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Making choices

Making choices. Dr. Yan Liu Department of Biomedical, Industrial & Human Factors Engineering Wright State University. Expected Monetary Value (EMV).

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Making choices

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  1. Making choices Dr. Yan Liu Department of Biomedical, Industrial & Human Factors Engineering Wright State University

  2. Expected Monetary Value (EMV) • One way to choose among risky alternatives is to pick the alternative with the highest expected value (EV). When the objective is measured in monetary values, the expected money value (EMV) is used • EV is the mean of a random variable that has a probability distribution function (Discrete Variable) (Continuous Variable)

  3. Payoff C1 O1 (p1) O2 A1 C2 (1-p1) C3 O3 (p2) A2 O4 C4 (1-p2) EMV(A1)=C1•p1+C2•(1-p1) EMV(A2)=C3•p2+C4•(1-p2)

  4. Solving Decision Trees • Decision Trees are Solved by “Rolling Back” the Trees • Start at the endpoints of the branches on the far right-hand side and move to left • When encountering a chance node, calculate its EV and replace the node with the EV • When encountering a decision node, choose the branch with the highest EV • Continue with the same procedures until a preferred alternative is selected for each decision node

  5. Lottery Ticket Example You have a ticket which will let you participate in a lottery that will pay off $10 with a 45% chance and nothing with a 55% chance. Your friend has a ticket to a different lottery that has a 20% chance of paying $25 and an 80% chance of paying nothing. Your friend has offered to let you have his ticket if you will give him your ticket plus one dollar. Should you agree to trade? Win $24 EMV=$4 (0.2) $25 Trade Ticket EMV(Trade Ticket)=24•0.2+ (-1)•0.8=$4 Lose Ticket Result -$1 (0.8) $0 -$1 Win $10 EMV=$4.5 (0.45) $10 Keep Ticket EMV(Keep Ticket)=100•0.45+ (0)•0.55=$4.5 Lose Ticket Result $0 (0.55) $0 Conclusion: You should keep your ticket !

  6. Product-Switching Example A company needs to decide whether to switch to a new product or not. The product that the company is currently making provides a fixed payoff of $150,000. If the company switches to the new product, its payoff depends on the level of sales. It is estimated that there are about 30% chance of high-level sales ($300,000 payoff), 50% chance of medium-level sales ($100,000 payoff), and 20% chance of low-level sales (losing $100,000). A survey which costs $20,000 can be performed to provide information regarding the sales to be expected. If the survey shows high-level sales, then there are about 60% chance of high-level sales and 40% chance of medium-level sales when the company sells the product. On the other hand, if the survey shows low-level sales, then there are about 60%chance of medium-level sales and 40% chance of low-level sales when the company sells the product.

  7. Old $150,000 $130,000 Survey High High $300,000 (0.6) $280,000 New (0.5) $100,000 (0.4) Medium $80,000 $150,000 Old Perform Survey $130,000 Survey Low Medium $100,000 (0.6) $80,000 (0.5) New -$20,000 Low -$100,000 (0.4) -$120,000 $150,000 Don’t Perform Old $150,000 New $300,000 (0.3) High $300,000 Medium $100,000 (0.5) $100,000 -$100,000 (0.2) Low -$100,000

  8. Old $150,000 $130,000 D3 EMV= $200,000 Survey High High $300,000 (0.6) $280,000 EMV= $165,000 (0.5) New EMV(U3) =0.6•280,000+0.4•80,000=$200,000 $100,000 (0.4) Medium U3 $80,000 Old $150,000 $130,000 D4 Perform Survey U1 Survey Low EMV= $0 Medium $100,000 (0.6) (0.5) $80,000 New EMV(U4) =0.6•80,000+0.4•(-120,000)=$0 -$20,000 Low -$100,000 (0.4) -$120,000 U4 D1 EMV(U1) =0.5•200,000+0.5•130,000=$165,000 D2 $150,000 Don’t Perform Old $150,000 New $300,000 (0.3) High $300,000 $100,000 (0.5) Medium $100,000 -$100,000 (0.2) Low U2 EMV= $120,000 -$100,000 EMV(U2) =0.3•300,000+0.5•(100,000)+0.2•(-100,000)=$120,000 Conclusion: Perform survey. If survey shows high-level sales, then switch the new product ; otherwise, stay with the old product

