Decision Tree Analysis
Decision Analysis
•
Managers often must make decisions in environments that are fraught with
uncertainty.
•
Some Examples
– A manufacturer introducing a new product into the marketplace
• What will be the reaction of potential customers?
• How much should be produced?
• Should the product be test-marketed?
• How much advertising is needed?
– A financial firm investing in securities
• Which are the market sectors and individual securities with the best prospects?
• Where is the economy headed?
• How about interest rates?
• How should these factors affect the investment decisions?
9-2
Decision Analysis
•
Managers often must make decisions in environments that are fraught with
uncertainty.
•
Some Examples
– A government contractor bidding on a new contract.
• What will be the actual costs of the project?
• Which other companies might be bidding?
• What are their likely bids?
– An agricultural firm selecting the mix of crops and livestock for the season.
• What will be the weather conditions?
• Where are prices headed?
• What will costs be?
– An oil company deciding whether to drill for oil in a particular location.
• How likely is there to be oil in that location?
• How much?
• How deep will they need to drill?
• Should geologists investigate the site further before drilling?
9-3
The Goferbroke Company Problem
•
The Goferbroke Company develops oil wells in unproven territory.
•
A consulting geologist has reported that there is a one-in-four chance of oil on
a particular tract of land.
•
Drilling for oil on this tract would require an investment of about $100,000.
•
If the tract contains oil, it is estimated that the net revenue generated would be
approximately $800,000.
•
Another oil company has offered to purchase the tract of land for $90,000.
Question: Should Goferbroke drill for oil or sell the tract?
9-4
Prospective Profits
Profit
Status of Land
Oil
Dry
Drill for oil
$700,000
–$100,000
Sell the land
90,000
90,000
Chance of status
1 in 4
3 in 4
Alternative
9-5
Decision Analysis Terminology
•
The decision maker is the individual or group responsible for making the
decision.
•
The alternatives are the options for the decision to be made.
•
The outcome is affected by random factors outside the control of the decision
maker. These random factors determine the situation that will be found when
the decision is executed. Each of these possible situations is referred to as a
possible state of nature.
•
The decision maker generally will have some information about the relative
likelihood of the possible states of nature. These are referred to as the prior
probabilities.
•
Each combination of a decision alternative and a state of nature results in some
outcome. The payoff is a quantitative measure of the value to the decision
maker of the outcome. It is often the monetary value.
9-6
Prior Probabilities
State of Nature
Prior Probability
The tract of land contains oil
0.25
The tract of land is dry (no oil)
0.75
9-7
Payoff Table (Profit in $Thousands)
State of Nature
Alternative
Oil
Dry
Drill for oil
700
–100
Sell the land
90
90
0.25
0.75
Prior probability
9-8
The Maximax Criterion
•
The maximax criterion is the decision criterion for the eternal optimist.
•
It focuses only on the best that can happen.
•
Procedure:
– Identify the maximum payoff from any state of nature for each alternative.
– Find the maximum of these maximum payoffs and choose this alternative.
State of Nature
Alternative
Oil
Dry
Maximum in Row
Drill for oil
700
–100
700  Maximax
Sell the land
90
90
90
9-9
The Maximin Criterion
•
The maximin criterion is the decision criterion for the total pessimist.
•
It focuses only on the worst that can happen.
•
Procedure:
– Identify the minimum payoff from any state of nature for each alternative.
– Find the maximum of these minimum payoffs and choose this alternative.
State of Nature
Alternative
Oil
Dry
Minimum in Row
Drill for oil
700
–100
–100
Sell the land
90
90
90  Maximin
9-10
The Maximum Likelihood Criterion
•
The maximum likelihood criterion focuses on the most likely state of nature.
•
Procedure:
– Identify the state of nature with the largest prior probability
– Choose the decision alternative that has the largest payoff for this state of nature.
State of Nature
Alternative
Oil
Dry
Drill for oil
700
–100
Sell the land
90
90
0.25
0.75
Prior probability
–100
90  Step 2: Maximum

Step 1: Maximum
9-11
Bayes’ Decision Rule
•
Bayes’ decision rule directly uses the prior probabilities.
•
Procedure:
– For each decision alternative, calculate the weighted average of its payoff by
multiplying each payoff by the prior probability and summing these products. This
is the expected payoff (EP).
– Choose the decision alternative that has the largest expected payoff.
A
1
2
3
4
5
6
7
8
B
C
D
E
F
Bayes' Decision Rule for the Goferbroke Co.
Payoff Table
Alternative
Drill
Sell
Prior Probability
State of Nature
Oil
Dry
700
-100
90
90
0.25
Expected
Payoff
100
90
0.75
9-12
Bayes’ Decision Rule
•
Features of Bayes’ Decision Rule
– It accounts for all the states of nature and their probabilities.
– The expected payoff can be interpreted as what the average payoff would become
if the same situation were repeated many times. Therefore, on average, repeatedly
applying Bayes’ decision rule to make decisions will lead to larger payoffs in the
long run than any other criterion.
•
Criticisms of Bayes’ Decision Rule
– There usually is considerable uncertainty involved in assigning values to the prior
probabilities.
– Prior probabilities inherently are at least largely subjective in nature, whereas
sound decision making should be based on objective data and procedures.
– It ignores typical aversion to risk. By focusing on average outcomes, expected
(monetary) payoffs ignore the effect that the amount of variability in the possible
outcomes should have on decision making.
