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

8

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

9

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

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

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

22

Decision Tree

23

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.

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

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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)

30

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)

35

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