Chapter 12 (Decision Analysis)
IS 320
Irwin/McGraw-Hill
Payoff table
Future Demand
Low
…….
Moderate
High
Profits in $ millions
Small
10
10
10
Medium
7
12
12
-4
2
16
Proposed Capacity
…
Very large
McGraw-Hill/Irwin
Payoff table
States of Nature
“payoff”
d1
d2
Courses of
Action
…
dM
McGraw-Hill/Irwin
s1
s2
…
sN
Decision making under risk
States of Nature
“payoff”
d1
d2
Courses of
Action
…
McGraw-Hill/Irwin
dM
p1
p2
…
pN
s1
s2
…
sN
Decision making under certainty
State of Nature
“payoff”
d1
d2
Courses of
Action
…
dM
McGraw-Hill/Irwin
s1
Decisions Under Certainty
•
State of nature is certain (one state).
•
Select decision that yields highest return (e.g., linear programming, integer
programming).
•
Examples:
–
–
–
–
Product mix
Diet problem
Distribution
Scheduling
McGraw-Hill/Irwin
Decisions Under Uncertainty (or Risk)
•
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?
McGraw-Hill/Irwin
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?
McGraw-Hill/Irwin
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 $700,000.
•
Another oil company has offered to purchase the tract of land for $90,000.
Question: Should Goferbroke drill for oil or sell the tract?
McGraw-Hill/Irwin
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
McGraw-Hill/Irwin
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.
McGraw-Hill/Irwin
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
McGraw-Hill/Irwin
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
McGraw-Hill/Irwin
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
McGraw-Hill/Irwin
90
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
McGraw-Hill/Irwin
90 ← Maximin
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
↑
Step 1: Maximum
McGraw-Hill/Irwin
–100
90 ← Step 2: Maximum
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.
McGraw-Hill/Irwin
Payoff Table
Alternative
Drill
Sell
Prior Probability
State of Nature
Oil
Dry
700
-100
90
90
0.25
0.75
Expected
Payoff
100
90
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.
McGraw-Hill/Irwin
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.
McGraw-Hill/Irwin
Decision Tree for Goferbroke
Payoff
700
Oil (0.25)
B
Drill
Dry (0.75)
-100
A
Sell
90
McGraw-Hill/Irwin
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 under the Tools menu.
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).
A
1
2
3
4
5
6
7
8
9
McGraw-Hill/Irwin
B C
D
E
F
G
Drill
0
0
0
1
0
Sell
0
0
0
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 Tools menu, and select “Change to event
node”.
McGraw-Hill/Irwin
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.
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
McGraw-Hill/Irwin
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
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.
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
McGraw-Hill/Irwin
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
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
McGraw-Hill/Irwin
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
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
McGraw-Hill/Irwin
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
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
McGraw-Hill/Irwin
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
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
McGraw-Hill/Irwin
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.
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
McGraw-Hill/Irwin
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
Incorporating New Information
•
Often, a preliminary study can be done to better determine the true state of
nature.
•
Examples:
– Market surveys
– Test marketing
– Seismic testing (for oil)
Question: What is the value of this information?
McGraw-Hill/Irwin
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).
McGraw-Hill/Irwin
Expected Value of Perfect Information (EVPI)
State of Nature
Decision
Oil
Dry
Drill for oil
700
–100
Sell the land
90
90
0.25
0.75
Prior Probability
Suppose they had a test that could predict ahead of time whether the side would be
wet or dry.
• Expected Payoff = (0.25)(700) + (0.75)(90) = 242.5
• Expected Value of Perfect Information (EVPI)
= Expected Payoff (with perfect info) – Expected Payoff (without info)
= 242.5 – 100
= 142.5
McGraw-Hill/Irwin
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
0.25
0.75
Prior Probability
10
11
EP (with perfect info)
242.5
McGraw-Hill/Irwin
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
242.5
McGraw-Hill/Irwin
Drill
0.75
-100
Dry
2
0
90
Sell
90
90
90
Using New Information to Update the Probabilities
•
The prior probabilities of the possible states of nature often are quite
subjective in nature. They may only be rough estimates.
•
It is frequently possible to do additional testing or surveying (at some expense)
to improve these estimates. The improved estimates are called posterior
probabilities.
McGraw-Hill/Irwin
Imperfect Information (Seismic Test)
Suppose a seismic test is available that would better (but not perfectly) indicate
whether or not the site was wet or dry.
– Favorable result usually means the site has oil (but not always)
– Unfavorable results usually means the site is dry (but not always)
Record of 100 Past Seismic Test Sites
Seismic
Result
Actual State of Nature
Oil
Dry
Total
Favorable
15
15
30
Unfavorable
10
60
70
Total
25
75
100
McGraw-Hill/Irwin
Seismic Survey for Goferbroke
•
Goferbroke can obtain improved estimates of the chance of oil by conducting a
detailed seismic survey of the land, at a cost of $30,000.
•
Possible findings from a seismic survey:
– FSS: Favorable seismic soundings; oil is fairly likely.
– USS: Unfavorable seismic soundings; oil is quite unlikely.
•
P(finding | state) = Probability that the indicated finding will occur,
given that the state of nature is the indicated one.
P(finding | state)
State of Nature
Favorable (FSS)
Unfavorable (USS)
Oil
P(FSS | Oil) = 0.6
P(USS | Oil) = 0.4
Dry
P(FSS | Dry) = 0.2
P(USS | Dry) = 0.8
McGraw-Hill/Irwin
Decision Tree for the Full Goferbroke Co. Problem
Oil
f
Drill
Dry
c
Sell
Unfavorable
Oil
b
Do seismic survey
g
Drill
Dry
Favorable
d
Sell
a
Oil
h
Drill
No seismic survey
e
Sell
McGraw-Hill/Irwin
Dry
Calculating Joint Probabilities
•
Each combination of a state of nature and a finding will have a joint
probability determined by the following formula:
P(state and finding) = P(state) P(finding | state)
•
P(Oil and FSS) = P(Oil) P(FSS | Oil) = (0.25)(0.6) = 0.15.
