Presentation on theme: "Making choices Dr. Yan Liu"— Presentation transcript:
1 Making choices Dr. Yan Liu Department of Biomedical, Industrial & Human Factors EngineeringWright 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 usedEV is the mean of a random variable that has a probability distribution function(Discrete Variable)(Continuous Variable)
4 Solving Decision Trees Decision Trees are Solved by “Rolling Back” the TreesStart at the endpoints of the branches on the far right-hand side and move to leftWhen encountering a chance node, calculate its EV and replace the node with the EVWhen encountering a decision node, choose the branch with the highest EVContinue 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$24EMV=$4$25(0.2)Trade TicketEMV(Trade Ticket)=24•0.2+ (-1)•0.8=$4LoseTicket Result-$1-$1$0(0.8)Win$10EMV=$4.5$10(0.45)Keep TicketEMV(Keep Ticket)=100•0.45+ (0)•0.55=$4.5LoseTicket Result$0$0(0.55)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.
8 Old $150,000 $130,000 Survey High High $300,000 (0.6) $280,000 (0.5) EMV= $200,000Survey HighHigh$300,000 (0.6)$280,000EMV=$165,000(0.5)NewEMV(U3) =0.6•280, •80,000=$200,000U3Medium$100,000 (0.4)$80,000Old$150,000Perform SurveyU1D4$130,000Survey LowEMV=$0Medium$100,000 (0.6)(0.5)$80,000NewEMV(U4) =0.6•80, •(-120,000)=$0-$20,000Low-$100,000 (0.4)U4-$120,000D1EMV(U1) =0.5•200, •130,000=$165,000D2Don’t PerformOld$150,000$150,000NewHigh$300,000 (0.3)$300,000Medium$100,000 (0.5)$100,000-$100,000 (0.2)EMV=$120,000U2Low-$100,000EMV(U2) =0.3•300, •(100,000)+0.2•(-100,000)=$120,000Conclusion: Perform survey. If survey shows high-level sales, then switch the new product ; otherwise, stay with the old product
9 Decision Path and Strategy 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.D1U1D2A1A2O1O2A3A4Path 1 ( A1 )A1O1Path 2 ( A2O1 )Decision Paths:A3D1U1Path 3 ( A2O2A3 )A2D2O2Path 4 ( A2O2A4 )A4
10 Decision Path and Strategy (Cont.) Decision StrategyThe collection of decision paths connected to one branch of the immediate decision by selecting one alternative from each decision node along that pathD1U1D2A1A2O1O2A3A4Strategy 1 (A1): Decision path A1A1O1Decision Strategies:Strategy 2 (A2A3):Decision paths A2O2A3, A2O1D1A3U1A2D2O2A4Strategy 3 (A2A4):Decision paths A2O2A4, A2O1
11 Risk Profiles Problems with Expected Value (EV) What is Risk Profile EV does not indicate all the possible consequencesThe 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 ProfileA graph that shows the probabilities associated with possible consequences given a particular decision strategyIndicates the relative risk levels of strategiesSteps of Deriving Risk Profiles from Decision TreesIdentify the decision strategiesFor 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 consequencesSummarize the probability of occurrence for each consequence
12 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)Decision Tree of the Lottery Ticket ExampleKeep TicketTrade TicketWin$24(0.2)Lose-$1$10$0(0.8)(0.45)(0.55)Pr(Payoff)Trade TicketKeep TicketPayoff($)Risk Profiles of the Lottery Ticket Example
13 Decision Tree of the Product-Switching Example Old$130,000Survey HighHigh(0.6)$280,000New(0.5)Medium(0.4)Perform Survey$80,000Old$130,000Survey LowMedium(0.6)(0.5)$80,000NewLow(0.4)-$120,000Don’t PerformOld$150,000NewHigh(0.3)$300,000Medium(0.5)$100,000Low(0.2)-$100,000Decision Tree of the Product-Switching Example1) Don’t perform survey and keep the old productDecision Strategies:2) Don’t perform survey and switch to the new product3) Perform survey, and if survey is high then keep the old product4) Perform survey, and if survey is high then switch to the new product
14 Don’t Perform New Medium High Low (0.3) (0.5) (0.2) -$100,000 $100,000 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 productPayoffs$300,000$100,000-$100,000Probabilities0.30.50.2Don’t PerformNewMediumHighLow(0.3)(0.5)(0.2)-$100,000$100,000$300,000Strategy 3): Perform survey and if survey high then keep the old productPerform SurveySurvey HighSurvey Low(0.5)Old$130,000$130,000 (100%)Strategy 4): Perform survey and if survey high then switch to the new productPerform SurveySurvey HighSurvey Low(0.5)New$130,000Medium(0.4)$280,000(0.6)High$80,000Payoffs$280,000$130,000$80,000Probabilities0.30.50.2
