Chapter 9
DECISION ANALYSIS
CHAPTER 9
DECISION ANALYSIS
Review Questions
9.1-1 The decision alternatives are to drill for oil or to sell the land.
9.1-2 The consulting geologist believes that there is 1 chance in 4 of oil on the tract of land.
9.1-3 Max does not put much faith in the assessment.
9.1-4 A detailed seismic survey of the land could be done to obtain more information.
9.1-5 The possible states of nature are the possible outcomes of the random factors that affect the
payoff that would be obtained from a decision alternative.
9.1-6 Prior probabilities are the estimated probabilities of the states of nature prior to obtaining
additional information through a test or survey.
9.1-7 The payoffs are quantitative measures of the outcomes from a decision alternative and a
state of nature. Payoffs are generally expressed in monetary terms.
9.2-1 The maximax criterion identifies the maximum payoff for each decision alternative and
chooses the decision alternative with the maximum of these maximum payoffs. The
maximax criterion is for the eternal optimist.
9.2-2 The maximax citerion completely ignores the prior probabilities and ignores all payoffs
except for the largest one.
9.2-3 The maximin criterion identifies the minimum payoff for each decision alternative and
chooses the decision alternative with the maximum of these minimum payoffs. The
maximin criterion is for the total pessimist.
9.2-4 The maximin criterion ignores the prior probabilities and ignores all payoffs except the
maximin payoff.
9.2-5 The maximum likelihood criterion focuses on the most likely state of nature, the one with
the largest prior probability.
9.2-6 Criticisms of the maximum likelihood criterion include: 1) this criterion chooses an
alternative without considering its payoffs for states of nature other than the most likely
one, 2) for alternatives that are not chosen, this criterion ignores their payoffs for states of
nature other than the most likely one, 3) if the differences in the payoffs for the most likely
state of nature are much less than for another somewhat likely state of nature, then it might
make sense to focus on this latter state of nature instead, and 4) if there are many states of
nature and they are nearly equally likely, then the probability that the most likely state of
nature will be the true one is fairly low.
9.2-7 Bayes’ decision rule says to choose the alternative with the largest expected payoff.
Chapter 9 - Page 1
Chapter 9
DECISION ANALYSIS
9.2-8 The expected payoff is calculated by multiplying each payoff by the prior probability of the
corresponding state of nature and then summing these products.
9.2-9 Criticisms of Bayes’ decision rule include: 1) there usually is considerable uncertainty
involved in assigning values to prior probabilities, 2) prior probabilities inherently are at
least largely subjective in nature, whereas sound decision making should be based on
objective data and procedures, and 3) by focusing on average outcomes, expected payoffs
ignore the effect that the amount of variability in the possible outcomes should have on the
decision making.
9.3-1 A decision tree is a graphical display of the progression of decisions and random events to
be considered.
9.3-2 A decision node indicates that a decision needs to be made at that point in the process. An
event node indicates that a random event occurs at that point.
9.3-3 Decision nodes are represented by squares while circles represent event nodes.
9.4-1 Sensitivity analysis might be helpful to study the effect if some of the numbers included in
the model are not correct.
9.4-2 It assures that each piece of data is in only one place and it makes it easy for anyone to
interpret the model, even if they don’t understand TreePlan or decision trees.
9.4-3 A data table displays results of selected output cells for various trial values of a data cell.
9.4-4 If there is less than a 23.75% chance of oil, they should sell. If it’s more, they should drill.
9.5-1 Perfect information means knowing for sure which state of nature is the true state of nature.
9.5-2 The expected payoff with perfect information is calculated by multiplying the maximum
payoff for each alternative by the prior probability of the corresponding state of nature.
9.5-3 The decision tree should be started with a chance node whose branches are the various
states of nature.
9.5-4 EVPI = EP (with perfect information) – EP (without more information)
9.5-5 If the cost of obtaining more information is more than the expected value of perfect
information then it is not worthwhile to obtain more information.
9.5-6 If the cost of obtaining more information is less than the expected value of perfect
information then it might be worthwhile to obtain more information.
9.5-7 In the Goferbroke problem the EVPI >C so it might be worthwhile to do the seismic
survey.
9.6-1 Posterior probabilities are revised probabilities of the states of nature after doing a test or
survey to improve the prior probabilities.
9.6-2 The possible findings are favorable with oil being fairly likely, or unfavorable with oil
being quite unlikely.
9.6-3 Conditional probabilities need to be estimated.
Chapter 9 - Page 2
Chapter 9
DECISION ANALYSIS
9.6-4 The five kinds of probabilities considered are prior, conditional, joint, unconditional, and
posterior.
9.6-5 P(state and finding) = P(state) * P(finding | state).
9.6-6 P(finding) = sum of P(state and finding) for each state.
9.6-7 P(state | finding) = P(state and finding) / P(finding).
9.6-8 Bayes’ theorem is used to calculate posterior probabilities.
9.7-1 A decision tree provides a graphical display of the progression of decisions and random
events for a problem.
9.7-2 A decision needs to be made at a decision node.
9.7-3 A random event will occur at a event node.
9.7-4 The probabilities of random events and the payoffs need to be inserted before beginning
analysis.
9.7-5 When performing the analysis, start at the right side of the decision tree and move left one
column at a time.
9.7-6 For each event node, calculate its expected payoff by multiplying the payoff of each branch
by the probability of that branch and then summing these products.
9.7-7 For each decision node, compare the expected payoffs of its branches and choose the
alternative whose branch has the largest expected payoff.
9.7-8 The expected value of sample information (EVSI) is the change in expected value that is
obtained when sample information is obtained. If the EVSI is greater than the cost of
obtaining the information, then it would be worthwhile to obtain the information.
9.8-1 Consolidate the data and results into one section of the spreadsheet.
9.8-2 Performing sensitivity analysis on a piece of data should require changing a value in only
one place on the spreadsheet.
9.8-3 A data table can consider changes in only one or two data cells.
9.8-4 One.
9.8-5 Yes. The spider graph can consider changes in many data cells at a time.
9.8-6 SensIt’s spider graph assumes that each data value varies by the same amount. Sensit’s
tornado diagram overcomes this limitation.
9.9-1 Utilities are intended to reflect the true value of an outcome to the decision-maker.
Chapter 9 - Page 3
Chapter 9
DECISION ANALYSIS
9.9-2
U(M)
U(M)
U(M)
M
Risk averse
M
Risk seeker
M
Risk neutral
9.9-3 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 alternative courses of action
if the two alternatives have the same expected utility.
9.9-4 The decision-maker is offered a lottery with the maximum payoff with probability p and the
minimum payoff with probability 1–p, as compared to a definite payoff of M.
9.9-5 The point of indifference is the value of p where the decision-maker is indifferent between
the two hypothetical alternatives.
9.9-6 The value obtained to evaluate each node of the tree is the expected utility.
9.9-7 Max decided to do the seismic survey and to sell if the result is unfavorable or drill if the
result is favorable.
9.10-1
The Goferbroke problem contained the same elements as typical applications of decision
analysis but is oversimplified.
9.10-2
An influence diagram complements the decision tree for representing and analyzing
decision analysis problems.
9.10-3
Typical participants include management, an analyst, and a group facilitator.
9.10-4
A manager can go to a management consulting firm that specializes in decision analysis.
Chapter 9 - Page 4
Chapter 9
DECISION ANALYSIS
Problems
9.1
a) Max(A1) = 6, Max(A2) = 4, Max(A3) = 8. Maximax = 8 with alternative A3.
b) Min(A1) = 2, Min(A2) = 3, Min(A3) = 1. Maximin = 3 with alternative A2.
9.2
a) Max(A1) = 30, Max(A2) = 31, Max(A3) = 22, Max(A4) = 29. Maximax = 31 with A2.
b) Min(A1) = 20, Min(A2) = 14, Min(A3) = 22, Min(A4) = 21. Maximin = 22 with A3.
Chapter 9 - Page 5
Chapter 9
DECISION ANALYSIS
9.3
a)
Alternative
Buy 10 cases
Buy 11 cases
Buy 12 cases
Buy 13 cases
Prior
Probability
State of Nature
Sell 11 cases
Sell 12
cases
$50
$50
$55
$55
$52
$60
$49
$57
0.4
0.3
Sell 10
cases
$50
$47
$44
$41
0.2
Sell 13
cases
$50
$55
$60
$65
0.1
b) Max(Buy 10) = $50, Max(Buy 11) = $55, Max(Buy 12) = $60, Max(Buy 13) = $65.
Maximax = $65 with buying 13 cases.
c) Min(Buy 10) = $50, Min(Buy 11) = $47, Min(Buy 12) = $44, Min(Buy 13) = $41.
Maximin = $50 with buying 10 cases.
d) The most likely state of nature is to sell 11 cases. Under this state, she should buy 11
cases with a payoff of $55.
e)
A
B
1
Purchase Price
2
Selling Price
3
4 Payoff Table
5
6
Alternative
7
Buy 10
8
Buy 11
9
Buy 12
10
Buy 13
11
12
Prior Probability
C
$3
$8
Sell
10
$50
$47
$44
$41
0.2
D
E
State of Nature
Sell
Sell
11
12
$50
$50
$55
$55
$52
$60
$49
$57
0.4
0.3
F
G
Sell
13
$50
$55
$60
$65
0.1
Jean should buy 12 cases. The maximum expected payoff is $53.60.
Chapter 9 - Page 6
H
Expected
Payoff
$50.00
$53.40
$53.60
$51.40
Chapter 9
DECISION ANALYSIS
f) (i) 0.2 and 0.5
A
B
1
Purchase Price
2
Selling Price
3
4 Payoff Table
5
6
Alternative
7
Buy 10
8
Buy 11
9
Buy 12
10
Buy 13
11
12
Prior Probability
C
$3
$8
Sell
10
$50
$47
$44
$41
0.2
D
E
State of Nature
Sell
Sell
11
12
$50
$50
$55
$55
$52
$60
$49
$57
0.2
0.5
F
G
Sell
13
$50
$55
$60
$65
H
Expected
Payoff
$50.00
$53.40
$55.20
$53.00
0.1
Jean should purchase 12 cases. The maximum expected payoff is $55.20.
(ii) 0.3 and 0.4
A
B
Purchase Price
Selling Price
1
2
3
4 Payoff Table
5
6
Alternative
7
Buy 10
8
Buy 11
9
Buy 12
10
Buy 13
11
12
Prior Probability
C
$3
$8
Sell
10
$50
$47
$44
$41
0.2
D
E
State of Nature
Sell
Sell
11
12
$50
$50
$55
$55
$52
$60
$49
$57
0.3
0.4
F
G
Sell
13
$50
$55
$60
$65
H
Expected
Payoff
$50.00
$53.40
$54.40
$52.20
0.1
Jean should purchase 12 cases. The maximum expected payoff is $54.40.
(iii) 0.5 and 0.2
A
B
1
Purchase Price
2
Selling Price
3
4 Payoff Table
5
6
Alternative
7
Buy 10
8
Buy 11
9
Buy 12
10
Buy 13
11
12
Prior Probability
C
$3
$8
Sell
10
$50
$47
$44
$41
0.2
D
E
State of Nature
Sell
Sell
11
12
$50
$50
$55
$55
$52
$60
$49
$57
0.5
0.2
F
G
Sell
13
$50
$55
$60
$65
0.1
Jean should purchase 11 cases. The maximum expected payoff is $53.40.