•
P(Oil and USS) = P(Oil) P(USS | Oil) = (0.25)(0.4) = 0.1.
•
P(Dry and FSS) = P(Dry) P(FSS | Dry) = (0.75)(0.2) = 0.15.
•
P(Dry and USS) = P(Dry) P(USS | Dry) = (0.75)(0.8) = 0.6.
McGraw-Hill/Irwin
Probabilities of Each Finding
•
Given the joint probabilities of both a particular state of nature and a particular
finding, the next step is to use these probabilities to find each probability of
just a particular finding, without specifying the state of nature.
P(finding) = P(Oil and finding) + P(Dry and finding)
•
P(FSS) = 0.15 + 0.15 = 0.3.
•
P(USS) = 0.1 + 0.6 = 0.7.
McGraw-Hill/Irwin
Calculating the Posterior Probabilities
•
The posterior probabilities give the probability of a particular state of nature,
given a particular finding from the seismic survey.
P(state | finding) = P(state and finding) / P(finding)
•
P(Oil | FSS) = 0.15 / 0.3 = 0.5.
•
P(Oil | USS) = 0.1 / 0.7 = 0.14.
•
P(Dry | FSS) = 0.15 / 0.3 = 0.5.
•
P(Dry | USS) = 0.6 / 0.7 = 0.86.
McGraw-Hill/Irwin
Probability Tree Diagram
Prior
Probabilities
Conditional
Probabilities
Joint
Probabilities
Posterior
Probabilities
P(state)
P(finding | state)
P(state and finding)
P(state | finding)
0.6
FSS, given Oil
0.25
Oil
0.75
Dry
0.4
USS, given Oil
0.2
FSS, given Dry
0.8
USS, given Dry
0.25(0.6) = 0.15
Oil and FSS
0.15 = 0.5
0.3
Oil, given FSS
0.25(0.4) = 0.1
Oil and USS
0.1 = 0.14
0.7
Oil, given USS
0.75(0.2) = 0.15
Dry and FSS
0.15 = 0.5
0.3
Dry, given FSS
0.75(0.8) = 0.6
Dry and USS
0.6 = 0.86
0.7
Dry, given USS
Unconditional probabilities: P(FSS) = 0.15 + 0.15 = 0.3
P(finding)
McGraw-Hill/Irwin
P(USS) = 0.1 + 0.6 = 0.7
Posterior Probabilities
P(state | finding)
Finding
Oil
Dry
Favorable (FSS)
P(Oil | FSS) = 1/2
P(Dry | FSS) = 1/2
Unfavorable (USS)
P(Oil | USS) = 1/7
P(Dry | USS) = 6/7
McGraw-Hill/Irwin
Template for Posterior Probabilities
B
C
3 Data:
Prior
4
State of
Nature
Probability
5
Oil
0.25
6
Dry
0.75
7
8
9
10
11
12 Posterior
13 Probabilities:
Finding
P(Finding)
14
FSS
0.3
15
USS
0.7
16
17
18
19
McGraw-Hill/Irwin
D
E
F
P(Finding | State)
Finding
FSS
0.6
0.2
USS
0.4
0.8
P(State | Finding)
State of Nature
Oil
0.5
0.1429
Dry
0.5
0.8571
G
H
Decision Tree for the Full Goferbroke Co. Problem
Oil
f
Drill
Dry
c
Sell
Unfavorable
Oil
b
Do seismic survey
g
Drill
Dry
Favorable
d
Sell
a
Oil
h
Drill
No seismic survey
e
Sell
McGraw-Hill/Irwin
Dry
Decision Tree with Probabilities and Payoffs
Payoff
f
0
Dry(0.857)
Drill
-100
c
Unfavorable
Sell
90
Oil (0.5)
b
d
a
90
Sell
Drill
No seismic survey
e
Sell
-100
90
-130
670
-130
60
h
0
McGraw-Hill/Irwin
0
Dry (0.5)
Drill
-100
Favorable
(0.3)
-30
800
g
0
670
60
0
Do seismic survey
Oil (0.143)
800
Oil (0.25)
800
0
Dry (0.75)
700
-100
90
The Final Decision Tree
Payoff
-15.7
f
Drill
60
c
Unfavorable
123
b
-30
Favorable (0.3)
123
a
Drill
270
d
Sell
-100
Drill
90
Sell
670
0
-130
60
Oil (0.25)
800
700
0
-100
100
e
800
Dry (0.5)
100
h
No seismic survey
Oil (0.5)
90
0
McGraw-Hill/Irwin
60
270
g
0
670
-130
90
0
Do seismic survey
800
0
Dry (0.857)
-100
Sell
Oil (0.143)
Dry (0.75)
-100
90
TreePlan for the Full Goferbroke Co. Problem
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
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
McGraw-Hill/Irwin
B C
D
E
F G
H
I
J K
L
M
N O
P
Q
R
S
Decision Tree for Goferbroke Co. Problem (With Survey)
0.143
Oil
670
Drill
800
-100 -15.714
0.7
Unfavorable
0.857
Dry
-130
2
0
670
0
-130
60
Sell
60
90
60
Do Survey
0.5
-30
123
Oil
670
Drill
800
-100
270
0.3
Favorable
0.5
Dry
-130
1
0
670
0
-130
270
1
Sell
123
60
90
60
0.25
Oil
700
Drill
800
-100
100
700
0.75
Dry
No Survey
-100
1
0
0
-100
100
Sell
90
90
90
Organizing the Spreadsheet for Sensitivity Analysis
A
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
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
B C
D
E
F G
H
I
J K
L
M
N O
P
Q
R
S
T
U
0.143
Oil
Cost of Survey
Cost of Drilling
Revenue if Oil
Revenue if Sell
Revenue if Dry
Prior Probability Of Oil
P(FSS|Oil)
P(USS|Dry)
670
Drill
800
-100 -15.714
0.7
Unfavorable
0
670
0.857
Dry
-130
2
0
V
-130
W
X
Y
Data
30
100
800
90
0
0.25
0.6
0.8
60
Sell
60
90
Do Survey?
