Lecture no41 decision tree analysis

Decision-Tree Analysis

Lecture No. 41
Chapter 12
Contemporary Engineering Economics
Copyright © 2016

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Decision Tree Analysis

•

A graphical tool for describing:

o
o
o

The actions available to the decision-maker
The events that can occur
The relationship between the actions and events

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Constructing a Decision Tree
A company is considering marketing a new product. Once the product is introduced, there is a 70% chance of
encountering a competitive product.
Two options are available for each situation.
Option 1 (with competitive product): Raise your price and see how your competitor responds. If the
competitor raises price, your profit will be $60. If they lower the price, you will lose $20.
Option 2 (without competitive product): You still have two options: raise your price or lower your price.

o
o

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Conditional Profits and Probabilities

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Rollback Procedure

•
•

To analyze a decision tree, we begin at the end of the tree and work backward.
For each chance node, we calculate the expected monetary value (EMV), and place it in the node
to indicate that it is the expected value calculated over all branches emanating from that node.

•

For each decision node, we select the one with the highest EMV (or minimum cost). Then those
decision alternatives not selected are eliminated from further consideration.

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Making Sequential Investment Decisions

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Decision Rules

o
o
o

Market the new product.
Whether or not you encounter a competitive product, raise your price.
The expected monetary value associated with marketing the new product is $44.

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Bill’s Decision Problem: $50,000 to Invest



Decision Problem

o

Buying a highly speculative stock (d1) with
three potential levels of return: High (50%),
Medium (9%), and Low (−30%)

o

Buying a risk-free U.S. Treasury bond (d2) with a
guaranteed 7.5% return



Seek advice from an expert?

o

Seek professional advice before making the
decision

o

Do not seek professional advice; do on his own

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Financial Data

o
o
o
o
o
o
o

Total amount available for investment: $50,000
Investment horizon: one year
Commission fee for stock trade: $100
Commission fee for bond trade: $150
Tax rate for long-term capital gains on stock: 20%
Tax rate for long-term capital gains on T. Bond: 0%
Bill’s discount rate (MARR) = 5%

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Decision Tree for Bill’s Investment Problem: Select Option 2

-

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Expected Value of Perfect Information (EVPI)

o

What is EVPI? This is equivalent to asking yourself how much you can improve your decision if you had
perfect information.

o

Mathematical relationship
EVPI = EPPI − EMV = EOL
where EPPI (Expected profit with perfect information) is the expected profit you could obtain if you had
perfect information, and EMV (Expected monetary value) is the expected profit you could obtain based
on your own judgment. This is equivalent to expected opportunity loss (EOL).

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Expected Value of Perfect Information
Decision Option

Potential Return Level

Opportunity Loss

(Prior Optimal)
Probability

Option1: Invest

Option 2: Invest in

in Stock

Bonds

Optimal Choice with

Associated with Investing

Perfect Information

in Bonds

High (A)

0.25

$16,510

$898

Stock

$15,612

Medium (B)

0.40

890

898

Bond

0

Low(C)

0.35

−13,967

898

Bond

0

EMV

−$405

$898

$3,903
EVPI = EPPI − EV

EPPI = (0.25)($16,510) + (0.40)($898)
+ (0.35)($898) = $4,801

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= $4,801 − $898
= $3,903

EOL = (0.25)($15,612)
+ (0.40)(0) + (0.35)(0)
= $3,903

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Bill’s Investment Problem with an Option of Getting Professional Advice

Updating Conditional Profit (or Loss) after Paying a Fee to the Expert (Fee =
$200)

Revised Decision Tree

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Conditional Probabilities of the Expert’s Prediction, Given a Potential Return on the
Stock

F

Given Level of Stock Performance

0.8

0.2

UF

What the Report

High

Medium

Low

Will Say

(A)

(B)

(C)

A
F
B

UF

Favorable (F)

0.80

0.65

0.20

Unfavorable (UF)

0.20

0.35

0.80

C
U

UF

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Nature’s Tree: Conditional Probabilities and Joint Probabilities

Nature’s Tree

Joint and Marginal Probabilities
P(A,F) = P(F|A)P(A) = (0.80)(0.25) = 0.20
P(A,UF|A)P(A) = (0.20)(0.25) = 0.05
P(B,F) = P(F|B)P(B) = (0.65)(0.40) = 0.26

P(B,UF) = P(UF|B)P(B) = (0.35)(0.40) = 0.14

P(F) = 0.20 + 0.26 + 0.07
= 0.53
P(UF) = 1 − P(F) = 1 − 0.53
= 0.47

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Joint and Marginal Probabilities
What the Report Will Say
Joint Probabilities

When Potential Level of Return Is Given

Marginal Probabilities of Return Level
Favorable (F)

Unfavorable (UF)

High (A)

0.20

0.05

0.25

Medium (B)

0.26

0.14

0.40

Low (C)

0.07

0.28

0.35

Marginal Probabilities of what the

0.53

0.47

1.00

report will say

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Posterior Probabilities
A

P(A/F)= ?

B

F

C
0.53

0.47
A
UF
B

C

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Determining Revised Probabilities
P(A|F) = P(A,F)/P(F) = 0.20/0.53 = 0.38

P(B|F) = P(B,F)/P(F) = 0.26/0.53 = 0.49

P(C|F) = P(C,F)/P(F) = 0.07/0.53 = 0.13

P(A|UF) = P(A,UF)/P(UF) = 0.05/0.47 = 0.30

P(B|UF) + P(B,UF)/P(UF) = 0.14/0.47 = 0.30

P(C|UF) = P(C,UF)/P(UF) = 0.28/0.47 = 0.59

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Posterior Probabilities
A

0.38

B
0.49
F

C
0.13
0.53

0.47
A
UF

0.11
B
0.30
C
0.59

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Decision Making After Seeing the Report

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EVPI After Taking the Sample

•

EVPI before taking the sample

EVPI = EPPI - EV = $3,903

•

EV after spending $200

EVPIe = EPPIe - EVe
= $16,348(0.25) + $729(0.40) + 698(0.35) − $2,836

•

= $1,786.90
Expected value of sample information (EVSI):

EVSI = $3,903 − $1,786.90 = $2,116.10
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Decision Tree Analysis
PROS

CONS

Describes the decision problem graphically so it is

EMV rule to select a decision at a decision node;

easier to understand

ignore the variability of financial outcome (riskneutral environment)
Trees can grow very quickly as we add more
decision options and event nodes.

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