Chapter 9 - Page 7
H
Expected
Payoff
$50.00
$53.40
$52.80
$50.60
Chapter 9
DECISION ANALYSIS
9.4
a) Max(Conservative) = $30 million
Max(Speculative) = $40 million
Max(Countercyclical) = $15 million
Maximax = $40 million with the speculative investment
b) Min(Conservative) = –$10 million
Min(Speculative) = –$30 million
Min(Countercyclical) = –$10 million
Maximin = –$10 million with either the conservative of countercyclical investment.
c) The stable economy is the most likely state of nature.
The speculative investment has the maximum payoff for this state ($10 million).
d) The countercyclical investment has the maximum expected payoff of $5 million.
1
2
3
4
5
6
7
8
A
Payoff Table ($million)
Alternative
(Investment)
Conservative
Speculative
Countercyclical
Prior Probability
B
C
D
State of Nature (Economy)
Improving
Stable
Worsening
30
5
-10
40
10
-30
-10
0
15
0.1
Chapter 9 - Page 8
0.5
0.4
E
F
Expected
Payoff
($million)
1.5
-3
5
Chapter 9
DECISION ANALYSIS
9.5
a) The countercyclical investment has the maximum expected payoff of $8 million.
1
2
3
4
5
6
7
8
A
Payoff Table ($million)
Alternative
(Investment)
Conservative
Speculative
Countercyclical
Prior Probability
B
C
D
E
State of Nature (Economy)
Improving
Stable
Worsening
30
5
-10
40
10
-30
-10
0
15
0.1
0.3
F
Expected
Payoff
($million)
-1.5
-11
8
0.6
b) The speculative investment has the maximum expected payoff of $5 million.
1
2
3
4
5
6
7
8
A
Payoff Table ($million)
Alternative
(Investment)
Conservative
Speculative
Countercyclical
Prior Probability
B
C
D
State of Nature (Economy)
Improving
Stable
Worsening
30
5
-10
40
10
-30
-10
0
15
0.1
Chapter 9 - Page 9
0.7
0.2
E
F
Expected
Payoff
($million)
4.5
5
2
Chapter 9
DECISION ANALYSIS
c&d)
0.1
Improving
30
30
30
0.7
Stable
Conservative
5
0
4.5
5
5
0.2
Worsening
-10
-10
-10
0.1
Improving
40
40
40
0.7
Stable
Speculative
2
5
10
0
5
10
10
0.2
Worsening
-30
-30
-30
0.1
Improving
-10
-10
-10
0.7
Stable
Counter-cyclical
0
0
2
0
0
0.2
Worsening
15
15
Chapter 9 - Page 10
15
Chapter 9
DECISION ANALYSIS
e)
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
B C
D
E
F G
H
0.1
Improving
I
J
K
L
30
30
30
0.5
Stable
Conservative
M
Payoff Table ($million)
Alternative
(Investment)
Conservative
Speculative
Countercyclical
N
O
P
State of Nature (Economy)
Improving
Stable
Worsening
30
5
-10
40
10
-30
-10
0
15
5
0
1.5
5
5
Prior Probability
0.4
Worsening
Investment
-10
-10
-10
0.1
Improving
40
40
40
0.5
Stable
Speculative
3
5
10
0
-3
10
10
0.4
Worsening
-30
-30
-30
0.1
Improving
-10
-10
-10
0.5
Stable
Countercyclical
0
0
5
0
0
0.4
Worsening
15
15
15
Chapter 9 - Page 11
Expected Payoff
($million)
0.1
Countercyclical
5
0.5
0.4
Chapter 9
DECISION ANALYSIS
f) Part a)
M
N
O
P
2 Payoff Table ($million)
3
Alternative
State of Nature (Economy)
4
(Investment)
Improving
Stable
Worsening
5
Conservative
30
5
-10
6
Speculative
40
10
-30
7
Countercyclical
-10
0
15
8
9
Prior Probability
0.1
0.3
0.6
10
11
Investment
Countercyclical
12
13
Expected Payoff
8
14
($million)
Part b)
2
3
4
5
6
7
8
9
10
11
12
13
14
M
Payoff Table ($million)
Alternative
(Investment)
Conservative
Speculative
Countercyclical
Prior Probability
Investment
Expected Payoff
($million)
N
O
State of Nature (Economy)
Improving
Stable
Worsening
30
5
-10
40
10
-30
-10
0
15
0.1
0.7
Speculative
5
g)
M
16
17
18
19
20
21
22
23
24
25
26
27
28
29
Probability of
Stable Economy
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
P
N
Optimal
Investment
Speculative
Countercyclical
Countercyclical
Countercyclical
Countercyclical
Countercyclical
Countercyclical
Countercyclical
Speculative
Speculative
Speculative
Chapter 9 - Page 12
O
Expected
Payoff
($million)
5
12.5
11
9.5
8
6.5
5
3.5
5
9
13
0.2
Chapter 9
DECISION ANALYSIS
h)
15
10
5
0
0
Expected Payoff
-5
($million)
0.2
0.4
0.6
0.8
1
-10
Conservative
Speculative
Countercyclical
-15
-20
-25
Probability of Stable Economy
Counter-cyclical and conservative cross at approximately p=0.62.
Conservative and speculative cross at approximately p = 0.68.
9.6
a) Max(A1) = 80, Max(A2) = 50, Max(A3) = 60.
Maximax = $80 thousand when choosing alternative A1.
b) Min(A1) = 25, Min(A2) = 30, Min(A3) = 40.
Maximin = $40 thousand when choosing alternative A3.
c) S2 is the most likely outcome. For this state, the maximum payoff of $50 thousand
occurs with alternative A2.
d) Alternative A3 has the highest expected payoff of $48 thousand.
1
2
3
4
5
6
7
8
A
Payoff Table ($thousand)
Alternative
A1
A2
A3
Prior Probability
B
C
State of Nature
S1
S2
80
25
30
50
60
40
0.4
0.6
Chapter 9 - Page 13
D
E
Expected
Payoff
($thousand)
47
42
48
Chapter 9
DECISION ANALYSIS
e)
0.4
State 1
80
Alternative 1
0
80
47
80
0.6
State 2
25
25
25
0.4
State 1
30
Alternative 2
30
30
3
48
0
42
0.6
State 2
50
50
50
0.4
State 1
60
Alternative 3
0
60
48
60
0.6
State 2
40
40
Chapter 9 - Page 14
40
Chapter 9
DECISION ANALYSIS
f) When the prior probability of S1 is 0.2, alternative A2 should be chosen, with an
expected payoff of $46 thousand.
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
B C
D
E
F G
H
I
J
K
L
M
N
O
0.2
State 1
Payoff Table ($thousand)
80
Alternative 1
80
80
36
0
Alternative
A1
A2
A3
0.8
State 2
State of Nature
S1
S2
80
25
30
50
60
40
25
25
25
0.2
Prior Probability
0.2
Best Alternative
2
Expected Payoff
($thousand)
46
0.8
State 1
30
Alternative 2
30
30
2
46
46
0
0.8
State 2
50
50
50
0.2
State 1
60
Alternative 3
60
60
0
44
0.8
State 2
40
40
40
When the prior probability of S1 is 0.6, alternative A1 should be chosen, with an
expected payoff of $58 thousand.
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
B C
D
E
F G
H
I
J
K
L
M
N
O
0.6
State 1
Payoff Table ($thousand)
80
Alternative 1
0
80
58
80
Alternative
A1
A2
A3
0.4
State 2
State of Nature
S1
S2
80
25
30
50
60
40
25
25
25
0.6
Prior Probability
0.6
Best Alternative
1
Expected Payoff
($thousand)
58
State 1
30
Alternative 2
30
30
1
58
0
38
0.4
State 2
50
50
50
0.6
State 1
60
Alternative 3
0
60
52
60
0.4
State 2
40
40
40
Chapter 9 - Page 15
0.4
Chapter 9
DECISION ANALYSIS
g)
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
M
Prior
Probability
of S 1
0.20
0.24
0.28
0.32
0.36
0.40
0.44
0.48
0.52
0.56
0.60
N
Best
Alternative
2
2
2
3
3
3
3
1
1
1
1
1
O
Expected
Payoff
($thousand)
46
46
45.2
45.6
46.4
47.2
48
49.2
51.4
53.6
55.8
58
Chapter 9 - Page 16
Chapter 9
DECISION ANALYSIS
9.7
a) Max(A1) = $220 thousand, Max(A2) = $200 thousand.
Maximax = $220 thousand when choosing alternative A1.
b) Min(A1) = $110 thousand, Min(A2) = $150 thousand.
Maximin = $150 thousand when choosing alternative A2.
c) S1 is the most likely outcome. For this state, the maximum payoff of $220 thousand
occurs with alternative A1.
d) Alternative A1 has the highest expected payoff of $194 thousand.
1
2
3
4
5
6
7
A
Payoff Table ($thousand)
B
C
D
Alternative
A1
A2
S1
220
200
State of Nature
S2
170
180
S3
110
150
Prior Probability
0.6
0.3
0.1
E
F
Expected
Payoff
($thousand)
194
189
e & f)
0.6
State 1
220
220
220
0.3
State 2
Alternative 1
170
0
194
170
170
0.1
State 3
110
110
110
1
194
0.6
State 1
200
200
200
0.3
State 2
Alternative 2
180
0
189
180
180
0.1
State 3
150
150
Chapter 9 - Page 17
150
Chapter 9
DECISION ANALYSIS
g)
16
17
18
19
20
21
22
23
24
25
26
27
28
M
Prior
Probability
of S 1
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
N
Best
Alternative
1
2
2
2
1
1
1
1
1
1
O
Expected
Payoff
($thousand)
194
183
184
185
186.5
189
191.5
194
196.5
199
Let p = prior probability of S1. A1 and A2 cross when p is between 0.40 and 0.45. A
little experimentation reveals that the best alternative changes at p=0.433. They should
choose A2 when p ≤ 0.433, A1 when p > 0.433.
h)
16
17
18
19
20
21
22
23
24
25
26
27
28
M
Prior
Probability
of S 1
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
N
Best
Alternative
1
2
2
2
2
2
1
1
1
1
O
Expected
Payoff
($thousand)
194
174
176.5
179
181.5
184
188.5
194
199.5
205
Let p = prior probability of S1. A1 and A2 cross when p is between 0.50 and 0.55. A
little experimentation reveals that the best alternative changes at p=0.517. They should
choose A2 when p ≤ 0.517, A1 when p > 0.517.
Chapter 9 - Page 18
Chapter 9
DECISION ANALYSIS
i)
16
17
18
19
20
21
22
23
24
25
26
27
28
M
Prior
Probability
of S 2
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
N
Best
Alternative
1
2
2
2
1
1
1
1
1
1
O
Expected
Payoff
($thousand)
194
180
181.5
183
185
188
191
194
197
200
Let p = prior probability of S2. A1 and A2 cross when p is between 0.10 and 0.15. A
little experimentation reveals that the best alternative changes at p=0.517. They should
choose A2 when p ≤ 0.133, A1 when p > 0.133.
j) Alternative A1 should be chosen.
Chapter 9 - Page 19
Chapter 9
DECISION ANALYSIS
9.8
a)
Alternative
Crop 1
Crop 2
Crop 3
Crop 4
Prior
Probability
State of Nature (Weather)
Dry
Moderate
Damp
20
35
40
22.5
30
45
30
25
25
20
20
20
0.3
0.5
0.2
Chapter 9 - Page 20
Chapter 9
DECISION ANALYSIS
b) Crop 1 has the highest expected payoff of $31,500.