Action
Yes
60
Do Survey
If No
If Yes
0.5
-30
123
Oil
Drill
Drill
Sell
If Favorable
If Unfavorable
Prior
Probability
0.25
0.75
FSS
0.6
0.2
P(Finding | State)
Finding
USS
0.4
0.8
P(Finding)
0.3
0.7
P(State | Finding)
State of Nature
Oil
Dry
0.5
0.5
0.143 0.857
670
Drill
800
-100
270
0.3
Favorable
0.5
Expected Payoff
($thousands)
123
Dry
-130
1
0
670
0
-130
270
1
Sell
123
60
90
60
Data:
State of
Nature
Oil
Dry
0.25
Oil
700
Drill
800
-100
100
700
0.75
Dry
-100
No Survey
1
0
0
-100
100
Sell
90
90
McGraw-Hill/Irwin
90
Posterior
Probabilities:
Finding
FSS
USS
Z
AA
The Plot Option of SensIt
U
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
V
Cost of Survey
Cost of Drilling
Revenue if Oil
Revenue if Sell
Revenue if Dry
Prior Probability Of Oil
P(FSS|Oil)
P(USS|Dry)
Do Survey?
If No
W
Y
Action
Yes
If Yes
Drill
Drill
Sell
Expected Payoff
($thousands)
123
McGraw-Hill/Irwin
X
Data
30
100
800
90
0
0.25
0.6
0.8
If Favorable
If Unfavorable
SensIt Plot
Sensit - Sensitivity Analysis - Plot
700
600
500
400
300
200
100
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Prior Probability Of Oil
McGraw-Hill/Irwin
0.7
0.8
0.9
1
Optimal Policy
Let p = Prior probability of oil
If
p ≤ 0.168, then sell the land (no seismic survey).
If
0.169 ≤ p ≤ 0.308, then do the survey; drill if favorable, sell if not.
If
p ≥ 0.309, then drill for oil (no seismic survey).
McGraw-Hill/Irwin
The Spider Option of SensIt
U
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
V
Cost of Survey
Cost of Drilling
Revenue if Oil
Revenue if Sell
Revenue if Dry
Prior Probability Of Oil
P(FSS|Oil)
P(USS|Dry)
Do Survey?
If No
W
Y
Action
Yes
If Yes
Drill
Drill
Sell
Expected Payoff
($thousands)
123
McGraw-Hill/Irwin
X
Data
30
100
800
90
0
0.25
0.6
0.8
If Favorable
If Unfavorable
SensIt Spider Graph
Sensit - Sensitivity Analysis - Spider
136
134
132
130
128
126
Cost of Survey
Cost of Drilling
Revenue if Oil
Revenue if Sell
124
122
120
118
116
114
112
110
90%
92%
94%
96%
98%
100%
102%
% Change in Input Value
McGraw-Hill/Irwin
104%
106%
108%
110%
The Tornado Option of SensIt
U
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
Cost of Survey
Cost of Drilling
Revenue if Oil
Revenue if Sell
Revenue if Dry
Prior Probability Of Oil
P(FSS|Oil)
P(USS|Dry)
Do Survey?
If No
V
W
X
Y
Data
30
100
800
90
0
0.25
0.6
0.8
Low
28
75
600
85
Base
30
100
800
90
High
32
140
1000
95
Action
Yes
If Yes
Drill
Drill
Sell
Expected Payoff
($thousands)
123
McGraw-Hill/Irwin
If Favorable
If Unfavorable
SensIt Tornado Diagram
Sensit - Sensitivity Analysis - Tornado
Revenue if Oil 600
1000
Cost of Drilling
140
Revenue if Sell
75
85
Cost of Survey
32
90
100
110
120
95
28
130
Expected Payoff
McGraw-Hill/Irwin
140
150
160
Risk Attitude
•
Consider the following coin-toss gambles. How much would you sell each of
these gambles for?
•
Heads: You win $200
Tails: You lose $0
•
Heads: You win $300
Tails: You lose $100
•
Heads: You win $20,000
Tails: You lose $0
•
Heads: You win $30,000
Tails: You lose $10,000
McGraw-Hill/Irwin
Demand for Insurance
•
House Value = $150,000
•
Insurance Premium = $500
•
Probability of fire destroying house (in one year) = 1 / 1,000
Question: Should you buy insurance?
A
2
3
4
5
6
7
8
9
10
11
12
13
14
McGraw-Hill/Irwin
B C
D
E
F G
H
I
J
K
Buy Insurance
-500
-500
-500
2
0.001
-150
Fire
-150000
Self-Insure
-150000
0
-150
-150000
0.999
No Fire
0
0
0
Using Utilities to Better Reflect the Values of Payoffs
•
Thus far, when applying Bayes’ decision rule, we have assumed that the
expected payoff in monetary terms is the appropriate measure.
