Making choices
Dr. Yan LiuDepartment 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
)()(1
ir
n
ii yYPyYE
(Discrete Variable)
dyyfyYE
)()( (Continuous Variable)
3
EMV(A1)=C1•p1 +C2•(1-p1)
EMV(A2)=C3•p2 +C4• (1-p2)
A2
A1
O1 C1
O2
O3
O4
C2
C3
C4
(p1)
Payoff
(1-p1)
(p2)
(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
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?
Keep
Ticket
Trade
Ticket
Win
$24
$25LoseWin
Lose
-$1$10
$0
-$1 $0$10
$0
EMV(Trade Ticket)=24•0.2+ (-1)•0.8=$4
EMV(Keep Ticket)=100•0.45+ (0)•0.55=$4.5
EMV=$4
EMV=$4.5
Conclusion: You should keep your ticket !
Ticket
ResultTicket
Result
Lottery Ticket Example
(0.2)
(0.8)(0.45)
(0.55)
6
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.
Product-Switching Example
7
Don’t Perfor
m
Perform
Survey
Survey High
OldNew
-$100,000
-$20,0
00
$300,000 (0.3)
HighMediumLow
$100,000 (0.5)-$100,000 (0.2)
$100,000
$300,000
$150,000
Survey
Low
(0.5)
(0.5)
Old
$130,000
New Medi
um
Low
$100,000 (0.4)
-$100,000 (0.4)
$280,000
$300,000 (0.6)
High
$80,000
Old
$130,000
New
Medium
$100,000 (0.6)
$80,000-$120,000
$150,000
$150,000
$150,000
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Don’t Perfor
m
Perform
Survey
Survey High
OldNew
-$100,000
-$20,0
00
$300,000 (0.3)
HighMediumLow
$100,000 (0.5)-$100,000 (0.2)
$100,000
$300,000
$150,000
Survey
Low
(0.5)
(0.5)
Old
$130,000
New Medi
um
Low
$100,000 (0.4)
-$100,000 (0.4)
$280,000
$300,000 (0.6)
High
$80,000
Old
$130,000
New
Medium
$100,000 (0.6)
$80,000-$120,000
$150,000
$150,000
$150,000
EMV(U3) =0.6•280,000+0.4•80,000=$200,000U1
U2
U3
U4
D1
D2
D3
D4
EMV(U4) =0.6•80,000+0.4•(-120,000)=$0
EMV= $0
EMV= $200,0
00
EMV(U2) =0.3•300,000+0.5•(100,000)+0.2•(-100,000)=$120,000
EMV=
$120,000
EMV(U1) =0.5•200,000+0.5•130,000=$165,000
EMV=
$165,000
Conclusion: Perform survey. If survey shows high-level sales, then switch the new product ; otherwise, stay with the old product
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Decision Path and Strategy
Decision Path Represents a possible future scenario, starting from the left-most node to the
consequence at the end of a branch by selecting one alternative from a decision node and by following one outcome from a chance node.
Path 1 ( A1 )Path 2 ( A2O1 )
Path 3 ( A2O2A3 )Path 4 ( A2O2A4 )
D1 U1D2
A1
A2
O1
O2
A3
A4
A1
A2
D1 D2
U1
O1
O2
A3
A4
Decision Paths:
10
Decision Path and Strategy (Cont.)
Decision Strategy The collection of decision paths connected to one branch of the immediate
decision by selecting one alternative from each decision node along that path
Strategy 1 (A1): Decision path A1
Strategy 3 (A2A4): Decision paths A2O2A4, A2O1
Strategy 2 (A2A3): Decision paths A2O2A3, A2O1
A1
A2
D1D2
U1
O1
O2
A3
A4
Decision Strategies:
D1 U1D2
A1
A2
O1
O2
A3
A4
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Risk Profiles
Problems with Expected Value (EV) EV does not indicate all the possible consequences The statistical interpretation of EV as the average amount obtained by
“playing the game” a large number of times is not appropriate in rare cases (e.g. hazards in nuclear power plants)
What is Risk Profile A graph that shows the probabilities associated with possible consequences
given a particular decision strategy Indicates the relative risk levels of strategies
Steps of Deriving Risk Profiles from Decision Trees Identify the decision strategies For each strategy, collapse the decision tree by multiplying out the
probabilities on sequential chance branches (Don’t confuse it with solving decision trees!)