0.3
Dry
20
20
20
0.5
Moderate
Crop 1
35
0
31.5
35
35
0.2
Damp
40
40
40
0.3
Dry
22.5
22.5
22.5
0.5
Moderate
Crop 2
30
0
30.75
30
30
0.2
Damp
45
45
45
1
31.5
0.3
Dry
30
30
30
0.5
Moderate
Crop 3
25
0
26.5
25
25
0.2
Damp
25
25
25
0.3
Dry
20
20
20
0.5
Moderate
Crop 4
20
0
20
20
20
0.2
Damp
20
20
20
Chapter 9 - Page 21
Chapter 9
DECISION ANALYSIS
c) When the prior probability of moderate weather is 0.2, Crop 2 has the highest expected
payoff of $35,250.
1
2
3
4
5
6
7
8
9
A
Payoff Table ($thousand)
Alternative
Crop 1
Crop 2
Crop 3
Crop 4
Prior Probability
B
C
D
E
State of Nature (Weather)
Dry
Moderate
Damp
20
35
40
22.5
30
45
30
25
25
20
20
20
0.3
0.2
F
Expected
Payoff
($thousand)
33
35.25
26.5
20
0.5
When the prior probability of moderate weather is 0.3, Crop 2 has the highest expected
payoff of $33,750.
1
2
3
4
5
6
7
8
9
A
Payoff Table ($thousand)
Alternative
Crop 1
Crop 2
Crop 3
Crop 4
Prior Probability
B
C
D
E
State of Nature (Weather)
Dry
Moderate
Damp
20
35
40
22.5
30
45
30
25
25
20
20
20
0.3
0.3
F
Expected
Payoff
($thousand)
32.5
33.75
26.5
20
0.4
When the prior probability of moderate weather is 0.4, Crop 2 has the highest expected
payoff of $32,250.
1
2
3
4
5
6
7
8
9
A
Payoff Table ($thousand)
Alternative
Crop 1
Crop 2
Crop 3
Crop 4
Prior Probability
B
C
D
State of Nature (Weather)
Dry
Moderate
Damp
20
35
40
22.5
30
45
30
25
25
20
20
20
0.3
0.4
Chapter 9 - Page 22
0.3
E
F
Expected
Payoff
($thousand)
32
32.25
26.5
20
Chapter 9
DECISION ANALYSIS
When the prior probability of moderate weather is 0.6, Crop 1 has the highest expected
payoff of $31,000.
1
2
3
4
5
6
7
8
9
A
Payoff Table ($thousand)
Alternative
Crop 1
Crop 2
Crop 3
Crop 4
Prior Probability
B
C
D
State of Nature (Weather)
Dry
Moderate
Damp
20
35
40
22.5
30
45
30
25
25
20
20
20
0.3
0.6
Chapter 9 - Page 23
0.1
E
F
Expected
Payoff
($thousand)
31
29.25
26.5
20
Chapter 9
DECISION ANALYSIS
9.9
When x = 50, alternative A3 has the highest expected payoff of $5,600.
A
1
x=
2
3 Payoff Table ($hundred)
4
5
Alternative
A1
6
A2
7
A3
8
9
10
Prior Probability
B
50
C
D
S1
100
25
35
State of Nature
S2
50
40
150
S3
10
90
30
0.4
0.2
0.4
E
F
Expected
Payoff
($hundred)
54
54
56
When x = 75, alternative A1 has the highest expected payoff of $7,400.
A
1
x=
2
3 Payoff Table ($hundred)
4
5
Alternative
A1
6
A2
7
A3
8
9
10
Prior Probability
B
75
C
D
S1
150
25
35
State of Nature
S2
50
40
225
S3
10
90
30
0.4
0.2
0.4
E
Barbara Miller should pay a maximum of $1,800 to increase x to 75.
Chapter 9 - Page 24
F
Expected
Payoff
($hundred)
74
54
71
Chapter 9
DECISION ANALYSIS
9.10
a) Alternative A2 has the highest expected payoff of $1,000.
1
2
3
4
5
6
7
8
A
B
Payoff Table ($thousands)
C
D
Alternative
A1
A2
A3
S1
4
0
3
State of Nature
S2
0
2
0
S3
0
0
1
Prior Probability
0.2
0.5
0.3
E
F
Expected
Payoff
($thousands)
0.8
1
0.9
b) With perfect information, choose A1 for when the state is S1, A2 when the state is S2, and
A3 when the state is S3.
EP(with perfect information) = (0.2)(4) + (0.5)(2) + (0.3)(1) = $2,100
EVPI = EP(with perfect information) – EP (without more information)
= $2,100 – $1,000 = $1,100.
Chapter 9 - Page 25
Chapter 9
DECISION ANALYSIS
c)
Alternative 1
4
4
0.2
State 1
4
Alternative 2
1
0
4
0
0
0
Alternative 3
3
3
3
Alternative 1
0
0
0.5
State 2
0
Alternative 2
2
2.1
0
2
2
2
2
Alternative 3
0
0
0
Alternative 1
0
0
0.3
State 3
0
Alternative 2
3
0
1
0
0
0
Alternative 3
1
1
1
EVPI = EP(with perfect information) – EP (without more information)
= $2,100 – $1,000 = $1,100.
d) Since the information will cost $1,000 and the value is no more than $1,100, it might be
worthwhile to spend the money.
Chapter 9 - Page 26
Chapter 9
DECISION ANALYSIS
9.11
a) Alternative A1 has the highest expected payoff of $35.
1
2
3
4
5
6
7
A
Payoff Table
Alternative
A1
A2
A3
Prior Probability
B
S1
$50
$0
$20
C
State of Nature
S2
$100
$10
$40
D
S3
-$100
-$10
-$40
0.5
0.3
0.2
E
F
Expected
Payoff
$35
$1
$14
b) With perfect information, choose A1 for when the state is S1, A1 when the state is S2, and
A2 when the state is S3.
EP(with perfect information) = (0.5)($50) + (0.3)($100) + (0.2)(–$10) = $53
EVPI = EP(with perfect information) – EP (without more information)
= $53 – $35 = $18
Chapter 9 - Page 27
Chapter 9
DECISION ANALYSIS
c)
Alternative 1
50
50
0.5
State 1
50
Alternative 2
1
0
50
0
0
0
Alternative 3
20
20
20
Alternative 1
100
100
0.3
State 2
100
Alternative 2
1
53
0
100
10
10
10
Alternative 3
40
40
40
Alternative 1
-100
-100
0.2
State 3
-100
Alternative 2
2
0
-10
-10
-10
-10
Alternative 3
-40
-40
-40
EVPI = EP(with perfect information) – EP (without more information)
= $53 – $35 = $18
d) Betsy should consider spending up to $18 to obtain more information.
Chapter 9 - Page 28
Chapter 9
DECISION ANALYSIS
9.12
a) Alternative A3 has the highest expected payoff of $35,000.
1
2
3
4
5
6
7
8
A
B
Payoff Table ($thousands)
C
D
Alternative
A1
A2
A3
S1
-100
-10
10
State of Nature
S2
10
20
10
S3
100
50
60
Prior Probability
0.2
0.3
0.5
E
F
Expected
Payoff
($thousands)
33
29
35
b) If S1 occurs for certain then choose alternative A3 (payoff is $10,000).
If S1 does not occur for certain then the chance of S2 occurring is 3/8 and the chance of
S3 occurring is 5/8. So choose A1 (expected payoff is $66,250).
A1:
(3/8)(10) + (5/8)(100) = 66.25
A2:
(3/8)(20) + (5/8)(50) = 38.75
A3:
(3/8)(10) + (5/8)(60) = 41.25
EP(with information) = (0.2)(10) + (0.8)(66.25) = 55
EVI = EP (with information) – EP (without more information)
= 55 – 35 = $20,000
The maximum amount you should pay for the information is $20,000.
The decision with this information would be to choose A3 if S1 will occur. Otherwise
choose A1. The expected payoff is $55,000 (excluding the payment for information).
c) If S2 occurs for certain then choose alternative A2 (payoff is $20,000).
If S2 does not occur for certain then the chance of S1 occurring is 2/7 and the chance of
S3 occurring is 5/7. So choose A3 (expected payoff is $45,714).
A1:
(2/7)(–100) + (5/7)(100) = 42.857
A2:
(2/7)(–10) + (5/7)(50) = 32.857
A3:
(2/7)(10) + (5/7)(60) = 45.714
EP(with information) = (0.3)(20) + (0.7)(42.857) = 38
EVI = EP (with information) – EP (without more information)
= 38 – 35 = $3,000
The maximum amount you should pay for the information is $3,000.
The decision with this information would be to choose A2 if S2 will occur. Otherwise
choose A3. The expected payoff is $38,000 (excluding the payment for information).
Chapter 9 - Page 29
Chapter 9
DECISION ANALYSIS
d) If S3 occurs for certain then choose alternative A1 (payoff is $100,000).
If S3 does not occur for certain then the chance of S1 occurring is 2/5 and the chance of
S2 occurring is 3/5. So choose A3 (expected payoff is $10,000).
A1:
(2/5)(–100) + (3/5)(10) = –34
A2:
(2/5)(–10) + (3/5)(20) = 8
A3:
(2/5)(10) + (3/5)(10) = 10
EP(with information) = (0.5)(100) + (0.5)(10) = 55
EVI = EP (with information) – EP (without more information)
= 55 – 35 = $20,000
The maximum amount you should pay for the information is $20,000.
The decision with this information would be to choose A1 if S3 will occur. Otherwise
choose A3. The expected payoff is $55,000 (excluding the payment for information).
e) With perfect information, choose A3 for when the state is S1, A2 when the state is S2, and
A1 when the state is S3.
EP(with perfect information) = (0.2)(10) + (0.3)(20) + (0.5)(100) = $58,000
EVPI = EP(with perfect information) – EP (without more information)
= 58 – 35 = $23,000
A maximum of $23,000 should be paid for the information. With perfect information,
choose A3 for when the state is S1, A2 when the state is S2, and A1 when the state is S3.
The resulting expected payoff is $58,000.
f) The maximum amount you should ever pay for testing is $23,000.
Chapter 9 - Page 30
Chapter 9
DECISION ANALYSIS
9.13
a)
Prior
Probabilities
P (state)
Conditional
Probabilities
P(finding | state)
0.8
Joint
Probabilities
P(state and finding)
Posterior
Probabilities
P(state | finding)
0.2
Oil and FSS
0.4
Oil, given FSS
0.05
Oil and USS
0.1
Oil, given USS
0.3
Dry and FSS
0.6
Dry, given FSS
0.45
Dry and USS
0.9
Dry, given USS
FSS, given Oil
USS, given Oil
0.25
Oil
0.75
0.2
Dry
0.4
FSS, given Dry
USS, given Dry
0.6
b)
B
C
3 Data:
4
State of
Prior
5
Nature
Probability
6
Oil
0.25
7
Dry
0.75
8
9
10
11
12 Posterior
13 Probabilities:
14
Finding
P(Finding)
15
FSS
0.5
16
USS
0.5
17
18
19
D
E
FSS
0.8
0.4
USS
0.2
0.6
F
P(Finding | State)
Finding
P(State | Finding)
State of Nature
Oil
0.4
0.1
Chapter 9 - Page 31
Dry
0.6
0.9
G
H
Chapter 9
DECISION ANALYSIS
c & d) The optimal policy is to do a seismic survey and sell if it is unfavorable or drill if
it is favorable.