•
In many situations, this is inappropriate.
•
Suppose an individual is offered the following choice:
– Accept a 50-50 chance of winning $100,000.
– Receive $40,000 with certainty.
•
Many would pick $40,000, even though the expected payoff on the 50-50
chance of winning $100,000 is $50,000. This is because of risk aversion.
•
A utility function for money is a way of transforming monetary values to an
appropriate scale that reflects a decision maker’s preferences (e.g., aversion to
risk).
McGraw-Hill/Irwin
A Typical Utility Function for Money
U(M)
1
0.75
0.5
0.25
0
McGraw-Hill/Irwin
$10,000
$30,000
$60,000
$100,000 M
Shape of Utility Functions
U(M)
U(M)
(a) Risk averse
McGraw-Hill/Irwin
M
U(M)
(b) Risk seeker
M
(c) Risk neutral
M
Creating a Utility Function
(Equivalent Lottery Method)
1. Set U(Min) = 0.
2. Set U(Max) = 1.
3. To find U(x):
Choose p such that you are indifferent between the following:
a) A payment of x for sure.
b) A payment of Max with probability p and a payment of Min with probability 1–p.
4. U(x) = p.
McGraw-Hill/Irwin
Utility Functions
•
When a utility function for money is incorporated into a decision analysis
approach, it must be constructed to fit the current preferences and values of the
decision maker.
•
Fundamental Property: Under the assumptions of utility theory, the decision
maker’s utility function for money has the property that the decision maker is
indifferent between two alternatives if the two alternatives have the same
expected utility.
•
When the decision maker’s utility function for money is used, Bayes’ decision
rule replaces monetary payoffs by the corresponding utilities.
•
The optimal decision (or series of decisions) is the one that maximizes the
expected utility.
McGraw-Hill/Irwin
Illustration of Fundamental Property
By the fundamental property, a decision maker with the utility function belowright will be indifferent between each of the three pairs of alternatives below-left.
U(M)
• 25% chance of $100,000
• $10,000 for sure
Both have E(Utility) = 0.25.
• 50% chance of $100,000
• $30,000 for sure
Both have E(Utility) = 0.5.
• 75% chance of $100,000
• $60,000 for sure
Both have E(Utility) = 0.75.
1
0.75
0.5
0.25
0
McGraw-Hill/Irwin
$10,000
$30,000
$60,000
$100,000 M
The Lottery Procedure
1. We are given three possible monetary payoffs—M1, M2, M3 (M1 < M2 < M3).
The utility is known for two of them, and we wish to find the utility for the
third.
2. The decision maker is offered the following two alternatives:
a) Obtain a payoff of M3 with probability p.
Obtain a payoff of M1 with probability (1–p).
b) Definitely obtain a payoff of M2.
3. What value of p makes you indifferent between the two alternatives?
4. Using this value of p, write the fundamental property equation,
E(utility for a) = E(utility for b)
so
p U(M3) + (1–p) U(M1) = U(M2).
5. Solve this equation for the unknown utility.
McGraw-Hill/Irwin
Procedure for Constructing a Utility Function
1. List all the possible monetary payoffs for the problem, including 0.
2. Set U(0) = 0 and then arbitrarily choose a utility value for one other payoff.
3. Choose three of the payoffs where the utility is known for two of them.
4. Apply the lottery procedure to find the utility for the third payoff.
5. Repeat steps 3 and 4 for as many other payoffs with unknown utilities as
desired.
6. Plot the utilities found on a graph of the utility U(M) versus the payoff M.
Draw a smooth curve through these points to obtain the utility function.
McGraw-Hill/Irwin
Generating the Utility Function for Max Flyer
•
The possible monetary payoffs in the Goferbroke Co. problem are –130, –100,
0, 60, 90, 670, and 700 (all in $thousands).
•
Set U(0) = 0.
•
Arbitrarily set U(–130) = –150.
McGraw-Hill/Irwin
Finding U(700)
•
The known utilities are U(–130) = –150 and U(0) = 0.
The unknown utility is U(700).
•
Consider the following two alternatives:
a) Obtain a payoff of 700 with probability p.
Obtain a payoff of –130 with probability (1–p).
b) Definitely obtain a payoff of 0.
•
What value of p makes you indifferent between these two alternatives?
Max chooses p = 0.2.
•
By the fundamental property of utility functions, the expected utilities of the two
alternatives must be equal, so
pU(700) + (1–p)U(–130) = U(0)
0.2U(700) + 0.8(–150) = 0
0.2U(700) – 120 = 0
0.2U(700) = 120
U(700) = 600
McGraw-Hill/Irwin
Finding U(–100)
•
The known utilities are U(–130) = –150 and U(0) = 0.
The unknown utility is U(–100).
•
Consider the following two alternatives:
a) Obtain a payoff of 0 with probability p.
Obtain a payoff of –130 with probability (1–p).
b) Definitely obtain a payoff of –100.
•
What value of p makes you indifferent between these two alternatives?
Max chooses p = 0.3.
•
By the fundamental property of utility functions, the expected utilities of the two
alternatives must be equal, so
pU(0) + (1–p)U(–130) = U(–100)
0.3(0) + 0.7(–150) = U(–100)
U(–100) = –105
McGraw-Hill/Irwin
Finding U(90)
•
The known utilities are U(700) = 600 and U(0) = 0.
The unknown utility is U(90).
•
Consider the following two alternatives:
a) Obtain a payoff of 700 with probability p.
Obtain a payoff of 0 with probability (1–p).
b) Definitely obtain a payoff of 90.
•
What value of p makes you indifferent between these two alternatives?
Max chooses p = 0.15.
