Approach EV(d2) = EV(d3) =

33. Expected Value Approach So, we choose D1 (drill), since it his the highest EV.

34. Homework status Before the next class, you should complete the following homework problems in chapter 4: 5, 9a, 9b, 9c, 9d

35. Decision Tree Often a decision tree is used to help organize the calculations, especially in complex decision problems The decision tree on the next slide illustrates.

36. Decision Tree Create the decision tree using the steps on the following slides.

37. Decision Tree Layout decision tree (left to right) in chronological order. Use squares to represent decision nodes Use circles to represent state-of-nature nodes At decision (square) nodes, create and label one branch to the right for each d i

38. Decision Tree At state-of-nature (circle) nodes, create and label one branch to the right for each s j List payoff, V ij, associated with each endpoint Note probabilities on branches to the right of state-of-nature nodes.

39. Decision Tree (Optional: Arbitrarily label the nodes for reference.) Then you can make expected value calculations for each state-of-nature node. These calculations are made, by working from right to left across the tree. Expected value at a decision node is the value of the best decision available at that node.

40. TreePlan TreePlan is an Excel Add-In which facilitates calculations on decision trees. It is available for download from Blackboard (under Tools).

41. The Value of Information What is it worth (to the decision maker) to have more information about which state will occur? Tack Tossing Game (or) HIV Infection Example

42. Expected Value of Perfect Information Frequently information is available which can improve the probability estimates for the states of nature. The expected value of perfect information (EVPI) is the increase in the expected profit that would result if one knew with certainty which state of nature would occur.

43. Expected Value of Perfect Information The EVPI provides an upper bound on the expected value of any sample or survey information. What would it be worth to KNOW which state would occur before making a decision?

44. Expected Value of Perfect Information Step 1: Determine the optimal decision for each state of nature. (This tells the decision strategy which uses perfect information.) Step 2: Compute the expected value of this decision strategy which uses perfect information. Step 3: Subtract the EV of the optimal decision (i.e. the one found previously) from the amount determined in step (2).

45. 1. If s1 will occur, choose _____. If s2 will occur, choose _____. If s3 will occur, choose _____. If s4 will occur, choose _____. 2. EV of this strategy is ______. 3. EVPI= _____. Expected Value of Perfect Information

46. Homework status Before the next class, you should complete the following homework problems in chapter 4: 14, 21a

47. Risk Analysis Risk analysis helps the decision maker recognize the difference between: the expected value of a decision alternative, and the payoff that might actually occur The risk profile for a decision alternative shows the possible payoffs for the decision alternative along with their associated probabilities.

48. Risk Analysis Risk Profile for D1: Drill (presented in a tabular form) PayoffProbability $-5,000,000.40 $5,250,000.35 $9,750,000.15 $16,500,000.10

49. Risk Analysis Risk Profile for D1: Drill (presented in a graphical form) Probability Payoff 1.6.4.2.8 $-5 mil $9.75 mil $16.5 mil$5.25 mil

50. Risk Analysis Suppose probabilities had been: 0, 0,.22/41, 19/41 Note that D1 (Drill) would still be optimal (So would D2 (Uncond. Lease)—It’s a tie.) Consider the Risk Profile for D2: Uncond. Lease

51. Risk Analysis for D2 (with modified probabilities) Probability Payoff 1.6.4.2.8 $500,000

52. Risk Analysis for D1 (with modified probabilities Probability Payoff 1.6.4.2.8 $-5 mil$5.25 mil

53. Homework status Before the next class, you should complete the following homework problems in chapter 4: 7

54. Sensitivity Analysis Sensitivity analysis can be used to determine how changes to the following inputs affect the recommended decision alternative: probabilities for the states of nature values of the payoffs

55. Sensitivity Analysis (for probabilities) Two approaches: 1. Trial and Error 2. Solve for the “breakpoint” (can do this when there are 2 states of nature)

56. Payoff Table MCF of gas  Probability Decision ↓ 5 million (s1).1 3.5 million (s2).15 2.5 million (s3) p 0 (s4) 1-p Drill (d1) $16.5 million $9.75 million $5.25 million $-5 million Uncond. Lease (d2) $.5 million Cond. Lease (d3) $11.25 million $7.875 million $0

57. Sensitivity Analysis (for probabilities) Suppose the only states are S3 and S4 Let p represent the probability of S3 Then: EV(D1) = p(5,250,000) + (1-p)(-5,000,000) = 10,250,000p - 5,000,000 EV(D2) = p(500,000) + (1-p)(500,000) = 500,000 EV(D3) = p(0) + (1-p)(0) = 0 For low values of p, which decision is best?