0.1
Oil
670
Drill
800
-100
-50
0.5
Unfavorable
0.9
Dry
-130
2
0
670
0
-130
60
Sell
60
90
60
Do seismic survey
0.4
-30
125
Oil
670
Drill
800
-100
190
0.5
Favorable
0.6
Dry
-130
1
0
670
0
-130
190
1
Sell
125
60
90
60
0.25
Oil
700
Drill
800
-100
100
700
0.75
Dry
No seismic survey
-100
1
0
0
-100
100
Sell
90
90
90
Chapter 9 - Page 32
Chapter 9
DECISION ANALYSIS
9.14
a)
U
V
W
X
Y
Z
3
Data
Low Base High
4
Cost of Survey
30
28
30
32
5
Cost of Drilling
100
75
100
140
6
Revenue if Oil
800
600
800
1000
7
Revenue if Sell
90
85
90
95
8
Revenue if Dry
0
9
Prior Probability Of Oil
0.25
10
P(FSS|Oil)
0.6
11
P(USS|Dry)
0.8
12
13
14
Action
15
Do Survey?
Yes
16
17
If No
If Yes
18
19
Drill
Drill If Favorable
20
Sell If Unfavorable
21
22
23
24
Expected Payoff
25
($thousands)
26
125
27
28
29
30 Data:
P(Finding | State)
31
State of
Prior
Finding
32
Nature
Probability
FSS USS
33
Oil
0.25
0.8
0.2
34
Dry
0.75
0.4
0.6
35
36
37
38
39 Posterior
P(State | Finding)
40 Probabilities:
State of Nature
41
Finding
P(Finding)
Oil
Dry
42
FSS
0.5
0.4
0.6
43
USS
0.5
0.1
0.9
44
45
46
Chapter 9 - Page 33
AA
Chapter 9
DECISION ANALYSIS
b)
Sensit - Sensitivity Analysis - Plot
160
150
140
Expected
130
Payoff
120
110
100
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.7
0.8
0.9
1
P(FSS|Oil)
Sensit - Sensitivity Analysis - Plot
190
180
170
160
Expected 150
Payoff 140
130
120
110
100
0
0.1
0.2
0.3
0.4
0.5
0.6
P(FSS|Dry)
Chapter 9 - Page 34
Chapter 9
DECISION ANALYSIS
c)
Sensit - Sensitivity Analysis - Spider
145
140
135
Cost of Survey
Cost of Drilling
Revenue if Oil
Revenue if Sell
130
Expected
Payoff 125
Value
120
115
110
105
90%
94%
98%
102%
106%
110%
% Change in Input Value
Sensit - Sensitivity Analysis - Tornado
Revenue if Oil
600
Cost of Drilling
1000
140
75
Revenue if Sell
85
95
Cost of Survey
32
28
90
100
110
120
130
Expected Payoff
Chapter 9 - Page 35
140
150
160
170
Chapter 9
DECISION ANALYSIS
9.15
a)
Alternative
Extend Credit
Don’t Extend Credit
Prior Probabilities
Poor Risk
-$15,000
$0
0.2
State of Nature
Average Risk
$10,000
$0
0.5
Good Risk
$20,000
$0
0.3
b) Extending credit maximizes the expected payoff ($8,000).
1
2
3
4
5
6
7
A
Payoff Table ($thousands)
Alternative
Extend Credit
Don't Extend Credit
Prior Probability
B
C
D
E
State of Nature (Credit Record)
Poor
Average
Good
-15
10
20
0
0
0
0.2
0.5
F
Expected
Payoff
($thousands)
8
0
0.3
c) With perfect information, you would extend credit if their credit record is average or
good, and don’t extend credit if their credit record is poor.
EP(with perfect information) = (0.2)(0) + (0.5)(10) + (0.3)(20) = $11,000
EVPI = EP(with perfect information) – EP (without more information)
= $11,000 – $8,000 = $3,000.
This indicates that the credit-rating organization should not be used.
Chapter 9 - Page 36
Chapter 9
DECISION ANALYSIS
d) PF = Poor Finding
PS = Poor State
Prior
Probabilities
P (state)
AF = Average Finding
AS = Average State
Conditional
Probabilities
P(finding | state)
Joint
Probabilities
P(state and finding)
GF = Good Finding
GS = Good State
Posterior
Probabilities
P(state | finding)
0.1
PS and PF
0.2778
PS, given PF
0.4
AF, given PS
0.08
PS and AF
0.1778
PS, given AF
0.1
0.02
PS and GF
0.1053
PS, given GF
0.2
AS and PF
0.5556
AS, given PF
0.25
AS and AF
0.5556
AS, given AF
0.05
AS and GF
0.2632
AS, given GF
0.06
GS and PF
0.1667
GS, given PF
0.12
GS and AF
0.2667
GS, given AF
0.12
GS and GF
0.6316
GS, given GF
PF, given PS
GF, given PS
0.5
0.2 PS
PF, given AS
0.5
AS
0.5
AF, given AS
GF, given AS
0.3
0.4
0.1
GS
PF, given GS
0.2
0.4
AF, given GS
GF, given GS
0.4
Chapter 9 - Page 37
Chapter 9
DECISION ANALYSIS
e)
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
Template for Posterior Probabilities
Data:
State of
Nature
Poor
Average
Good
Prior
Probability
0.2
0.5
0.3
Posterior
Probabilities:
Finding
P(Finding)
Poor
0.36
Average
0.45
Good
0.19
Poor
0.5
0.4
0.2
P(Finding | State)
Finding
Average
Good
0.4
0.1
0.5
0.1
0.4
0.4
Poor
0.278
0.178
0.105
P(State | Finding)
State of Nature
Average
Good
0.556
0.167
0.556
0.267
0.263
0.632
Chapter 9 - Page 38
G
H
Chapter 9
DECISION ANALYSIS
f & g)
Vincent should not get the credit rating and simply extend credit.
Chapter 9 - Page 39
Chapter 9
DECISION ANALYSIS
0.278
Poor
-20000
-15000
-20000
0.556
Average
Extend Credit
5000
0
-290
0.36
10000
5000
0.166
Good
Poor
1
0
15000
-290
20000
15000
Don't Extend Credit
-5000
0
-5000
0.178
Poor
-20000
-15000
-20000
0.556
Average
Extend Credit
5000
0
3210
0.45
Average
Get credit rating
10000
5000
0.266
Good
1
-5000
2992.15
0
15000
3210
20000
15000
Don't Extend Credit
-5000
0
-5000
0.105
Poor
-20000
-15000
-20000
0.263
Average
Extend Credit
5000
0
8695
0.19
10000
5000
0.632
Good
Good
2
1
8000
0
15000
8695
20000
15000
Don't Extend Credit
-5000
0
-5000
0.2
Poor
-15000
-15000
-15000
0.5
Average
Extend Credit
10000
0
8000
10000
10000
0.3
Don't get credit rating
Good
1
0
20000
8000
20000
20000
Don't Extend Credit
0
0
0
Chapter 9 - Page 40
Chapter 9
DECISION ANALYSIS
h) EVSI = Value with credit rating (ignoring cost) – expected value without
= 8000 – 8000
=0
The sample information provides no value. This is clear because the same decision is
made regardless of the findings of the credit report. (Note there is some rounding error.
Without the rounding error, the expected payoff with the credit rating is 3000 rather
than 2992.15.)
Chapter 9 - Page 41
Chapter 9
DECISION ANALYSIS
9.16
a) Alternative A1 maximizes the expected payoff ($100).
1
2
3
4
5
6
A
Payoff Table
Alternative
A1
A2
B
C
State of Nature
S1
S2
$400
-$100
$0
$100
Prior Probability
0.4
D
E
Expected
Payoff
$100
$60
0.6
b)
Alternative 1
0.4
State 1
400
400
400
1
0
400
Alternative 2
0
0
0
220
Alternative 1
0.6
State 2
-100
-100
-100
2
0
100
Alternative 2
100
100
100
EVPI = EP (with perfect info) – EP (without more info) = $220 – $100 = $120
This indicates that it might be worthwhile to do the research.
c) P(state and finding) = P(state) P(finding | state)
i)
P(Predict S1 and Actual S1) = (0.4)(0.6) = 0.24
ii) P(Predict S1 and Actual S2) = (0.4)(0.4) = 0.16
iii) P(Predict S2 and Actual S1) = (0.6)(0.2) = 0.12
iv) P(Predict S2 and Actual S2) = (0.6)(0.8) = 0.48
d) P(Predict S1) = 0.24 + 0.12 = 0.36
P(Predict S2) = 0.16 + 0.48 = 0.64
e) P(state | finding) = P(state and finding) / P(finding)
P(Actual S1 | Predict S1) = 0.24 / 0.36 = 0.667
P(Actual S1 | Predict S2) = 0.16 / 0.64 = 0.250
P(Actual S2 | Predict S1) = 0.12 / 0.36 = 0.333
P(Actual S2 | Predict S2) = 0.48 / 0.64 = 0.750
Chapter 9 - Page 42
Chapter 9
DECISION ANALYSIS
f)
B
C
3 Data:
4
State of
Prior
5
Nature
Probability
6
Actual S1
0.4
7
Actual S2
0.6
8
9
10
11
12 Posterior
13 Probabilities:
14
Finding
P(Finding)
15 Predict S1
0.36
16 Predict S2
0.64
17
18
19
D
E
F
P(Finding | State)
Finding
Predict S1 Predict S2
0.6
0.4
0.2
0.8
G
H
P(State | Finding)
State of Nature
Actual S1 Actual S2
0.667
0.333
0.250
0.750
g) If S1 is predicted, then choosing alternative A1 maximizes the expected payoff
($233.33).
1
2
3
4
5
6
A
Payoff Table
Alternative
A1
A1
Prior Probability
B
C
State of Nature
S1
S2
$400
-$100
$0
$100
0.6667
D
E
Expected
Payoff
$233.33
$33.33
0.3333
h) If S2 is predicted, then choosing alternative A2 maximizes the expected payoff ($75).
1
2
3
4
5
6
A
Payoff Table
Alternative
A1
A1
Prior Probability
B
C
State of Nature
S1
S2
$400
-$100
$0
$100
0.25
D
E
Expected
Payoff
$25
$75
0.75
i) Expected payoff given research is (0.36)($233.33) + (0.64)($75) – $100 = $32.
j) The optimal policy is to do no research and simply choose A1.
Chapter 9 - Page 43
Chapter 9
DECISION ANALYSIS
k)
0.667
S1 occurs
300
Choose A1
0
400
133.5
0.36
Predict S1
300
0.333
S2 occurs
-200
-100
-200
1
0
133.5
0.667
S1 occurs
-100
Choose A2
0
0
-66.7
-100
0.333
S2 occurs
0
Research
-100
100
32.06
0
0.25
S1 occurs
300
Choose A1
0
400
-75
0.64
Presdict S2
300
0.75
S2 occurs
-200
-100
-200
2
0
-25
0.25
S1 occurs
-100
Choose A2
0
-100
2
100
0
-25
0.75
S2 occurs
0
100
0
0.4
S1 occurs
400
Choose A1
400
100
400
0.6
S2 occurs
-100
No Reasearch
-100
-100
1
0
100
0.4
S1 occurs
0
Choose A2
0
0
60
0
0.6
S2 occurs
100
100
Chapter 9 - Page 44
100
Chapter 9
DECISION ANALYSIS
9.17
a through d)
Prior
Probabilities
P (state)
Conditional
Probabilities
P(finding | state)
Joint
Probabilities
P(state and finding)
0.095
user and positive
0.95
positive, given user
Posterior
Probabilities
P(state | finding)
0.6786
user, given positive
negative, given user
0.05
0.1
0.005
user and negative
user
nonuser
0.9
0.045
nonuser and positive
0.05
positive, given nonuser
0.0058
user, given negative
0.3214
nonuser, given positive
negative given nonuser
0.95
0.855
nonuser and negative
0.9942
nonuser, given negative
e)
B
C
3 Data:
4
State of
Prior
5
Nature
Probability
6
User
0.1
7
Nonuser
0.9
8
9
10
11
12 Posterior
13 Probabilities:
14
Finding
P(Finding)
15
Positive
0.14
16
Negative
0.86
17
18
19
D
E
Positive
0.95
0.05
F
P(Finding | State)
Finding
Negative
0.05
0.95
User
0.679
0.006
P(State | Finding)
State of Nature
Nonuser
0.321
0.994
Chapter 9 - Page 45
G
H
Chapter 9
DECISION ANALYSIS
9.18
a)
State of Nature
Successful
Unsuccessful
$1,500,000
–$1,800,000
0
0
0.667
0.333
Alternative
Develop new product
Don’t develop new product
Prior Probabilities
b) Choosing to develop the product maximizes the expected payoff ($400,000).