•
By the fundamental property of utility functions, the expected utilities of the two
alternatives must be equal, so
pU(700) + (1–p)U(0) = U(90)
0.15(600) + 0.85(0) = U(90)
U(90) = 90
McGraw-Hill/Irwin
Max’s Utility Function for Money
U(M)
monetary value line
700
600
utility function
500
400
300
200
100
0
-200 -100
-100
-200
McGraw-Hill/Irwin
100
200 300 400
500
Thousands of dollars
600
700
M
Utilities for the Goferbroke Co. Problem
McGraw-Hill/Irwin
Monetary Payoff, M
Utility, U(M)
–130
–150
–100
–105
0
0
60
60
90
90
670
580
700
600
Decision Tree with Utilities
A
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
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
B C
D
E
F G
H
I
J K
L
M
N O
P
Q
R
S
0.143
Oil
580
Drill
580
0
-45.61
0.7
Unfavorable
0.857
Dry
-150
2
0
580
-150
-150
60
Sell
60
60
60
Do Survey
0.5
0
106.5
Oil
580
Drill
580
0
215
0.3
Favorable
-150
-150
-150
215
1
Sell
106.5
McGraw-Hill/Irwin
0.5
Dry
1
0
580
60
60
60
0.25
Oil
600
Drill
600
0
71.25
600
0.75
Dry
No Survey
-105
2
0
-105
-105
90
Sell
90
90
90
Exponential Utility Function
•
The procedure for constructing U(M) requires making many difficult decisions
about probabilities.
•
An alternative approach assumes a certain form for the utility function and
adjusts this form to fit the decision maker as closely as possible.
•
A popular form is the exponential utility function
U(M) = R (1 – e–M/R)
where R is the decision maker’s risk tolerance.
•
An easy way to estimate R is to pick the value that makes you indifferent
between the following two alternatives:
a) A 50-50 gamble where you gain R dollars with probability 0.5 and lose R/2 dollars
with probability 0.5.
b) Neither gain nor lose anything.
McGraw-Hill/Irwin
Using TreePlan with an Exponential Utility Function
•
Specify the value of R in a cell on the spreadsheet.
•
Give the cell a range name of RT (TreePlan refers to this term as the risk
tolerance).
•
Click on the Option button in the TreePlan dialogue box and select the “Use
Exponential Utility Function” option.
McGraw-Hill/Irwin
Decision Tree with an Exponential Utility Function
A
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
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
B C
D
E
F G
H
I
J K
L
M
N O
P
Q
R
S
0.143
Oil
670
Drill
800
-100
0.7
Unfavorable
-57.052
-0.0815
0.857
Dry
-130
2
0
670
0.602
0
60
0.0791
-130
-0.196
Sell
60
90
Do Survey
60
0.0791
0.5
-30
90.0036
0.1163
Oil
670
Drill
800
-100 165.23116
0.203
0.3
Favorable
0.5
Dry
-130
1
0
670
0.602
0
165.231
0.203
1
-130
-0.196
Sell
90
0.1163
60
90
60
0.0791
0.25
Oil
700
Drill
800
-100
32.7511
0.0440
700
0.618
0.75
Dry
No Survey
-100
2
0
0
90
0.11629
-100
-0.147
Sell
90
90
Risk Tolerance (RT)
McGraw-Hill/Irwin
728
90
0.1163
Decisions Under Uncertainty (or Risk)
•
State of nature is uncertain (several possible states)
•
Examples
– Drilling for oil
• Uncertainty: Oil found? How much? How deep? Selling Price?
• Decision: Drill or not?
– Developing a new product
• Uncertainty: R&D Cost, demand, etc.
• Decisions: Design, quantity, produce or not?
– Newsvendor problem
• Uncertainty: Demand
• Decision: Stocking levels
– Producing a movie
• Uncertainty: Cost, gross, etc.
• Decisions: Develop? Arnold or Keanu?
McGraw-Hill/Irwin
Oil Drilling Problem
•
Consider the problem faced by an oil company that is trying to decide whether
to drill an exploratory oil well on a given site.
•
Drilling costs $200,000.
•
If oil is found, it is worth $800,000.
•
If the well is dry, it is worth nothing.
State of Nature
Decision
Wet
Dry
Drill
600
–200
0
0
Do not drill
McGraw-Hill/Irwin
Decision Criteria
State of Nature
Decision
Wet
Dry
Drill
600
–200
0
0
Do not drill
Which decision is best?
•
“Optimist”
•
“Pessimist”
•
“Second–Guesser”
•
“Joe Average”
McGraw-Hill/Irwin
Bayes’ Decision Rule
•
Suppose that the oil company estimates that the probability that the site is
“Wet” is 40%.
State of Nature
Decision
Wet
Dry
Drill
600
–200
0
0
0.4
0.6
Do not drill
Prior Probability
•
Expected value of payoff (Drill) = (0.4)(600) + (0.6)(–200) = 120
•
Expected value of payoff (Do not drill) = (0.4)(0) + (0.6)(0) = 0
Bayes’ Decision Rule: Choose the decision that maximizes the expected payoff (Drill).
McGraw-Hill/Irwin
Features of Bayes’ Decision Rule
•
Accounts not only for the set of outcomes, but also their probabilities.
•
Represents the average monetary outcome if the situation were repeated
indefinitely.
•
Can handle complicated situations involving multiple related risks.
McGraw-Hill/Irwin
Using a Decision Tree to Analyze Oil Drilling Problem
Wet
0.4
600
Dry
0.6
-200
Drill
Do not drill
0
Folding Back:
• At each event node (circle): calculate expected value (SUMPRODUCT of
payoffs and probabilities for each branch).
