Making choices

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
EMV(A1)C1p1 C2(1-p1)
EMV(A2)C3p2 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
EMV4
25
(0.2)
Trade Ticket
EMV(Trade Ticket)240.2 (-1)0.84
Lose
Ticket Result
-1
0
(0.8)
-1
Win
10
EMV4.5
10
(0.45)
Keep Ticket
EMV(Keep Ticket)1000.45 (0)0.554.5
Lose
Ticket 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 60chance
of medium-level sales and 40 chance of low-level
sales when the company sells the product.
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Old
150,000
130,000
Survey High
300,000 (0.6)
High
280,000
New
(0.5)
100,000 (0.4)
Medium
80,000
Old
150,000
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
Dont Perform
Old
150,000
New
300,000 (0.3)
High
300,000
Medium
100,000 (0.5)
100,000
Low
-100,000 (0.2)
-100,000
8
Old
150,000
130,000
D3
EMV 200,000
Survey High
300,000 (0.6)
High
280,000
EMV 165,000
(0.5)
New
EMV(U3) 0.6280,0000.480,000200,000
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) 0.680,0000.4(-120,000)0
-20,000
Low
-100,000 (0.4)
-120,000
U4
D1
EMV(U1) 0.5200,0000.5130,000165,000
D2
150,000
Dont Perform
Old
150,000
New
300,000 (0.3)
High
300,000
100,000 (0.5)
Medium
100,000
Low
-100,000 (0.2)
U2
EMV 120,000
-100,000
EMV(U2) 0.3300,0000.5(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
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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 )
A1
Path 2 ( A2O1 )
O1
Decision Paths
A3
Path 3 ( A2O2A3 )
A2
Path 4 ( A2O2A4 )
O2
A4
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
A1
O1
Strategy 2 (A2A3) Decision paths A2O2A3, A2O1
Decision Strategies
A3
A2
O2
A4
Strategy 3 (A2A4) Decision paths A2O2A4, A2O1
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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 (Dont confuse it with solving
  decision trees!)
• Keep track of all possible consequences
• Summarize the probability of occurrence for each
  consequence

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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
(0.6)
High
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
Dont Perform
Old
150,000
(0.3)
High
New
300,000
Medium
(0.5)
100,000
Low
(0.2)
-100,000
Decision Tree of the Product-Switching Example
1) Dont perform survey and keep the old product
2) Dont 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
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Strategy 1) Dont perform survey and keep the
old product
150,000 (100)
Strategy 2) Dont 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
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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
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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

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
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Dominance (Cont.)

• Stochastic Dominance
• If for any x, Pr(Payoff xstrategy B)
  Pr(Payoff xstrategy 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.
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Decision Elements

• Objectives (and Measures)

• 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 Make

• Which job to take (In-town job or forest job)

• Uncertain Events

• Amount of fun
• Amount of work ( of hours per week)

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Influence Diagram
Amount of Fun
Fun
Overall Satisfaction
Job Decision
Salary
Amount of Work
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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
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• 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
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• 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
E(Fun of Forest job) 0.10(100)0.25(90)0.40(60)
0.20(25)0.05(0) 61.5
EV
E(Fun of In-Town job) 60
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• 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
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Sams 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.)

• 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
Ks1.5Kf , Because KsKf1 ? Ks0.6, Kf0.4
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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
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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.
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1. Solving the decision tree
EV(U2)7.5
EV(U2)00.5150.57.5
EV(U1)5.08
O21
O11
O22
EV(U1)80.2740.735.08
O12
EV(U3)100.4500.554.5
O31
O32
EV(U3)4.5
In conclusion, according to the EV, we should
choose A, and if O11 occurs, then choose A1
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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)
(0.27)
0
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
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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