1
2
3
4
5
6
7
A
Payoff Table ($millions)
B
C
D
State of Nature
Successful Unsuccessful
1.5
-1.8
0
0
Alternative
Develop Product
Don't Develop Product
Prior Probability
0.667
E
Expected
Payoff
($millions)
0.4
0
0.333
c) With perfect information, Telemore should develop the product if it would be
successful, and don’t if it will be unsuccessful.
EP(perfect information) = (0.667)(1.5) + (0.333)(0) = $1 million.
EVPI = EP(with perfect information) – EP(without more information)
= $1,000,000 – $400,000 = $600,000.
This indicates that consideration should be given to conducting the market survey.
d)
B
3 Data:
4
State of
5
Nature
6
Successful
7
Unsuccessful
8
9
10
11
12 Posterior
13 Probabilities:
14
Finding
15
Predict Successful
16 Predict Unsuccessful
17
18
19
C
D
Prior
Probability
0.667
0.333
Predict Successful
0.8
0.3
E
P(Finding | State)
Finding
Predict Unsuccessful
0.2
0.7
Successful
0.842
0.364
P(State | Finding)
State of Nature
Unsuccessful
0.158
0.636
P(Finding)
0.633
0.367
Chapter 9 - Page 46
F
G
H
Chapter 9
DECISION ANALYSIS
e) They should conduct the survey, and develop the product if the survey predicts the
product will be successful. The expected payoff is $520,000.
0.8421
Successful
1.4
Develop Product
1.5
0
0.1579
Unsuccessful
0.8789
0.63333
Predict Success
1.4
-1.9
1
-1.8
-1.9
0 0.8789474
Don't Develop
-0.1
0
-0.1
Conduct Survey
-0.1
0.3636
Successful
0.52
1.4
Develop Product
1.5
0
0.6364
Unsuccessful
-0.7
0.36667
Predict Unsuccessful
-1.9
2
0
1.4
-1.8
-1.9
-0.1
1
Don't Develop
0.52
-0.1
0
-0.1
0.6667
Successful
1.5
Develop Product
0
1.5
0.4
1.5
0.3333
Unsuccessful
No Survey
-1.8
1
0
-1.8
-1.8
0.4
Don't Develop
0
0
0
Chapter 9 - Page 47
Chapter 9
DECISION ANALYSIS
f)
Sensit - Sensitivity Analysis - Spider
0.75
0.7
0.65
0.6
Expected 0.55
Payoff
0.5
Value
0.45
Profit if Success
Loss if Unsuccessful
Cost of Survey
0.4
0.35
0.3
75%
80%
85%
90%
95%
100% 105%
110%
115%
120% 125%
% Change in Input Value
Sensit - Sensitivity Analysis - Tornado
Profit if Success 1.125
1.875
Loss if Unsuccessful
2.25
Cost of Survey
1.35
0.125
0.3
0.35
0.4
0.45
0.075
0.5
0.55
Expected Payoff
Chapter 9 - Page 48
0.6
0.65
0.7
0.75
Chapter 9
DECISION ANALYSIS
9.19
a)
State of Nature
p=0.05
p=0.25
–$1,500
–$1,500
–$750
–$3,750
0.8
0.2
Alternative
Screen
Don’t screen
Prior Probabilities
b) Choosing not to screen maximizes the expected payoff. The expected cost is $1,350.
1
2
3
4
5
6
A
Payoff Table
B
C
State of Nature
p = 0.05
p = 0.25
-$1,500
-$1,500
-$750
-$3,750
Alternative
Screen
Don't Screen
Prior Probability
0.8
D
E
Expected
Payoff
-$1,500
-$1,350
0.2
c) With perfect information, they would screen if p = 0.25, and don’t screen if p = 0.05.
EP(with perfect information) = (0.8)(–$750) + (0.2)(–$1,500) = –$900
EVPI = EP(with perfect information) – EP(without more information)
= (–$900) – (–$1,350) = $450.
This indicates that consideration should be given to inspecting the single item.
d)
B
3 Data:
4
State of
5
Nature
6
p = 0.05
7
p = 0.25
8
9
10
11
12 Posterior
13 Probabilities:
14
Finding
15
Defective
16
Nondefective
17
18
19
C
D
Prior
Probability
0.8
0.2
Defective
0.05
0.25
E
F
P(Finding | State)
Finding
Nondefective
0.95
0.75
p = 0.05
0.444
0.835
P(State | Finding)
State of Nature
p = 0.25
0.556
0.165
P(Finding)
0.09
0.91
Chapter 9 - Page 49
G
H
Chapter 9
DECISION ANALYSIS
e) The optimal policy is not to pre-screen or screen.
0.444
p=0.05
-1625
Screen
-1500
0
-1625
0.09
Defective
-1625
0.556
p=0.25
-1625
-1500
-1625
1
0
-1625
0.444
p=0.05
-875
Don't screen
0
-750
-2543
-875
0.556
p=0.25
-3875
Pre-screen
-125
-3750
-1392.95
-3875
0.835
p=0.05
-1625
Screen
-1500
0
-1625
0.91
Nondefective
-1625
0.165
p=0.25
-1625
-1500
-1625
2
0
-1370
0.835
p=0.05
-875
Don't screen
-750
-875
2
-1350
0
-1370
0.165
p=0.25
-3875
-3750
-3875
0.8
p=0.05
-1500
Screen
0
-1500
-1500
-1500
0.2
p=0.25
-1500
Don't Pre-screen
-1500
-1500
2
0
-1350
0.8
p=0.05
-750
Don't screen
0
-750
-1350
-750
0.2
p=0.25
-3750
-3750
-3750
f) EVSI = EP(with information ignoring cost of information) – EP(without information)
= (–1392.95 + 125) – (–1350) = (–1267.95) – (–1350) = $82.95.
If the prescreening inspection cost is less than $82.96, then it would be worthwhile to
use.
Chapter 9 - Page 50
Chapter 9
DECISION ANALYSIS
9.20
a)
State of Nature
Sell 10,000
Sell 100,000
$0
$54 million
$15 million
$15 million
Alternative
Build Computers
Sell Rights
b)
Sell 10,000
0
Build Computers
Sell 100,000
54
Sell Rights
15
c) They should build computers, with an expected payoff of $27 million.
0.5
Sell 10,000
0
Build Computers
-6
6
27
0
0.5
Sell 100,000
54
1
60
54
27
Sell Rights
15
15
15
Chapter 9 - Page 51
Chapter 9
DECISION ANALYSIS
d)
Sensit - Sensitivity Analysis - Plot
55
50
45
Expected 40
35
Payoff 30
25
20
15
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Prior Probability of Selling 10,000
e)
54
Expected
Profit
($millions)
Crossover
point
Build
15
Sell
0.2
0.4
0.6
0.8
1.0
Prior Probability of Selling 10,000
f) Let p = prior probability of selling 10,000.
For Build:
EP
For Sell:
EP
= p(0) + (1 – p)(54)
= –54p + 54
= p(15) + (1 – p)(15)
= 15
Build and Sell cross when –54p + 54 = 15 or 54p = 39 or p = 0.722
They should build when p ≤ 0.722, and sell when p > 0.722.
Chapter 9 - Page 52
1
Chapter 9
DECISION ANALYSIS
9.21
a) With perfect information, they should build computers if they will sell 100,000 of them,
and sell the rights if they could only sell 10,000 computers.
EP(with perfect information) = (0.5)(54) + (0.5)(15) = $34.5 million
EVPI = EP(with perfect information) – EP without more information)
= 34.5 – 27 = $7.5 million.
b) Since the market research will cost $1 million it might be worthwhile to perform it.
c)
Prior
Probabilities
P (state)
Conditional
Probabilities
P(finding | state)
Joint
Probabilities
P(state and finding)
0.333
0.667
sell 10k and sell 10k
sell 10k given sell 10k
Posterior
Probabilities
P(state | finding)
0.667
sell 10k given sell 10k
sell 100k given sell 10k
0.5
sell 10k
sell 100k
0.5
0.333
0.167
sell 10k and sell 100k
0.167
sell 100k and sell 10k
0.333
sell 10k given sell 100k
0.333
sell 10k given sell 100k
0.333
sell 100k given sell 10k
sell 100k given sell 100k
0.667
0.333
sell 100k and sell 100k
Chapter 9 - Page 53
0.667
sell 100k given sell 100k
Chapter 9
DECISION ANALYSIS
d)
B
3 Data:
4
State of
5
Nature
6
Sell 10,000
7
Sell 100,000
8
9
10
11
12 Posterior
13 Probabilities:
14
Finding
15
Predict Sell 10,000
16 Predict Sell 100,000
17
18
19
C
D
Prior
Probability
0.5
0.5
Predict Sell 10,000
0.667
0.333
E
P(Finding | State)
Finding
Predict Sell 100,000
0.333
0.667
Sell 10,000
0.667
0.333
P(State | Finding)
State of Nature
Sell 100,000
0.333
0.667
P(Finding)
0.5
0.5
Chapter 9 - Page 54
F
G
H
Chapter 9
DECISION ANALYSIS
9.22
a) The optimal policy is to do no market research and build the computers. The expected
payoff is $27 million.
0.6667
Sell 10,000
-1
Build computers
-6
6
17
0.5
Predict sell 10,000
0.3333
Sell 100,000
53
1
0
-1
60
53
17
Sell rights
14
15
14
Market research
-1
0.3333
Sell 10,000
26
-1
Build computers
-6
6
35
0.5
Predict sell 100,000
0.6667
Sell 100,000
53
1
0
-1
60
53
35
2
Sell rights
27
14
15
14
0.5
Sell 10,000
0
Build computers
-6
6
27
0
0.5
Sell 100,000
No market research
54
1
0
60
54
27
Sell rights
15
15
15
b) EVSI = EP(with information) – EP(without information)
= 27 – 27 = 0.
The information has no value. This is easy to see, as the same decision is made
regardless of the prediction of the market research.
Chapter 9 - Page 55
Chapter 9
DECISION ANALYSIS
c) If the rights can be sold for $16.5 or $13.5 million, the optimal policy is still to build
the computers with an expected payoff of $27 million.
If the cost of setting up the assembly line is $5.4 million or $6.6 million, the optimal
policy is still to build the computers with an expected payoff of $27.6 or $26.4 million,
respectively.
If the difference between the selling price and variable cost of each computer is $540 or
$660, the optimal policy is still to build the computers with an expected payoff of $23.7
or $33.3 million, respectively.
For each combination of financial data, the expected payoff is as shown below. In all
cases, the optimal policy is to build the computers (without market research).