• At each decision node (square): choose “best” branch (maximum value).
McGraw-Hill/Irwin
Using TreePlan to Analyze Oil Drilling Problem
1. Choose Decision Tree under the Tools menu.
2. Click on “New Tree” and it will draw a default tree with a single decision node
and two branches, as shown below.
A
1
2
3
4
5
6
7
8
9
B C
D
E
F
G
Decision 1
0
0
0
0
0
1
0
Decision 2
0
3. Label each branch. Replace “Decision 1” with “Drill” (cell D2). Replace
“Decision 2” with “Do not drill” (cell D7).
4. To replace the terminal node of the drill branch with an event node, click on
the terminal node (cell F3) and then choose Decision Tree under the Tools
menu. Click on “Change to event node,” choose two branches, then click OK.
McGraw-Hill/Irwin
Using TreePlan to Analyze Oil Drilling Problem
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
B C
D
E
F G
H
I
J
K
0.5
Event 3
0
Drill
0
0
0
0
0.5
Event 4
0
1
0
0
0
Do not drill
0
0
0
5. Change the labels “Event 3” and “Event 4” to “Wet” and “Dry”, respectively.
6. Change the default probabilities (cells H1 and H6) from 0.5 and 0.5 to the
correct values of 0.4 and 0.6.
7. Enter the partial payoffs under each branch: (-200) for “Drill” (D6), 0 for “Do
not drill” (D14), 800 for “Wet” (H4), and 0 for “Dry” (H9). The terminal value
cash flows are calculated automatically from the partial cash flows.
McGraw-Hill/Irwin
Final Decision Tree
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
B C
D
E
F G
H
I
J
K
0.4
Wet
600
Drill
800
-200
120
600
0.6
Dry
-200
1
0
-200
120
McGraw-Hill/Irwin
Do not drill
0
0
0
Features of TreePlan
•
Terminal values (payoff) are calculated automatically from the partial payoffs
(K3 = D6+H4, K8 = D6+H9, K13 = D14).
•
Foldback values are calculated automatically (I4 = K3, I9 = K8, E6 = H1*I4 +
H6*I9, E14 = K13, A10 = Max(E6,E14)).
•
Optimal decisions are indicated inside decision node squares (labeled by
branch number from top to bottom, e.g., branch #1 = Drill, branch #2 = Do not
drill).
•
Changes in the tree can be made by clicking on a node and choosing Decision
Tree under the Tools menu (change type of node, # of branches, etc.)
•
Clicking “Options…” in the Decision Tree dialogue box allows the choice of
Maximize Profit or Minimize Cost.
McGraw-Hill/Irwin
Making Sequential Decisions
•
Consider a pharmaceutical company that is considering developing an
anticlotting drug.
•
They are considering two approaches
– A biochemical approach (more likely to be successful)
– A biogenetic approach (more radical)
•
While the biogenetic approach is not nearly as likely to succeed, if would
likely capture a much larger portion of the market if it did.
R&D Choice
Investment
Outcomes
Profit
(excluding R&D)
Biochemical
$10 million
Large success
Small success
$90 million
$50 million
0.7
0.3
Biogenetic
$20 million
Success
Failure
$200 million
$0 million
0.2
0.8
McGraw-Hill/Irwin
Probability
Biochemical vs. Biogenetic
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
0.7
Large Success
80
Biochemical
-10
90
68
80
0.3
Small Success
40
50
40
1
68
McGraw-Hill/Irwin
0.2
Success
180
Biogenetic
-20
200
20
180
0.8
Failure
-20
0
-20
Simultaneous Development
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
N
O
Market BC
0.14
Large Success (BC), Success (BG)
60
90
60
2
0
170
Market BG
170
200
170
Market BC
0.06
Small Success (BC), Success (BG)
Simultaneous Development
1
72.4
McGraw-Hill/Irwin
50
20
2
0
-30
20
170
72.4
Market BG
170
200
0.56
Large Success (BC), Failure (BG)
170
Market BC
1
0
60
60
90
0.24
Small Success (BC), Failure (BG)
60
Market BC
1
0
20
20
50
20
Biochemical First
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
30
31
32
33
34
35
36
37
38
39
B C
D
E
F G
H
I
J K
L
M
N O
P
Q
R S
T
U
V
W
Market BC
80
90
80
0.7
Large Success (BC)
Market BC
2
0
0.2
Success (BG)
82
60
90
60
2
0
170
Market BG
Pursue BG
170
200
-20
82
0.8
Failure (BG)
Biochemical First
Market BC
1
72.4
McGraw-Hill/Irwin
170
1
-10
72.4
0
60
60
90
60
Market BC
40
50
40
0.3
Small Success (BC)
Market BC
2
0
0.2
Success (BG)
50
20
50
20
2
0
170
Market BG
170
Pursue Biogenetic
200
-20
170
50
0.8
Failure (BG)
Market BC
1
0
20
20
50
20
Biogenetic First
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
30
31
32
33
34
35
36
37
38
39
B C
D
E
F G
H
I
J K
L
M
N O
P
Q
R S
T
U
V
W
Market BC
0.7
Large Success (BC)
60
90
60
2
0
170
Market BG
170
Pursue BC
-10
200
170
170
Market BC
0.3
Small Success (BC)
0.2
Success (BG)
20
2
2
0
20
50
0
170
180
Market BG
170
200
170
Market BG
180
Biogenetic First
200
180
1
74.4
McGraw-Hill/Irwin
-20
74.4
0.7
Large Success (BC)
Market BC
1
Pursue BC
-10
0
48
0.8
Failure (BG)
60
90
0.3
Small Success (BC)
60
Market BC
1
1
0
60
0
20
20
50
20
48
Don't Pursue BC
-20
0
-20
Incorporating New Information
•
Often, a preliminary study can be done to better determine the true state of
nature.