58. Sensitivity Analysis (for probabilities) We can graph EV(D1), EV(D2), and EV(D3) as functions of p:

59. Sensitivity Analysis (for probabilities) p EV EV(D1) EV(D3) EV(D2) 1.8.6.4.2

60. Sensitivity Analysis (for probabilities) For what values of p is D2 the optimal decision? Set EV(D2) = EV(D1) to find the high end of the interval Verify that D2 is best when 0 ≤ p ≤ about.53658

61. Sensitivity Analysis (for payoffs) Similar calculations may be made to perform sensitivity analysis for the payoff values. Example (again assuming there are only 2 states): Perform sensitivity analysis on the State 3 payoff for D1, assuming P(S3) =.4 (so P(S4) must =.6) That is, assuming all other payoff values remain unchanged, what is the optimal decision for various values of the payoff, V(D2,S1)? Verify (on your own) that $8,750,000 is the value at which the best decision changes from D2 to D1.

62. Payoff Table MCF of gas  Probability Decision ↓ 5 million (s1).1 3.5 million (s2).15 2.5 million (s3).4 0 (s4).6 Drill (d1) $16.5 million $9.75 million $V million$-5 million Uncond. Lease (d2) $.5 million Cond. Lease (d3) $11.25 million $7.875 million $0

63. Homework status Before the next class, you should complete the following homework problems in chapter 4: 9e, 11

64. Bayes’ Theorem and Posterior Probabilities Knowledge of sample (survey) information can often be used to revise the probability estimates for the states of nature. Prior to obtaining this information, the probability estimates for the states of nature are called prior probabilities.

65. Bayes’ Theorem and Posterior Probabilities With knowledge of conditional probabilities for the outcomes or indicators of the sample or survey information, these prior probabilities can be revised by employing Bayes' Theorem. The outcomes of this analysis are called posterior probabilities (or branch probabilities) for decision trees.

66. Optimal has the option of hiring a geological consulting company to perform a seismic study of the property. This study would provide more accurate information about the structure of the shale beneath the property, and might allow Optimal to make better estimates about the likelihood of the states. What is the most Optimal should be willing to pay for such a seismic study?. Sample Information

67. The specific geological consulting firm Optimal is considering will return one of 4 indicators about the structure of the shale beneath the property: I1 indicates a very favorable formation I2 indicates a favorable formation I3 indicates an unfavorable formation I4 indicates a highly unfavorable formation Sample Information

68. The geological consulting firm has performed 100 similar studies, and reports the following track record: Sample Information When this state occurred: The seismic study gave this indicator this many times: Total I1I2I3I4 S1741012 S2932216 S31163424 S4913151148

69. Based on these frequencies, we can infer that: P(I1 | S1) = 7/12 P(I1 | S2) = 9/16 P(I1 | S3) = 11/24 Etc. These conditional probabilities are called “The Reliability of the Indicators” Should Optimal purchase the seismic study? Sample Information

70. Computing Posterior Probabilities Step 1: Label each possible outcome of the survey or “indicator” I k. For each state of nature, multiply the prior probability, P(S j ), by the “conditional probability” (“reliability of the indicator”)—this gives the joint probabilitiy for the states and indicator = P(S j and I k )

71. Step 2: Step 2: Sum these joint probabilities over all states -- this gives the marginal probability for the indicator, P(. Sum these joint probabilities over all states -- this gives the marginal probability for the indicator, P(I k ). P(I k ) =  j P(I k | S j ) P(S j ) Computing Posterior Probabilities

72. Step 3: Step 3: For each state, divide its joint probability by the marginal probability for the indicator -- this gives the posterior probability, P(S j | k ). For each state, divide its joint probability by the marginal probability for the indicator -- this gives the posterior probability, P(S j | I k ). P(S j  I k ) =[P(I k |S j ) P(S j )] / P(I k ) Computing Posterior Probabilities

73. I1: Very Favorable Formation State Prior Conditional Joint Posterior s1.10 7/12.0583.1543 s2.15 9/16.0844.2231 s3.35 11/24.1604.4242 s4.40 3/16.0750.1983 Total.3781 1.00000 P(I1) =.3781 Posterior Probabilities

74. We must build a table similar to this one for each indicator. (Verify your posterior probability calculations by looking at the calculations in the Optimal Drilling Excel model on Blackboard.) Using these posterior probabilities, we may compute the expected value of the seismic survey (EVSI).