Sell Rights
$13.5 million
$13.5 million
$13.5 million
$13.5 million
$16.5 million
$16.5 million
$16.5 million
$16.5 million
Cost of
Assembly Line
$5.4 million
$5.4 million
$6.6 million
$6.6 million
$5.4 million
$5.4 million
$6.6 million
$6.6 million
Chapter 9 - Page 56
Selling Price –
Variable Cost
$540
$660
$540
$660
$540
$660
$540
$660
Expected
Payoff
$24.3 million
$30.9 million
$23.1 million
$29.7 million
$24.3 million
$30.9 million
$23.1 million
$29.7 million
Chapter 9
DECISION ANALYSIS
d)
30.0
25.0
20.0
Expected
15.0
Payoff
10.0
5.0
0.0
13.5
14
14.5
15
15.5
16
16.5
Sell Rights Payoff
Sensit - Sensitivity Analysis - Plot
28.0
27.5
Expected
27.0
Payoff
26.5
26.0
5.4
5.5
5.6
5.7
5.8
5.9
6
6.1
6.2
6.3
6.4
6.5
6.6
Assembly Line Cost
Sensit - Sensitivity Analysis - Plot
31.0
30.0
29.0
Expected 28.0
27.0
Payoff 26.0
25.0
24.0
23.0
0.00054
0.00056
0.00058
0.00060
Marginal Profit
Chapter 9 - Page 57
0.00062
0.00064
0.00066
Chapter 9
DECISION ANALYSIS
e)
Sensit - Sensitivity Analysis - Spider
31.0
30.0
29.0
Expected 28.0
Payoff 27.0
Value 26.0
25.0
24.0
23.0
90%
Sell Rights Payoff
Assembly Line Cost
Marginal Profit
95%
100%
105%
110%
% Change in Input Value
Sensit - Sensitivity Analysis - Tornado
Marginal Profit 0.00054
0.00066
Assembly Line Cost
6.6
Sell Rights Payoff
23.0
5.4
13.5
16.5
24.0
25.0
26.0
27.0
Expected Payoff
Chapter 9 - Page 58
28.0
29.0
30.0
31.0
Chapter 9
DECISION ANALYSIS
9.23
a and b)
0.4
2500
2500
580
0.6
0.2
-700
2
-700
900
900
820
900
0.8
1
800
820
800
750
750
9.24
10
0.5
10
2
0.5
15
0
0
15
0.5
2.5
30
30
0.5
-10
-10
2
8
-5
0.4
-5
2
0.3
5
40
40
5
0.7
8
-10
-10
0.6
10
10
Chapter 9 - Page 59
Chapter 9
DECISION ANALYSIS
9.25
a)
Alternative
Hold campaign
Don’t hold campaign
Prior Probabilities
State of Nature
Winning Season
Losing Season
$3 million
–$2 million
0
0
0.6
0.4
b) Choosing to hold the campaign maximizes the expected payoff ($1 million).
1
2
3
4
5
6
7
A
Payoff Table ($millions)
B
C
State of Nature
Alternative Winning Season Losing Season
Hold Campaign
3
-2
Don't Hold Campaign
0
0
Prior Probability
0.6
D
E
Expected
Payoff
($millions)
1
0
0.4
c) With perfect information, Leland University should hold the campaign if they will have
a winning season and don’t hold the campaign if they will have a losing season.
EP(with perfect information) = (0.6)(3) + (0.4)(0) = $1.8 million
EVPI = EP (with perfect info) – EP (without more info)
= $1.8 million – $1 million = $800,000.
Chapter 9 - Page 60
Chapter 9
DECISION ANALYSIS
d)
Prior
Probabilities
P (state)
Conditional
Probabilities
P(finding | state)
0.75
Joint
Probabilities
P(state and finding)
0.45
win and win
win, given win
Posterior
Probabilities
P(state | finding)
0.818
win. given win
lose, given win
0.25
0.6
0.15
win and lose
Win
0.4
Lose
0.25
win, given lose
0.333
win, given lose
0.1
lose and win
0.182
lose, given win
0.3
0.667
lose and lose
lose, given lose
lose, given lose
0.75
e)
B
3 Data:
4
State of
5
Nature
6 Winning Season
7
Losing Season
8
9
10
11
12 Posterior
13 Probabilities:
14
Finding
15
Predict W
16
Predict L
17
18
19
C
D
Prior
Probability
0.6
0.4
Predict W
0.75
0.25
E
P(Finding | State)
Finding
Predict L
0.25
0.75
Winning Season
0.818
0.333
P(State | Finding)
State of Nature
Losing Season
0.182
0.667
P(Finding)
0.55
0.45
Chapter 9 - Page 61
F
G
H
Chapter 9
DECISION ANALYSIS
f & g) Leland University should hire William. If he predicts a winning season then they
should hold the campaign, if he predicts a losing season then they should not hold the
campaign.
0.8182
Win
2.9
Hold campaign
0
3
1.99091
0.55
Predict win
0.1818
Lose
-2.1
1
0
2.9
-2
-2.1
1.99091
Don't hold campaign
-0.1
0
-0.1
Hire William
-0.1
0.3333
Win
1.05
2.9
Hold campaign
3
0 -0.43333
0.45
Predict lose
0.6667
Lose
-2.1
2
0
2.9
-2
-2.1
-0.1
1
Don't hold campaign
1.05
-0.1
0
-0.1
0.6
Win
3
Hold campaign
0
3
1
3
0.4
Lose
Don't hire William
-2
1
0
-2
-2
1
Don't hold campaign
0
0
0
h) EVSI = EP(with information) – EP(without information)
= 1.15 – 1 = 0.15.
The fee can be as high as $150,000 and the information would still be worthwhile.
Chapter 9 - Page 62
Chapter 9
DECISION ANALYSIS
9.26
a & b)
(Note: this decision tree continues on the next page.)
0.7
Growth
144
Stocks Y2
0
24
133.2
0.7
Growth
144
0.3
Recession
108
-12
108
1
20
133.2
0.7
Growth
126
Bonds Y2
0
6
127.8
126
0.3
Recession
132
12
Stocks Y1
100
132
0.2
Growth
122.94
108
18
108
0.7
Recession
Stocks Y2
81
0
82.8
-9
81
0.1
Depression
0.3
Recession
45
-45
45
2
-10
99
0.2
Growth
94.5
4.5
94.5
0.7
Recession
Bonds Y2
99
0
99
9
99
0.1
Depression
1
122.94
108
18
Chapter 9 - Page 63
108
Chapter 9
DECISION ANALYSIS
0.7
Growth
126
Stocks Y2
0
21
116.55
0.7
Growth
126
0.3
Recession
94.5
-10.5
94.5
1
5
116.55
0.7
Growth
110.25
Bonds Y2
5.25
0 111.825
110.25
0.3
Recession
115.5
10.5
Bonds Y1
100
115.5
0.2
Growth
117.885
132
22
132
0.7
Recession
Stocks Y2
99
0
101.2
-11
99
0.1
Depression
0.3
Recession
55
-55
55
2
10
121
0.2
Growth
115.5
5.5
115.5
0.7
Recession
Bonds Y2
121
0
121
11
121
0.1
Depression
132
22
132
The comptroller should invest in stocks the first year. If there is growth during the first
year then she should invest in stocks again the second year. If there is a recession
during the first year then she should invest in bonds for the second year. The expected
payoff is $122.94 million.
Chapter 9 - Page 64
Chapter 9
DECISION ANALYSIS
9.27
a & b) The optimal policy is to wait until Wednesday to buy if the price is $9 on
Tuesday. If the price is $10 or $11 on Tuesday then buy on Tuesday.
Buy
900
900
900
0.3
Close at $9
0.4
10% Lower
2
0
810
891
810
810
0.3
Unchanged
Wait
900
0
891
900
900
0.3
10% Higher
990
990
990
Buy
1000
1000
1000
0.3
Close at $10
0.2
10% Lower
1
1007.3
0
900
1000
900
900
0.2
Unchanged
Wait
1000
0
1040
1000
1000
0.6
10% Higher
1100
1100
1100
Buy
1100
1100
1100
0.4
Close at $11
0.1
10% Lower
1
0
990
1100
990
990
0.2
Unchanged
Wait
1100
0
1166
1100
1100
0.7
10% Higher
1210
1210
Chapter 9 - Page 65
1210
Chapter 9
DECISION ANALYSIS
9.28
The optimal policy is to sample the fruit and buy if it is excellent and reject if it is
unsatisfactory.
B
3 Data:
4
State of
5
Nature
6
Satisfactory Box
7 Unsatisfactory Box
8
9
10
11
12 Posterior
13 Probabilities:
14
Finding
15
Excellent
16
Not Excellent
17
18
19
C
D
Prior
Probability
0.9
0.1
Excellent
0.8
0.3
P(Finding)
0.75
0.25
E
P(Finding | State)
Finding
Not Excellent
0.2
0.7
P(State | Finding)
State of Nature
Satisfactory Box Unsatisfactory Box
0.960
0.040
0.720
0.280
Chapter 9 - Page 66
F
G
H
Chapter 9
DECISION ANALYSIS
0.96
Satisfactory
200
Buy
200
0
152
0.75
Excellent
0.04
Unsatisfactory
-1000
1
0
200
-1000
-1000
152
Reject
0
0
0
Sample
0
0.72
Satisfactory
114
200
Buy
200
0
-136
0.25
Not Excellent
0.28
Unsatisfactory
-1000
2
0
200
-1000
-1000
0
1
Reject
114
0
0
0
0.9
Satisfactory
200
Buy
200
0
80
200
0.1
Unsatisfactory
Don't sample
-1000
1
0
-1000
-1000
80
Reject
0
0
0
Chapter 9 - Page 67
Chapter 9
DECISION ANALYSIS
9.29
a)
Alternative
Introduce new product
Don’t introduce new product
Prior Probabilities
1
2
3
4
5
6
7
A
Payoff Table ($millions)
State of Nature
Successful
Unsuccessful
$40 million
–$15 million
0
0
0.5
0.5
B
Alternative
Test Market Approves
Test Market Doesn't Approve
Prior Probability
C
D
State of Nature
Successful
Unsuccessful
40
-15
0
0
0.5
E
Expected
Payoff
($millions)
12.5
0
0.5
Choose to introduce the new product (expected payoff is $12.5 million).
b) With perfect information, Morton Ward should introduce the product if it will be
successful, and don’t introduce the product if it won’t.
EP(with perfect information) = (0.5)(40) + (0.5)(0) = $20 million.
EVPI = EP (with perfect info) – EP (without more info) = 20 – 12.5 = $7.5 million.
Chapter 9 - Page 68
Chapter 9
DECISION ANALYSIS
c) The optimal policy is not to test but to introduce the new product. The expected payoff
is $12.5 million.
B
3 Data:
4
State of
5
Nature
6
Successful
7
Unsuccessful
8
9
10
11
12 Posterior
13 Probabilities:
14
Finding
15
Approved
16 Not Approved
17
18
19
C
D
Prior
Probability
0.5
0.5
Approved
0.8
0.25
E
P(Finding | State)
Finding
Not Approved
0.2
0.75
Successful
0.762
0.211
P(State | Finding)
State of Nature
Unsuccessful
0.238
0.789
P(Finding)
0.525
0.475
Chapter 9 - Page 69
F
G
H
Chapter 9
DECISION ANALYSIS
0.762
Successful
38
Introduce product
40
0
0.238
Unsuccessful
24.905
0.525
Approved
-17
1
0
38
-15
-17
24.9048
Don't introduce product
-2
0
-2
Test
-2
0.211
Successful
12.125
38
Introduce product
0 -5.4211
0.475
Not approved
40
38
0.789
Unsuccessful
-17
2
0
-15
-17
-2
2
Don't introduce product
12.5
-2
0
-2
0.5
Successful
40
Introduce product
0
40
12.5
40
0.5
Unsuccessful
Don't test
-15
1
0
-15
-15
12.5
Don't introduce product
0
0
0
d) EVSI = EP(with information) – EP(without information)
= 14.125 – 12.5 = 1.625.