•
Examples:
– Market surveys
– Test marketing
– Seismic testing (for oil)
Question: What is the value of this information?
McGraw-Hill/Irwin
Oil Drilling Problem
Consider again the problem faced by an oil company that is trying to decide
whether to drill an exploratory oil well on a given site. Drilling costs $200,000. If
oil is found, it is worth $800,000. If the well is dry, it is worth nothing. The prior
probability that the site is wet is estimated at 40%.
State of Nature
Decision
Wet
Dry
Drill
600
–200
0
0
0.4
0.6
Do not drill
Prior Probability
• Expected Payoff (Drill) = (0.4)(600) + (0.6)(–200) = 120
• Expected Payoff (Do not drill) = (0.4)(0) + (0.6)(0) = 0
McGraw-Hill/Irwin
Expected Value of Perfect Information (EVPI)
State of Nature
Decision
Wet
Dry
Drill
600
–200
0
0
0.4
0.6
Do not drill
Prior Probability
Suppose they had a test that could predict ahead of time whether the side would be
wet or dry.
• Expected Payoff = (0.4)(600) + (0.6)(0) = 240
• Expected Value of Perfect Information (EVPI)
= Expected Payoff (with perfect info) – Expected Payoff (without info)
= 240 – 120
= 120
McGraw-Hill/Irwin
Using TreePlan to Calculate EVPI
A
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
0.4
600
Wet
600
600
1
0
600
Do not drill
0
0
0
240
McGraw-Hill/Irwin
Drill
0.6
-200
Dry
-200
-200
2
0
0
Do not drill
0
0
0
Imperfect Information (Seismic Test)
Suppose a seismic test is available that would better (but not perfectly) indicate
whether or not the site was wet or dry.
– Good result usually means the site is wet (but not always)
– Bad results usually means the site is dry (but not always)
Record of 100 Past Seismic Test Sites
Actual State of Nature
Seismic
Result
Wet (W)
Dry (D)
Total
Good (G)
30
20
50
Bad (B)
10
40
50
Total
40
60
100
McGraw-Hill/Irwin
Decision Tree with Seismic Test
F G
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
McGraw-Hill/Irwin
H
I
J K
L
M
N O
P
Q
R
S
P(W | G) = ?
Wet
600
Drill
P(D | G) = ?
Dry
P(G) = ?
Good Test (G)
-200
Do not drill
0
P(W | B) = ?
Wet
600
Drill
P(D | B) = ?
Dry
P(B) = ?
Bad Test (B)
-200
Do not drill
0
Conditional Probabilities
•
Actual State of Nature
Seismic
Result
Wet (W)
Dry (D)
Total
Good (G)
30
20
50
Bad (B)
10
40
50
Total
40
60
100
Need probabilities of each test result:
– P(G) = 50 / 100 = 0.5
– P(B) = 50 / 100 = 0.5
•
Need conditional probabilities of each state of nature, given a test result:
–
–
–
–
P(W | G) = 30 / 50 = 0.6
P(D | G) = 20 / 50 = 0.4
P(W | B) = 10 / 50 = 0.2
P(D | B) = 40 / 50 = 0.8
McGraw-Hill/Irwin
Expected Value of Sample Information (EVSI)
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
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
B C
D
E
F G
H
I
J K
L
M
N O
P
Q
R
S
0.6
Wet
600
Drill
800
-200
280
0.5
Good Test (G)
0.4
Dry
-200
1
0
600
0
-200
280
Do not drill
0
0
0
Expected Value of
Sample Information
Do Seismic Test
0.2
0
140
Wet
600
Drill
800
-200
-40
0.5
Bad Test (B)
= 140 – 120
0.8
Dry
-200
2
0
600
0
-200
0
1
Do not drill
140
0
0
0
0.4
Wet
600
Drill
800
-200
120
600
0.6
Dry
Forego test
-200
1
0
0
-200
120
Do not drill
0
0
McGraw-Hill/Irwin
= EVSI
0
= 20.
Revising Probabilities
•
Suppose they don’t have the “Record of Past 100 Seismic Test Sites”.
•
Vendor of test certifies:
– Wet sites test “good” three quarters of the time.
– Dry sites test “bad” two thirds of the time
P(G | W) = 3/4
P(B | W) = 1/4
P(B | D) = 2/3
P(G | D) = 1/3
Is this the information needed in the decision tree?
McGraw-Hill/Irwin
Revising Probabilities (Probability Tree Diagram)
Prior
Probabilities
Conditional
Probabilities
Joint
Probabilities
P(state)
P(finding | state)
P(finding & state)
Good, given Wet
Good and Wet
(0.4)(0.75) = 0.3
Wet, given Good
0.3 / 0.5 = 0.6
Bad and Wet
(0.4)(0.25) = 0.1
Wet, given Bad
0.1 / 0.5 = 0.2
0.75
Wet
0.4
0.6
Dry
Posterior
Probabilities
P(State | Finding)
0.25
Bad, given Wet
Good, given Dry
Good and Dry Dry, given Good
(0.6)(0.33) = 0.2
0.2 / 0.5 = 0.4
0.333
0.667
Bad, given Dry
Bad and Dry
(0.6)(0.67) = 0.2
Dry, given Bad
0.4 / 0.5 = 0.8
P(Good) = 0.3 + 0.2 = 0.5
P(Bad) = 0.1 + 0.4 = 0.5
McGraw-Hill/Irwin
Template for Posterior Probabilities
B
3 Data:
4
State of
5
Nature
6
Wet
7
Dry
8
9
10
11
12 Posterior
13 Probabilities:
14
Finding
15
Good
16
Bad
17
18
19
C
Prior
Probability
0.4
0.6
P(Finding)
0.5
0.5
D
F
Good
0.75
0.333
P(Finding | State)
Finding
Bad
0.25
0.667
Wet
0.6
0.2
P(State | Finding)
State of Nature
Dry
0.4
0.8
Template available on textbook CD.