  9. A1 O1 A3 D1 U1 A2 D2 O2 A4 D1 U1 D2 Decision Path and Strategy • Decision Path • Represents a possible future scenario, starting from the left-most node to the consequence at the end of a branch by selecting one alternative from a decision node and by following one outcome from a chance node. Path 1 ( A1 ) A1 O1 Path 2 ( A2O1 ) Decision Paths: A3 Path 3 ( A2O2A3 ) A2 Path 4 ( A2O2A4 ) O2 A4

  10. A1 O1 A3 D1 U1 A2 D2 O2 A4 D1 U1 D2 Decision Path and Strategy (Cont.) • Decision Strategy • The collection of decision paths connected to one branch of the immediate decision by selecting one alternative from each decision node along that path Strategy 1 (A1): Decision path A1 A1 O1 Strategy 2 (A2A3): Decision paths A2O2A3, A2O1 Decision Strategies: A3 A2 O2 A4 Strategy 3 (A2A4): Decision paths A2O2A4, A2O1

  11. Risk Profiles • Problems with Expected Value (EV) • EV does not indicate all the possible consequences • The statistical interpretation of EV as the average amount obtained by “playing the game” a large number of times is not appropriate in rare cases (e.g. hazards in nuclear power plants) • What is Risk Profile • A graph that shows the probabilities associated with possible consequences given a particular decision strategy • Indicates the relative risk levels of strategies • Steps of Deriving Risk Profiles from Decision Trees • Identify the decision strategies • For each strategy, collapse the decision tree by multiplying out the probabilities on sequential chance branches (Don’t confuse it with solving decision trees!) • Keep track of all possible consequences • Summarize the probability of occurrence for each consequence

  12. (0.2) Win $24 Trade Ticket Lose (0.8) -$1 (0.45) Win $10 Keep Ticket Lose (0.55) $0 Decision Tree of the Lottery Ticket Example Decision strategies: • Trade ticket: • 2) Keep ticket: $24(0.2), -$1(0.8) $10(0.45), $0(0.55) Pr(Payoff) Trade Ticket Keep Ticket Payoff($) Risk Profiles of the Lottery Ticket Example

  13. Old $130,000 Survey High High (0.6) $280,000 New (0.5) Medium (0.4) $80,000 Perform Survey Old $130,000 Survey Low Medium (0.6) $80,000 (0.5) New Low (0.4) -$120,000 Don’t Perform Old $150,000 High (0.3) New $300,000 Medium (0.5) $100,000 Low (0.2) -$100,000 Decision Tree of the Product-Switching Example 1) Don’t perform survey and keep the old product 2) Don’t perform survey and switch to the new product Decision Strategies: 3) Perform survey, and if survey is high then keep the old product 4) Perform survey, and if survey is high then switch to the new product

  14. Probabilities 0.3 0.5 0.2 Payoffs $300,000 $100,000 -$100,000 (0.3) High $300,000 New (0.5) Don’t Perform Medium $100,000 (0.2) Low -$100,000 Survey High Old $130,000 Perform Survey (0.5) Survey Low $130,000 (0.5) (0.6) High Probabilities 0.3 0.5 0.2 Payoffs $280,000 $130,000 $80,000 $280,000 Survey High New (0.4) Medium Perform Survey $80,000 (0.5) Survey Low $130,000 (0.5) Strategy 1): Don’t perform survey and keep the old product $150,000 (100%) Strategy 2): Don’t perform survey and switch to the new product Strategy 3): Perform survey and if survey high then keep the old product $130,000 (100%) Strategy 4): Perform survey and if survey high then switch to the new product

  15. Pr(Payoff) Payoff($) Strategy 1 Strategy 3 Strategy 2 Strategy 4 Risk Profiles of the Product-Switch Example

  16. Pr(Payoff≤x) Trade Ticket Keep Ticket Payoff($) Cumulative Risk Profiles • A graph that shows the cumulative probabilities associated with possible consequences given a particular decision strategy Cumulative Risk Profiles of the Lottery Ticket Example