The cost of the test market can be as high as $1.625 million and still be worthwhile to
do.
e) If the net profit if successful is only $30 million, then the optimal policy is to conduct
the test market and only introduce the product if the test market approves. The expected
payoff is $8.125 million.
If the net profit if successful is $50 million, then the optimal policy is to skip the test
market and introduce the product, with an expected payoff of $17.5 million.
If the net loss if unsuccessful is only $11.25 million, then the optimal policy is to skip
the test market and introduce the product, with an expected payoff of $14.375 million.
Chapter 9 - Page 70
Chapter 9
DECISION ANALYSIS
If the net loss if unsuccessful is $18.75 million, then the optimal policy is to conduct the
test market and only introduce the product if the test market approves. The expected
payoff is $11.656 million.
For each combination of financial data, the expected payoff is as shown below. In all
cases, the optimal policy is to build the computers (without market research).
Net Profit if
Successful
$30 million
$30 million
$50 million
$50 million
Net Loss if
Unsuccessful
$11.25 million
$18.75 million
$11.25 million
$18.75 million
Optimal
Policy
Skip Test, Introduce Product
Test, Introduce if Approve
Skip Test, Introduce Product
Test, Introduce if Approve
Expected
Payoff
$9.375 million
$7.656 million
$19.375 million
$15.656 million
f)
Sensit - Sensitivity Analysis - Plot
18
16
Expected 14
Payoff 12
10
8
30
32
34
36
38
40
42
44
46
48
50
Net Profit if Successful
Sensit - Sensitivity Analysis - Plot
14.5
14
13.5
Expected
13
Payoff
12.5
12
11.5
11
12
13
14
15
16
Net Loss if Unsuccessful
Chapter 9 - Page 71
17
18
19
Chapter 9
DECISION ANALYSIS
g)
Sensit - Sensitivity Analysis - Spider
18
17
16
15
Expected 14
Payoff 13
Value 12
11
10
9
8
75%
Net Profit if Successful
Net Loss if Unsuccessful
85%
95%
105%
115%
125%
% Change in Input Value
Sensit - Sensitivity Analysis - Tornado
Net Profit if Successful 30
50
18.75
8
9
10
11
11.25
12
13
14
15
16
17
18
Expected Payoff
Both charts indicate that the expected profit is sensitive to both parameters, but is
somewhat more sensitive to changes in the profit if successful than to changes in the
loss if unsuccessful.
Chapter 9 - Page 72
Chapter 9
DECISION ANALYSIS
9.30
a) Chelsea should run in the NH primary. If she does well then she should run in the ST
primaries. If she does poorly then she should not run in the ST primaries. The expected
payoff is $666,667.
0.5714
Do well in ST
14.4
Run in ST
16
0 3.257
0.4667
Do well in NH
0.4286
Do poorly in ST
-11.6
1
0
14.4
-10
-11.6
3.2571
Don't run in ST
-1.6
0
-1.6
Run in NH
-1.6
0.25
Do well in ST
0.66667
14.4
Run in ST
0
16
-5.1
0.5333
Do poorly in NH
0.75
Do poorly in ST
-11.6
2
0
14.4
-10
-11.6
-1.6
1
Don't run in ST
0.66667
-1.6
0
-1.6
0.4
Do well in ST
16
Run in ST
0
16
0.4
16
0.6
Do poorly in ST
Don't run in NH
-10
1
0
-10
-10
0.4
Don't run in ST
0
0
0
b) If the payoff for a doing well in ST is only $12 million, Chelsea should not run in either
NH or ST, with an expected payoff of $0.
If the payoff for doing well in ST is $20 million, Chelsea should not run in NH, but
should run in ST, with an expected payoff of $2 million.
If the loss for doing poorly in ST is $7.5 million, Chelsea should not run in NH, but
should run in ST, with an expected payoff of $1.9 million.
If the loss for doing poorly in ST is $12.5 million, Chelsea should run in NH, and then
run in ST if she does well in NH, with an expected payoff of $166,667.
For each combination of financial data, the expected payoff is as shown below. In all
cases, the optimal policy is to build the computers (without market research).
Chapter 9 - Page 73
Chapter 9
DECISION ANALYSIS
Payoff if do
Well in ST
$12 million
$12 million
$20 million
$20 million
Loss if do
Poorly in ST
$7.5 million
$12.5 million
$7.5 million
$12.5 million
Optimal
Policy
Run in ST only
Don’t run in either
Run in ST only
Run in NH, Run in ST if do well
Expected
Payoff
$300,000
$0
$3.5 million
$1.233 million
c)
Sensit - Sensitivity Analysis - Plot
2.0
1.5
Expected
1.0
Payoff
0.5
0.0
12
13
14
15
16
17
18
19
20
Payoff if Win ST
Sensit - Sensitivity Analysis - Plot
2
1.5
Expected
Payoff
1
0.5
0
7
8
9
10
Loss if Lose ST
Chapter 9 - Page 74
11
12
13
Chapter 9
DECISION ANALYSIS
d)
Sensit - Sensitivity Analysis - Spider
2.0
1.5
Expected
Payoff 1.0
Value
0.5
Payoff if Win ST
Loss if Lose ST
0.0
75%
85%
95%
105% 115% 125%
% Change in Input Value
Sensit - Sensitivity Analysis - Tornado
Payoff if Win ST
12
20
12.5
0.0
7.5
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Expected Payoff
Both charts indicate that the expected payoff is sensitive to both parameters, although it
is slightly more sensitive to changes in the profit if she does well than to changes in the
loss if she does poorly.
Chapter 9 - Page 75
Chapter 9
DECISION ANALYSIS
9.31
a)
0.95455
Successful
1960
Hire
2000
0 1841.8
0.66
Pass
0.04545
Unsuccessful
-640
1
0
1960
-600
-640
1842
Don't hire
-40
0
-40
Test
-40
0.20588
Successful
1202
1960
Hire
2000
0 -104.7
0.34
Fail
0.79412
Unsuccessful
-640
2
0
1960
-600
-640
-40
2
Don't hire
1220
-40
0
-40
0.7
Successful
2000
Hire
2000
0
1220
2000
0.3
Unsuccessful
Don't test
-600
1
0
-600
-600
1220
Don't hire
0
0
0
Chapter 9 - Page 76
Chapter 9
DECISION ANALYSIS
B
3 Data:
4
State of
5
Nature
6
Successful
7 Not Successful
8
9
10
11
12 Posterior
13 Probabilities:
14
Finding
15
Pass Test
16
Fail Test
17
18
19
C
D
Prior
Probability
0.7
0.3
Pass Test
0.9
0.1
E
P(Finding | State)
Finding
Fail Test
0.1
0.9
Successful
0.9545
0.2059
P(State | Finding)
State of Nature
Not Successful
0.0455
0.7941
P(Finding)
0.6600
0.3400
The optimal policy is not to pay for testing and to hire Matthew.
b) If the fee is less than $22,000 then the testing is worthwhile.
Chapter 9 - Page 77
F
G
H
Chapter 9
DECISION ANALYSIS
9.32
a & b)
90.
They should do no seismic survey, and sell the land, with an expected utility of
0.143
Oil
440
Drill
440
0
-108.48
0.7
Unfavorable
0.857
Dry
-200
2
0
440
-200
-200
60
Sell
60
60
60
Do seismic survey
0.5
0
78
Oil
440
Drill
440
0
120
0.3
Favorable
0.5
Dry
-200
1
0
440
-200
-200
120
2
Sell
90
60
60
60
0.25
Oil
450
Drill
450
0
15
450
0.75
Dry
No seismic survey
-130
2
0
-130
-130
90
Sell
90
90
9.33
90
a) Choosing not to buy insurance maximizes the expected payoff ($249,840).
1
2
3
4
5
6
A
Payoff Table
Alternative
Insurance
No Insurance
Prior Probability
B
C
State of Nature
Earthquake No Earthquake
$249,820
$249,820
$90,000
$250,000
0.001
D
E
Expected
Payoff
$249,820
$249,840
0.999
b) U(insurance) = U(250,000-180) = 249,820 = 499.82
U(no insurance) = (0.999)U(250,000) + (0.001)U(90,000) = 499.8
The optimal policy is to buy insurance.
Chapter 9 - Page 78
Chapter 9
DECISION ANALYSIS
9.34
E(utility) of $19,000 = U(19) = 25 = 5
E(utility) of investment = (0.3)U(10) + (0.7)U(30) = (0.3) 16  (0.7) 36 = 5.4
Choose the investment to maximize expected utility.
9.35
U(10) = 0, U(30) = 1.
U(19) = p such that you are indifferent between 30 with probability p and 10 with
probability 1–p, compared to 19 for sure.
The parents are offering p = 0.7. If you are indifferent at this p, then U(19) = 0.7.
9.36
a) U(1) = p = 0.125.
b) E(5) = p = 0.5625.
c) Answers will vary.
Chapter 9 - Page 79
Chapter 9
DECISION ANALYSIS
9.37
a) Expected utility of A1 = pU(25) + (1 – p)U(36) = 5p + 6(1 – p) = 6 – p.
Expected utility of A2 = pU(100) + (1 – p)U(0) = 10p + 0 = 10p.
Expected utility of A3 = pU(0) + (1 – p)U(49) = 7(1 – p) = 7 – 7p.
Expected
Utility
10
A2
7
6
A1
A3
0.2
0.4
0.6
0.8
1
p
A1 and A3 cross when 6 – p = 7 – 7p, or p = 1/6.
A1 and A2 cross when 6 – p =10p, or p = 6/11.
Thus, A3 is best when p ≤ 1/6, A1 is best when 1/6 < p ≤ 6/11, and A2 is best when
p > 6/11.
Chapter 9 - Page 80
Chapter 9
DECISION ANALYSIS
b) When p = 0.25, alternative A1 is optimal with an expected utility of 0.4833.
0.25
State 1
25
Alternative 1
0
25
33.015
0.4833
25
0.393469
0.75
State 2
36
36
36
0.513248
0.25
State 1
100
Alternative 2
100
1
33.0149
0.4833
0
12.178
0.2162
100
0.864665
0.75
State 2
0
0
0
0
0.25
State 1
0
Alternative 3
0
0
31.604
0.4685
0
0
0.75
State 2
49
49
Chapter 9 - Page 81
49
0.6246889
p=
0.25
RT =
50
Chapter 9
DECISION ANALYSIS
When p = 0.5, alternative A1 is optimal with an expected utility of 0.4534.
0.5
State 1
25
Alternative 1
0
25
30.198
0.4534
25
0.393469
0.5
State 2
36
36
36
0.513248
0.5
State 1
100
Alternative 2
100
1
30.1981
0.45336
0
28.311
0.4323
100
0.864665
0.5
State 2
0
0
0
0
0.5
State 1
0
Alternative 3
0
0
18.723
0.3123
0
0
0.5
State 2
49
49
Chapter 9 - Page 82
49
0.6246889
p=
0.5
RT =
50
Chapter 9
DECISION ANALYSIS
When p = 0.75, alternative A2 is optimal with an expected utility of 0.6485.
0.75
State 1
25
Alternative 1
0
25
27.532
0.4234
25
0.393469
0.25
State 2
36
36
36
0.513248
0.75
State 1
100
Alternative 2
100
2
52.2771
0.6485
0
52.277
0.6485
100
0.864665
0.25
State 2
0
0
0
0
0.75
State 1
0
Alternative 3
0
0
8.4903
0.1562
0
0
0.25
State 2
49
49
Chapter 9 - Page 83
49
0.6246889
p=
0.75
RT =
50
Chapter 9
DECISION ANALYSIS
9.38
The optimal policy is not to test for disease A but to treat disease A. (Note: this decision
tree is continued on the next page.)