McGraw-Hill/Irwin
E
G
H
Risk Attitude
•
Consider the following coin-toss gambles. How much would you sell each of
these gambles for?
•
Heads: You win $200
Tails: You lose $0
•
Heads: You win $300
Tails: You lose $100
•
Heads: You win $20,000
Tails: You lose $0
•
Heads: You win $30,000
Tails: You lose $10,000
McGraw-Hill/Irwin
Demand for Insurance
•
House Value = $150,000
•
Insurance Premium = $500
•
Probability of fire destroying house (in one year) = 1 / 1,000
Question: Should you buy insurance?
A
2
3
4
5
6
7
8
9
10
11
12
13
14
McGraw-Hill/Irwin
B C
D
E
F G
H
I
J
K
Buy Insurance
-500
-500
-500
2
0.001
-150
Fire
-150000
Self-Insure
-150000
0
-150
-150000
0.999
No Fire
0
0
0
Utilities and Risk Aversion
Utility
1.00
Utility Curve
0.75
0.50
0.25
0
-200
-120
0
200
Monetary Values (Thousands of Dollars)
McGraw-Hill/Irwin
600
Payoff
Utility
$600,000
1.0
200,000
0.75
0
0.50
–120,000
0.25
–200,000
0
Oil Drilling Problem (Risk Aversion)
Risk Neutral:
Risk Averse:
Wet
0.4
$600
0.4
Dry
0.6
1
U($600)= 1
Dry
0.6
U(-$200) = 0
Drill
Drill
120
Wet
0.4
- $200
2
0.5
120
Do not drill
McGraw-Hill/Irwin
$0
Do not drill
U($0) = 0.5
Creating a Utility Function
(Equivalent Lottery Method)
1. Set U(Min) = 0.
2. Set U(Max) = 1.
3. To find U(x):
Choose p such that you are indifferent between the following:
a) A payment of x for sure.
b) A payment of Max with probability p and a payment of Min with probability 1–p.
4. U(x) = p.
McGraw-Hill/Irwin
Equivalent Lottery Method
•
Uncertain situation:
–$200 in worst case
$1,800 in best case
U(–$200) = 0
U($1,800) = 1
p
$1800
U($1800) = 1
-$200
U(-$200) = 0
Gamble
EU = p
1-p
•
U($800) =
•
U($200) =
•
U($400) =
•
U($600) =
McGraw-Hill/Irwin
Certain Equivalent
$x
U=?
Utility Curve
Utility
1.0
0.8
0.6
0.4
0.2
0
-$200
$200
$600
$1000
Monetary Value
McGraw-Hill/Irwin
$1400
$1800
Biochemical vs. Biogenetic First (Expected Payoff)
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
N O
P
Q
R
S
0.2
Success
180
200
180
0.7
Large Success
Biogenetic First
60
-20
74.4
Pursue Biochemical
90
-10
48
0.8
60
0.3
Small Success
Failure
20
1
0
50
20
48
1
74.4
Don't Pursue Biochemical
-20
0
-20
0.7
Large Success
80
Biochemical
-10
90
68
80
0.3
Small Success
40
50
McGraw-Hill/Irwin
40
Biochemical vs. Biogenetic First (with Utilities)
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
N O
P
Q
R
S
0.2
Success
1
200
1
0.7
Large Success
Biogenetic First
0.7
-20
0.688
Pursue Biochemical
-10
90
0.61
0.8
0.7
0.3
Small Success
Failure
0.4
1
0
50
0.4
0.61
2
0.74
Don't Pursue Biochemical
0
0
0
0.7
Large Success
0.8
Biochemical
-10
90
0.74
0.8
0.3
Small Success
0.6
50
McGraw-Hill/Irwin
0.6
Exponential Utility Function
Choose R so that you are indifferent between the following:
Gamble
$R
0.5
-$R/2
0.5
Certain Equivalent
U(M) = R(1 – e–M / R)
McGraw-Hill/Irwin
$0
Exponential Utility Function
U(M) = R(1 – e–M / R)
Utility
0
McGraw-Hill/Irwin
Monetary Value
Using an Exponential Utility Function with TreePlan
•
To use an exponential utility function in TreePlan, enter the R value in a cell
on the spreadsheet
•
Give this cell the range name RT (TreePlan calls this value the risk tolerance).
•
Choose “Use Exponential Utility Function” in the dialogue box shown below
(available by clicking on “Options…” in the Decision Tree dialogue box).
McGraw-Hill/Irwin
Biochemical vs. Biogenetic First
(with Exponential Utility)
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
30
31
32
B C
D
E
F G
H
I
J K
L
M
N O
P
Q
R
S
0.2
Success
180
200
180
0.8347
0.7
Large Success
Biogenetic First
60
-20
62.1963
0.46311
Pursue Biochemical
-10
0.8
46.237
0.3702
90
60
0.45119
0.3
Small Success
Failure
20
1
0
2
46.2373
0.37021
66.2373
0.48437
McGraw-Hill/Irwin
50
20
0.18127
Don't Pursue Biochemical
-20
0
-20
-0.2214
0.7
Large Success
80
Biochemical
-10
90
66.2373
0.48437
80
0.55067
0.3
Small Success
40
50
RT =
100
40
0.32968