15 Risk Profiles of the Product-Switch Example Payoff($)Pr(Payoff)Risk Profiles of the Product-Switch ExampleStrategy 1Strategy 2Strategy 3Strategy 4
16 Cumulative Risk Profiles A graph that shows the cumulative probabilities associated with possible consequences given a particular decision strategyPayoff($)Pr(Payoff≤x)Trade TicketKeep TicketCumulative 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 AMay also be concluded by drawing cumulative risk profilesPr(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 Astrategy BPayoff
18 Dominance (Cont.) Stochastic Dominance If for any x, Pr(Payoff ≤ x|strategy B) ≤ Pr(Payoff ≤ x|strategy A), then B stochastically dominates Astrategy Astrategy BPayoffPr(Payoff ≤ x)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 ExampleSam 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 Objectives (and Measures) Decision to Make Earning money (measured in $)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 MakeWhich job to take (In-town job or forest job)Uncertain EventsAmount of funAmount of work (# of hours per week)
21 Influence Diagram Overall Satisfaction Amount Fun Salary of Fun Fun JobDecisionSalaryAmountof Work
23 Analysis of the Salary Objective EMV(Salary of Forest job) = $2,600EMV:EMV(Salary of In-Town job) = 0.35(2730)+0.5(2320.5)+0.15( )= $2,422.88
24 Analysis of the Salary Objective EMV(Salary of Forest job) = $2,600EMV:EMV(Salary of In-Town job) = 0.35(2730)+0.5(2320.5)+0.15( )= $2,422.88Risk Profiles:Strategies:1) Forest Job 100% $2,6002) In-Town Job 35% $2,730; 50% $2,320.5; 15% $2,047.5Conclusion: For the salary objective, the forest job has higher EMV and has no riskCumulative 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 points to show quantitative meanings: 5(best) – 100 points, 4(Good) – 90 points, 3(Moderate) – 60 points, 2(bad) – 25 points, 1(worst) – 0 pointEV:E(Fun of Forest job) =0.10(100)+0.25(90)+0.40(60)+0.20(25)+0.05(0) = 61.5E(Fun of In-Town job) = 60
26 Cumulative Risk Profiles of the Fun Analysis of the Fun Objective (Cont.)Risk Profiles:Strategies:1) Forest Job 10% 100; 25% 90; 40% 60; 20% 30; 5% 02) In-Town Job 100% 60Conclusion: For the fun objective, the forest job has higher EV but is more riskyCumulative 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 AnalysisCombine multiple objectives into one overall objectiveStepsFirst, multiple objectives must have comparable scalesNext, assign weights to these objectives (the sum of all the weights should be equal to 1)Subjective judgmentPaying 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 scoreFinally, 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 funSet $2730 (the highest salary) = 100, and $ (the lowest salary) =0Then, Intermediate salary X is converted to: (X )∙100/( ) (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, henceKs=1.5Kf , Because Ks+Kf=1 Ks=0.6, Kf=0.4
31 EV: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) = 73.2EV(Overall Score of In-Town job)= 0.35(84)+0.50(48)+0.15(24) = 57Risk Profiles:The forest job stochastically dominates the in-town jobConclusion: The forest job is preferred to the in-town jobCumulative Risk Profiles of the Overall Scores
32 Exercise A B (0.27) A2 A1 $8 $0 $15 (0.5) (0.73) (0.45) (0.55) $4 $10 D1D2AB(0.27)A2A1$8$0$15(0.5)(0.73)(0.45)(0.55)$4$10U1U3U2O21O11O22O12O31O321. Solve the decision tree in the figure2. Create risk profiles and cumulative risk profiles for all possible strategies. Is one strategy stochastically dominant? Explain.
33 1. Solving the decision tree AB(0.27)A2A1$8$0$15(0.5)(0.73)(0.45)(0.55)$4$10U1U3U2EV(U2)=$7.5EV(U2)=0*0.5+15*0.5=$7.5EV(U1)=$5.08O21O11O22EV(U1)=8*0.27+4*0.73=$5.08O12EV(U3)=10*0.45+0*0.55=$4.5O31EV(U3)=$4.5O32In 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:Strategy 1: A - A1A1$8(0.27)D2$4 (0.73)$8 (0.27)U1A(0.73)$4D1(0.5)Strategy 2: A – A2(0.27)D2$0$0 (0.135)$4 (0.73)$15 (0.135)A2U2(0.5)U1$15A(0.73)$4D1Strategy 3: BD1(0.45)$10$0 (0.55)$10 (0.45)BU3(0.55)$0
35 2. Risk Profiles and Cumulative Risk Profiles (Cont.) Conclusion: No stochastic dominance existsStrategy A-A1Strategy A-A2Strategy BCumulative Risk Profiles