75. Expected Value of Sample Information The expected value of sample information (EVSI) is the additional expected profit possible through knowledge of the sample (or survey) information.

76. Expected Value of Sample Information Step 1: Determine the optimal decision strategy for the possible outcomes of the sample using the posterior probabilities for the states of nature.

77. Expected Value of Sample Information Step 2: Compute the expected value of that optimal decision strategy. (Remember to work from right to left across the tree.) Step 3: Now subtract the EV of the optimal decision obtained without using the sample information from the amount determined in step (2).

78. Optimal Decision Strategy, using the sample information:  If the outcome of the survey is I1 (very favorable), choose D1: Drill. (EV = $5,956,612)  If the outcome of the survey is I2 (favorable, choose D1: Drill. (EV = $2,883,603)  If the outcome of the survey is I3 (unfavorable), choose D3: Conditional Lease. (EV = $1,232,713)  If the outcome of the survey is I4 (highly unfavorable), choose D3: Conditional Lease. (EV = $875,000) EV of this decision strategy is.378125(5,956,612) +..257292(2,883,603)+.195833(1,232,713) +.16875(875,000) = $3,383,333. Best strategy using Sample Information

79. EVSI EV(Best decision strategy without SI) = $2,950,000. EV(Best decision strategy with SI) = $3,383,333 Difference is $433,333 = EVSI Conclusion: If the seismic survey costs less than $433,333, then buy it; otherwise, don’t.

80. Decision Tree Use the Decision Tree (created on paper or using TreePlan) to determine the Expected Values Remember to work from right to left across the tree.

81. Risk Profile Suppose the survey costs $250,000. Buy the survey. What is the payoff? If I1, D1, S1, Payoff = $16,500,00 If I1, D1, S2, Payoff = $9,750,000 If I1, D1, S3, Payoff = $5,250,000 If I1, D1, S4, Payoff = $-5,000,000 If I2, D1 S1, Payoff = $16,500,000 If I2, D1, S2, Payoff = $9,750,000 If I2 D1, S3, Payoff = $5,250,000 If I2, D1, S4, Payoff = $-5,000,000

82. Risk Profile If I3, D3, S1, Payoff = 11,250,000 If I3, D3, S2, Payoff = 7,875,000 If I3, D3, S3, Payoff = 0 If I3, D3, S4, Payoff = 0 If I4, D3, S1, Payoff = 11,250,000 If I4, D3, S2, Payoff = 7,875,000 If I4, D3, S3, Payoff = 0 If I4, D3, S4, Payoff = 0

83. Risk Profile PayoffProbability $-5,000,000.1833 (=.1983*.3781+.4211*.2573) $0.31875 (=.2234*.19583 +.6383*.19583 +.34568*.16875 +.54321*.16875) $5,250,000.24792 $7,875,000.0375 $9,750,000.1125 $11,250,000.0083 $16,500,000.09167 1.00000 The Risk Profile would look like this (in tabular form):

84. Homework status Before the next class, you should complete the following homework problems in chapter 4: 21b, 21c, 25a, 25b, 25c, 25d, 25e

85. Efficiency of Sample Information Efficiency of sample information is the ratio of EVSI to EVPI. As the EVPI provides an upper bound for the EVSI, efficiency is always a number between 0 and 1.

86. Efficiency of Sample Information The efficiency of the survey: EVSI/EVPI = ($433,333)/($2,200,000) =.197

87. Homework status Before the next class, you should complete the following homework problems in chapter 4: 25f

88. Summary of decision analysis on a tree using sample information 1. Layout decision tree (left to right) in chronological order. 2. List payoff associated with each endpoint 3. List prior probabilities on the part of the tree that doesn't use indicators.

89. Summary of decision analysis on a tree using sample information 4. Compute probabilities for indicator nodes, and put these on tree. 5. Compute posterior probabilities and put them on tree. 6. Determine the optimal decision strategy by computing the expected value at each node, working from right to left across the tree.

90. End of Chapter 4