  17. Dominance • Deterministic Dominance • If the worst payoff of strategy B is at least as good as that of the best payoff of strategy A, then strategy B deterministically dominates strategy A • May also be concluded by drawing cumulative risk profiles Pr(Payoff ≤ x) Draw a vertical line at the place where strategy B first leaves 0. If the vertical line corresponds to 100% for strategy A, then B deterministically dominates A. strategy B strategy A Payoff

  18. Pr(Payoff ≤ x) strategy B strategy A Payoff Dominance (Cont.) • Stochastic Dominance • If for any x, Pr(Payoff ≤ x|strategy B) ≤ Pr(Payoff ≤ x|strategy A), then B stochastically dominates A There is no crossing between the cumulative risk profiles of A and B, and the cumulative risk profile of B is located at the lower-right to that of A

  19. Making Decisions with Multiple Objectives Summer Job Example Sam has two job offers in hand. One job is to work as an assistant at a local small business. The job would pay a minimum wage ($5.25 per hour), require 30 to 40 hours per week, and have the weekends free. The job would last for three months, but the exact amount of work and hence the amount Sam could earn were uncertain. On the other hand, he could spend weekends with friends. The other job is to work for a conservation organization. This job would require 10 weeks of hard work and 40 hours weeks at $6.50 per hour in a national forest in a neighboring state. This job would involve extensive camping and backpacking. Members of the maintenance crew would come from a large geographic area and spend the entire 10 weeks together, including weekends. Sam has no doubts about the earnings of this job, but the nature of the crew and the leaders could make for 10 weeks of a wonderful time, 10 weeks of misery, or anything in between.

  20. Decision Elements • Earning money (measured in $) • Objectives (and Measures) • Having fun (measured using a constructed 5-point Likert scale; Table 4.5 at page 138) (5) Best: A large congenial group. Many new friendships made. Work is enjoyable, and time passes quickly. (4) Good: A small but congenial group of friends. The work is interesting, and time off work is spent with a few friends in enjoyable pursuits. (3) moderate: No new friends are made. Leisure hours are spent with a few friends doing typical activities. Pay is viewed as fair for the work done. (2) Bad: Work is difficult. Coworkers complain about the low pay and poor conditions. On some weekends it is possible to spend time with a few friends, but other weekends, are boring. (1) Worst: Work is extremely difficult, and working conditions are poor. Time off work is generally boring because outside activities are limited or no friends are available. • Decision to Make • Which job to take (In-town job or forest job) • Uncertain Events • Amount of fun • Amount of work (# of hours per week)

  21. Overall Satisfaction Fun Salary Influence Diagram Amount of Fun Fun Overall Satisfaction Job Decision Salary Amount of Work

  22. Decision Tree

  23. Analysis of the Salary Objective EMV(Salary of Forest job) = $2,600 EMV: EMV(Salary of In-Town job) = 0.35(2730)+0.5(2320.5)+0.15(2047.50)= $2,422.88

  24. Analysis of the Salary Objective EMV(Salary of Forest job) = $2,600 EMV: EMV(Salary of In-Town job) = 0.35(2730)+0.5(2320.5)+0.15(2047.50)= $2,422.88 Risk Profiles: Strategies: 1) Forest Job 100% $2,600 2) In-Town Job 35% $2,730; 50% $2,320.5; 15% $2,047.5 Conclusion: For the salary objective, the forest job has higher EMV and has no risk Cumulative Risk Profiles of the Salaries

  25. Analysis of the Fun Objective The ratings in the original 5-point Likert scale only indicate orders of the amount of fun without carrying quantitative meanings. Therefore, the original ratings are rescaled to 0 -100 points to show quantitative meanings: 5(best) – 100 points, 4(Good) – 90 points, 3(Moderate) – 60 points, 2(bad) – 25 points, 1(worst) – 0 point EV: E(Fun of Forest job) =0.10(100)+0.25(90)+0.40(60)+0.20(25)+0.05(0) = 61.5 E(Fun of In-Town job) = 60