Keep track of all possible consequences Summarize the probability of occurrence for each consequence
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Risk Profiles of the Lottery Ticket Example
Payoff($)
Pr(Payoff)
Trade Ticket
Keep Ticket
Decision Tree of the Lottery Ticket Example
Keep
Ticket
Trade Ticke
t
Win
$24
(0.2)Los
e-
$1$10$0
(0.8)
Win
(0.45)Los
e(0.55)
1) Trade ticket: 2) Keep ticket:
$24(0.2), -$1(0.8)$10(0.45), $0(0.55)
Decision strategies:
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Decision Strategies:
Decision Tree of the Product-Switching Example
Don’t Perfor
m
Perform
Survey
Survey High
OldNew
-$100,000
(0.3)HighMediumLow
(0.5)(0.2)
$100,000
$300,000
$150,000
Survey
Low
(0.5)
(0.5)
Old
New Medi
um
Low
(0.4)
(0.4)
$280,000
(0.6)
High
$80,000
OldNew
Medium
(0.6)
$80,000-$120,000
$130,000
$130,000
1) Don’t perform survey and keep the old product 2) Don’t perform survey and switch to the new product 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
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Strategy 1): Don’t perform survey and keep the old product
Strategy 2): Don’t perform survey and switch to the new product
Don’t Perfor
m
New
Medium
High
Low
(0.3)(0.5)
(0.2) -$100,000
$100,000
$300,000
Payoffs$300,000$100,000-$100,000
Probabilities0.30.50.2
Strategy 3): Perform survey and if survey high then keep the old product
Perform
Survey
Survey HighSurvey
Low
(0.5)(0.5)
Old
$130,000$130,0
00
$130,000 (100%)
Strategy 4): Perform survey and if survey high then switch to the new product
Perform
Survey
Survey
HighSurvey
Low
(0.5)(0.5)
New
$130,000
Medium
(0.4)
$280,000
(0.6)
High $80,
000
Payoffs$280,000$130,000$80,000
Probabilities0.30.50.2
$150,000 (100%)
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Payoff($)
Pr(Payoff)
Risk Profiles of the Product-Switch Example
Strategy 1 Strategy 2 Strategy 3 Strategy 4
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Cumulative Risk Profiles
A graph that shows the cumulative probabilities associated with possible consequences given a particular decision strategy
Payoff($)
Pr(Payoff≤x)
Trade TicketKeep Ticket
Cumulative Risk Profiles of the Lottery Ticket Example
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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
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 A
strategy B
Payoff
Pr(Payoff ≤ x)
18
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
strategy A
strategy B
Payoff
Pr(Payoff ≤ x)
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.
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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.
Earning money (measured in $)
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)
Decision Elements
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Job Decision
Overall Satisfaction
Fun
Salary
Amount of Fun
Amount of Work
Fun
Overall Satisfaction
Salary
Influence Diagram
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Decision Tree
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EMV(Salary of Forest job) = $2,600
EMV(Salary of In-Town job) = 0.35(2730)+0.5(2320.5)+0.15(2047.50)= $2,422.88EMV:
Analysis of the Salary Objective
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EMV(Salary of Forest job) = $2,600
EMV(Salary of In-Town job) = 0.35(2730)+0.5(2320.5)+0.15(2047.50)= $2,422.88EMV:
Analysis of the Salary Objective
Conclusion: For the salary objective, the forest job has higher EMV and has no risk
Cumulative Risk Profiles of the Salaries
Risk Profiles:
Strategies: 1) Forest Job 100% $2,600
2) In-Town Job 35% $2,730; 50% $2,320.5; 15% $2,047.5
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The ratings in the original 5-point Likert scale only indicate orders of the amount of fun without carrying quantitative meanings.
Analysis of the Fun Objective
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
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
EV:
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Cumulative Risk Profiles of the Fun
Conclusion: For the fun objective, the forest job has higher EV but is more risky
Risk Profiles:
Strategies: 1) Forest Job 10% 100; 25% 90; 40% 60; 20% 30; 5% 0
2) In-Town Job 100% 60
Analysis of the Fun Objective (Cont.)
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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.
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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
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Summer Job Example (Cont.)
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)
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
Convert the salary scale to the same 0 to 100 scale used to measure fun
Assign weights to salary and fun (Ks and Kf)
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Overall
Score88.6
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(Overall Score of In-Town job) = 0.35(84)+0.50(48)+0.15(24) = 57
Cumulative Risk Profiles of the Overall Scores
The forest job stochastically dominates the in-town job
Conclusion: The forest job is preferred to the in-town job
EV:
Risk Profiles:
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Exercise
D1
D2
A
B
(0.27)A2
A1 $8$0$15
(0.5)(0.5)
(0.73)
(0.45)
(0.55)
$4
$10
$0
U1
U3
U2
1. Solve the decision tree in the figure2. Create risk profiles and cumulative risk profiles for all possible strategies. Is one strategy stochastically dominant? Explain.
O11
O12
O21
O22
O31
O32
33
D1
D2
A
B
(0.27)A2
A1 $8$0$15
(0.5)(0.5)
(0.73)
(0.45)
(0.55)
$4
$10
$0
U1
U3
U2
EV(U2)=0*0.5+15*0.5=$7.5EV(U1)=8*0.27+4*0.73=$5.08EV(U3)=10*0.45+0*0.55=$4.5
EV(U2)=$7.5EV(U1)=
$5.08
EV(U3)=$4.5
In conclusion, according to the EV, we should choose A, and if O11 occurs, then choose A1
1. Solving the decision tree
O11
O12
O21O22
O31
O32
34
D1
D2
A
(0.27)
A1 $8
(0.73) $4U1
2. Risk Profiles and Cumulative Risk Profiles
Decision Strategies:Strategy 1: A - A1 $4
(0.73)$8 (0.27)
Strategy 2: A – A2
D1
D2
A
(0.27)A2
$0$15
(0.5)(0.5)
(0.73) $4U1
U2
$0 (0.135)$4 (0.73)$15 (0.135)
Strategy 3: B
D1
B(0.45)
(0.55)
$10
$0U3
$0 (0.55)$10 (0.45)
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2. Risk Profiles and Cumulative Risk Profiles (Cont.)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 2 4 6 8 10 12 14 16 18 20
Strategy A-A2
Strategy A-A1
Strategy B
Risk Profiles
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 2 4 6 8 10 12 14 16 18 20
Cumulative Risk Profiles
Conclusion: No stochastic dominance exists
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