9-13
Decision Trees
•
A decision tree can apply Bayes’ decision rule while displaying and analyzing
the problem graphically.
•
A decision tree consists of nodes and branches.
– A decision node, represented by a square, indicates a decision to be made. The
branches represent the possible decisions.
– An event node, represented by a circle, indicates a random event. The branches
represent the possible outcomes of the random event.
9-14
Decision Tree for Goferbroke
Payoff
700
Oil (0.25)
B
Drill
Dry (0.75)
-100
A
Sell
90
9-15
Using TreePlan
TreePlan, an Excel add-in developed by Professor Michael Middleton, can be used
to construct and analyze decision trees on a spreadsheet.
1. Choose Decision Tree on the Add-Ins tab (Excel 2007 or 2010) or the Tools menu
(for other versions).
2. Click on New Tree, and it will draw a default tree with a single decision node and
two branches, as shown below.
3. The labels in D2 and D7 (originally Decision 1 and Decision 2) can be replaced by
more descriptive names (e.g., Drill and Sell).
9-16
Using TreePlan
4. To replace a node (such as the terminal node of the drill branch in F3) by a
different type of node (e.g., an event node), click on the cell containing the node,
choose Decision Tree again from the Add-Ins tab (Excel 2007 or 2010) or Tools
menu (other versions of Excel), and select “Change to event node”.
9-17
Using TreePlan
5. Enter the correct probabilities in H1 and H6.
6. Enter the partial payoffs for each decision and event in D6, D14, H4, and H9.
9-18
TreePlan Results
•
The numbers inside each decision node indicate which branch should be
chosen (assuming the branches are numbered consecutively from top to
bottom).
•
The numbers to the right of each terminal node is the payoff if that node is
reached.
•
The number 100 in cells A10 and E6 is the expected payoff at those stages in
the process.
9-19
Consolidate the Data and Results
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
B C
D
E
F G
H
I
J
K
0.25
Oil
700
Drill
800
-100
100
700
0.75
Dry
-100
1
0
-100
100
Sell
90
90
90
Cost of Drilling
Revenue if Oil
Revenue if Sell
Revenue if Dry
Probability Of Oil
Data
100
800
90
0
0.25
Action
Drill
Expected Payoff
100
9-20
Sensitivity Analysis: Prior Probability of Oil = 0.15
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
B C
D
E
F G
H
I
J
K
0.15
Oil
700
Drill
800
-100
20
700
0.85
Dry
-100
2
0
-100
90
Sell
90
90
90
Cost of Drilling
Revenue if Oil
Revenue if Sell
Revenue if Dry
Probability Of Oil
Data
100
800
90
0
0.15
Action
Sell
Expected Payoff
90
9-21
Sensitivity Analysis: Prior Probability of Oil = 0.35
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
B C
D
E
F G
H
I
J
K
0.35
Oil
700
Drill
800
-100
180
700
0.65
Dry
-100
1
0
-100
180
Sell
90
90
90
Cost of Drilling
Revenue if Oil
Revenue if Sell
Revenue if Dry
Probability Of Oil
Data
100
800
90
0
0.35
Action
Drill
Expected Payoff
180
9-22
Using Data Tables to Do Sensitivity Analysis
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
B
C
D
E
F G
H
I
J
K
L
M
0.25
Oil
700
Drill
800
-100
100
700
0.75
Dry
-100
1
0
-100
100
Sell
90
90
90
Cost of Drilling
Revenue if Oil
Revenue if Sell
Revenue if Dry
Probability Of Oil
Data
100
800
90
0
0.25
Action
Drill
Expected Payoff
100
Probability
of Oil
0.15
0.17
0.19
0.21
0.23
0.25
0.27
0.29
0.31
0.33
0.35
Action
Drill
Expected
Payoff
100
Select these
cells (I18:K29),
before
choosing Table
from the Data
menu.
9-23
Data Table Results
The Effect of Changing the Prior Probability of Oil
I
16
17
18
19
20
21
22
23
24
25
26
27
28
29
Probability
of Oil
0.15
0.17
0.19
0.21
0.23
0.25
0.27
0.29
0.31
0.33
0.35
J
K
Action
Drill
Sell
Sell
Sell
Sell
Sell
Drill
Drill
Drill
Drill
Drill
Drill
Expected
Payoff
100
90
90
90
90
90
100
116
132
148
164
180
9-24
Checking Whether to Obtain More Information
•
Might it be worthwhile to spend money for more information to obtain better
estimates?
•
A quick way to check is to pretend that it is possible to actually determine the true
state of nature (“perfect information”).
•
EP (with perfect information) = Expected payoff if the decision could be made
after learning the true state of nature.
•
EP (without perfect information) = Expected payoff from applying Bayes’
decision rule with the original prior probabilities.
•
The expected value of perfect information is then
EVPI = EP (with perfect information) – EP (without perfect information).
9-25
Expected Payoff with Perfect Information
B
C
D
3 Payoff Table
State of Nature
4
Alternative
Oil
Dry
5
Drill
700
-100
6
Sell
90
90
7
Maximum Payoff
700
90
8
9
Prior Probability
0.25
0.75
10
11
EP (with perfect info)
242.5
9-26
Expected Payoff with Perfect Information
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
B C
D
E
F G
H
I
J
K
Drill
0.25
700
Oil
700
700
1
0
700
Sell
90
90
90
-100
-100
90
90
242.5
Drill
0.75
-100
Dry
2
0
90
Sell
90
9-27