  26. Analysis of the Fun Objective (Cont.) Risk Profiles: Strategies: 1) Forest Job 10% 100; 25% 90; 40% 60; 20% 30; 5% 0 2) In-Town Job 100% 60 Conclusion: For the fun objective, the forest job has higher EV but is more risky Cumulative Risk Profiles of the Fun

  27. Sam’s dilemma: Would he prefer a slightly higher salary for sure and take a risk on how much fun the summer will be? Or otherwise, would the in-town be better, playing it safe with the amount of fun and taking a risk on how much money will be earned? Therefore, Sam needs to make a trade-off between the objectives of maximizing fun and maximizing salary.

  28. Trade-off Analysis • Combine multiple objectives into one overall objective • Steps • First, multiple objectives must have comparable scales • Next, assign weights to these objectives (the sum of all the weights should be equal to 1) • Subjective judgment • Paying attention to the range of the attributes (the variables to be measured in the objectives) is crucial; Attributes having a wide range of possible values are usually important (why?) • Then, calculate the weighted average of consequences as an overall score • Finally, compare the alternatives using the overall score

  29. Summer Job Example (Cont.) • Convert the salary scale to the same 0 to 100 scale used to measure fun Set $2730 (the highest salary) = 100, and $2047.50 (the lowest salary) =0 Then, Intermediate salary X is converted to: (X-2047.50)∙100/(2730-2047.50) (Proportion Scoring) • Assign weights to salary and fun (Ks and Kf) Sam thinks increasing salary from the lowest to the highest is 1.5 times more important than improving fun from the worst to best, hence Ks=1.5Kf , Because Ks+Kf=1  Ks=0.6, Kf=0.4

  30. Overall Score 88.6 84.6 72.6 58.6 48.6 84.0 48.0 24.0

  31. EV(Overall Score of Forest job) =0.10(88.6)+0.25(84.6)+0.40(72.6)+0.20(58.6)+0.05(48.6) = 73.2 EV: EV(Overall Score of In-Town job) = 0.35(84)+0.50(48)+0.15(24) = 57 Risk Profiles: The forest job stochastically dominates the in-town job Conclusion: The forest job is preferred to the in-town job Cumulative Risk Profiles of the Overall Scores

  32. A1 $8 (0.5) $0 (0.27) D2 A2 U2 (0.5) $15 U1 A (0.73) $4 D1 (0.45) $10 B U3 (0.55) $0 Exercise O21 O11 O22 O12 O31 O32 1. Solve the decision tree in the figure 2. Create risk profiles and cumulative risk profiles for all possible strategies. Is one strategy stochastically dominant? Explain.

  33. A1 $8 (0.5) $0 (0.27) D2 A2 U2 (0.5) $15 U1 A (0.73) $4 D1 (0.45) $10 B U3 (0.55) $0 1. Solving the decision tree EV(U2)=$7.5 EV(U2)=0*0.5+15*0.5=$7.5 EV(U1)=$5.08 O21 O11 O22 EV(U1)=8*0.27+4*0.73=$5.08 O12 EV(U3)=10*0.45+0*0.55=$4.5 O31 O32 EV(U3)=$4.5 In conclusion, according to the EV, we should choose A, and if O11 occurs, then choose A1

  34. 2. Risk Profiles and Cumulative Risk Profiles Decision Strategies: A1 Strategy 1: A - A1 $8 (0.27) D2 $4 (0.73) $8 (0.27) U1 A (0.73) $4 D1 (0.5) $0 (0.135) $4 (0.73) $15 (0.135) $0 (0.27) D2 Strategy 2: A – A2 A2 U2 (0.5) $15 U1 A (0.73) $4 D1 Strategy 3: B D1 (0.45) $10 $0 (0.55) $10 (0.45) B U3 (0.55) $0

  35. 2. Risk Profiles and Cumulative Risk Profiles (Cont.) Risk Profiles Conclusion: No stochastic dominance exists Strategy A-A2 Strategy A-A1 Strategy B Cumulative Risk Profiles

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