0.5
Poor Health
0.8
Have A
0
7
10
17
7
0.5
Good Health
Treat A
27
30
-1
27
13
0.2
Have B
1
Die
-3
0
-3
0
0.5
Positive test result
-3
0.2
1
0
Die
13
0.8
Have A
0
-2
0
6
-2
0.8
Poor Health
8
Don't treat A
0
10
5.4
8
0.5
Die
0.2
Have B
0
-2
0
3
-2
0.5
Poor Health
Test for A
8
10
-2
8
8.3
0.5
Poor Health
0.2
Have A
0
7
10
17
7
0.5
Good Health
Treat A
27
30
-1
27
1
0.8
Have B
1
Die
-3
0
-3
0
0.5
Negative test result
0.2
2
0
-3
Die
3.6
0.2
Have A
0
-2
0
6
-2
0.8
Poor Health
8
Don't treat A
10
8
2
9
0
3.6
0.5
Die
0.8
Have B
0
-2
0
3
-2
0.5
Poor Health
8
10
Chapter 9 - Page 84
8
Chapter 9
DECISION ANALYSIS
0.5
Poor Health
9
0.5
Have A
0
10
19
9
0.5
Good Health
29
Treat A
30
-1
29
9
1
0.5
Have B
Die
-1
0
0
-1
0.2
Don't test for A
Die
1
0
-1
0
0.5
Have A
9
0
0
8
0
0.8
Poor Health
10
10
Don't treat A
0
10
0.5
6.5
Die
0
0.5
Have B
0
0
5
0
0.5
Poor Health
10
10
B
3 Data:
4
State of
5
Nature
6
Disease A
7
Disease B
8
9
10
11
12 Posterior
13 Probabilities:
14
Finding
15
Positive
16
Negative
17
18
19
C
D
Prior
Probability
0.5
0.5
Positive
0.8
0.2
P(Finding)
0.5
0.5
E
F
P(Finding | State)
Finding
Negative
0.2
0.8
P(State | Finding)
State of Nature
Disease A
Disease B
0.8
0.2
0.2
0.8
Chapter 9 - Page 85
10
G
H
Chapter 9
DECISION ANALYSIS
9.39
For the decision-maker to be indifferent between A1 and A2, U(x) must equal 3.5. So,
x1/3 = 3.5 or x = $42.88.
A1
$x
x1/3
0.6
$7,600
19
-$600
19
7
0.5
0.4
-$590
$0
-11
1
0 -11
7
-$141
A2
$16
-5
3.5
-5
0.5
$141
0
0
Chapter 9 - Page 86
Chapter 9
DECISION ANALYSIS
9.40
a)
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
0.1429
Oil
Q
R
S
T
U
V
Prior Prob. Of Oil
P(FSS|Oil)
P(USS|Dry)
Data
0.25
0.6
0.8
W
X
Y
Z
580
Drill
580
0
-45.714
0.7
Unfavorable
0.8571
Dry
-150
2
0
580
-150
-150
60
Sell
60
60
60
Do Survey
0.5
0
106.5
Oil
Do Survey?
Action
Yes
580
Drill
580
0
215
0.3
Favorable
If No
If Yes
Sell
Drill
Sell
Dry
If Favorable
If Unfavorable
-150
1
0
580
0.5
-150
-150
Expected Utility
106.5
215
1
Sell
106.5
60
60
60
0.25
Oil
Data:
State of
Nature
Oil
Dry
Prior
Probability
0.25
0.75
Posterior
Probabilities:
Finding
FSS
USS
P(Finding)
0.3
0.7
P(Finding | State)
Finding
USS
0.4
0.8
FSS
0.6
0.2
600
Drill
600
0
71.25
600
0.75
Dry
No Survey
-105
2
0
-105
-105
90
Sell
90
90
90
Chapter 9 - Page 87
P(State | Finding)
State of Nature
Oil
Dry
0.5
0.5
0.1429 0.86
AA
Chapter 9
DECISION ANALYSIS
b) When the prior probability of oil is 15%, the optimal policy is to do no survey and sell
the land, with an expected utility of 90.
U
V
Data
0.15
0.6
0.8
W
X
Y
Z
12
13 Prior Prob. Of Oil
14
P(FSS|Oil)
15
P(USS|Dry)
16
17
18
Action
19
Do Survey?
No
20
21
If No
If Yes
22
23
Sell
Drill If Favorable
24
Sell If Unfavorable
25
26
Expected Utility
27
90
28
29
30 Data:
P(Finding | State)
31
State of
Prior
Finding
32
Nature
Probability
FSS USS
33
Oil
0.15
0.6
0.4
34
Dry
0.85
0.2
0.8
35
36
37
38
39 Posterior
P(State | Finding)
40 Probabilities:
State of Nature
41
Finding
P(Finding)
Oil
Dry
42
FSS
0.26
0.3462 0.65
43
USS
0.74
0.0811 0.92
44
45
46
Chapter 9 - Page 88
AA
Chapter 9
DECISION ANALYSIS
When the prior probability of oil is 20%, the optimal policy is to do the survey and drill
if favorable or sell if unfavorable, with an expected utility of 90.86.
U
V
Data
0.2
0.6
0.8
W
X
Y
Z
12
13 Prior Prob. Of Oil
14
P(FSS|Oil)
15
P(USS|Dry)
16
17
18
Action
19
Do Survey?
Yes
20
21
If No
If Yes
22
23
Sell
Drill If Favorable
24
Sell If Unfavorable
25
26
Expected Utility
27
90.857143
28
29
30 Data:
P(Finding | State)
31
State of
Prior
Finding
32
Nature
Probability
FSS USS
33
Oil
0.2
0.6
0.4
34
Dry
0.8
0.2
0.8
35
36
37
38
39 Posterior
P(State | Finding)
40 Probabilities:
State of Nature
41
Finding
P(Finding)
Oil
Dry
42
FSS
0.28
0.4286 0.57
43
USS
0.72
0.1111 0.89
44
45
46
Chapter 9 - Page 89
AA
Chapter 9
DECISION ANALYSIS
When the prior probability of oil is 30%, the optimal policy is to do the survey and drill
if favorable or sell if unfavorable, with an expected utility of 120.19.
U
V
Data
0.3
0.6
0.8
W
X
Y
Z
12
13 Prior Prob. Of Oil
14
P(FSS|Oil)
15
P(USS|Dry)
16
17
18
Action
19
Do Survey?
Yes
20
21
If No
If Yes
22
23
Drill
Drill If Favorable
24
Sell If Unfavorable
25
26
Expected Utility
27
120.1875
28
29
30 Data:
P(Finding | State)
31
State of
Prior
Finding
32
Nature
Probability
FSS USS
33
Oil
0.3
0.6
0.4
34
Dry
0.7
0.2
0.8
35
36
37
38
39 Posterior
P(State | Finding)
40 Probabilities:
State of Nature
41
Finding
P(Finding)
Oil
Dry
42
FSS
0.32
0.5625 0.44
43
USS
0.68
0.1765 0.82
44
45
46
Chapter 9 - Page 90
AA
Chapter 9
DECISION ANALYSIS
When the prior probability of oil is 35%, the optimal policy is to skip the survey and
drill for oil, with an expected utility of 141.75.
U
V
Data
0.35
0.6
0.8
W
X
Y
Z
12
13 Prior Prob. Of Oil
14
P(FSS|Oil)
15
P(USS|Dry)
16
17
18
Action
19
Do Survey?
No
20
21
If No
If Yes
22
23
Drill
Drill If Favorable
24
Sell If Unfavorable
25
26
Expected Utility
27
141.75
28
29
30 Data:
P(Finding | State)
31
State of
Prior
Finding
32
Nature
Probability
FSS USS
33
Oil
0.35
0.6
0.4
34
Dry
0.65
0.2
0.8
35
36
37
38
39 Posterior
P(State | Finding)
40 Probabilities:
State of Nature
41
Finding
P(Finding)
Oil
Dry
42
FSS
0.34
0.6176 0.38
43
USS
0.66
0.2121 0.79
44
45
46
Chapter 9 - Page 91
AA
Chapter 9
DECISION ANALYSIS
Cases
9.1
a) The course of action that maximizes the expected payoff is to answer the $500,000
question alone. If you get that question correct, then use the phone-a-friend lifeline to
help answer to $1 million question. The expected payoff is $440,980.
$1 million (alone)
$1 million (Phone-a-Friend)
$500k (alone)
$500k (Phone-a-Friend)
Prob(Correct)
0.5
0.65
0.65
0.8
0.5
Correct
$1,000,000
Answer alone
0
1000000
$516,000
0.8
Correct
0.5
Incorrect
$32,000
1
0
$1,000,000
32000
$32,000
$516,000
Answer with
Phone-a-Friend
Don't Answer
$500,000
0
$419,200
500000
$500,000
0.2
Incorrect
$32,000
32000
$32,000
0.65
Correct
Answer with
Phone-a-Friend
0
$661,200
0.65
Correct
2
$1,000,000
0.35
Incorrect
$32,000
$440,980
1
0
$1,000,000
1000000
32000
$32,000
$661,200
Answer alone
Don't Answer
$500,000
0
$440,980
500000
$500,000
0.35
Incorrect
$32,000
32000
$32,000
Don't Answer
$250,000
250000
$250,000
b) Answers will vary depending on your level of risk aversion. Here is one possible
solution:
Set U(Maximum) = U($1 million) = 1.
Set U(Minimum) = U($32 thousand) = 0.
If getting $250 thousand for sure is equivalent to a 60% chance of getting $1 million
vs. a 40% chance of getting $32 thousand, then U($250 thousand) = p = 0.6.
If getting $500 thousand for sure is equivalent to a 90% chance of getting $1 million vs.
a 10% chance of getting $32 thousand, then U($250 thousand) = p = 0.9.
Chapter 9 - Page 92
Chapter 9
DECISION ANALYSIS
c) With the utilities derived in part b, the decision changes to using the phone-a-friend
lifeline to help answer the $500 thousand question, and then walk away.
$1 million (alone)
$1 million (Phone-a-Friend)
$500k (alone)
$500k (Phone-a-Friend)
Prob(Correct)
0.5
0.65
0.65
0.8
U($1 million) =
U($500k) =
U($250k) =
U($32k) =
1
0.9
0.6
0
0.5
Correct
1
Answer alone
0
0.5
0.8
Correct
2
0
1
$1,000,000
0.5
Incorrect
0
0
0
$32,000
0.9
Answer with
Phone-a-Friend
0
0
Don't Answer
0.72
0
0.9
$500,000
0.9
0.2
Incorrect
0
0
$32,000
0
0.65
Correct
Answer with
Phone-a-Friend
0
1
0
0.65
0.65
Correct
1
0.72
2
0
$1,000,000
0.35
Incorrect
0
0
0
$32,000
0.9
Answer alone
0
1
Don't Answer
0.585
0
0.9
0.9
$500,000
0.35
Incorrect
0
0
0
$32,000
Don't Answer
0
0.6
$250,000
0.6
Chapter 9 - Page 93
Chapter 9
DECISION ANALYSIS
9.2
a)
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
B
C
D
E
Template for Posterior Probabilities
Data:
State of
Nature
Well in Full Market
Poor in Full Market
Posterior
Probabilities:
Finding
Well in Test
Poor in Test
Prior
Probability
0.5
0.5
P(Finding | State)
Finding
Well in Test
Poor in Test
0.8
0.2
0.4
0.6
P(Finding)
0.6
0.4
P(State | Finding)
State of Nature
Well in Full Market
Poor in Full Market
0.666666667
0.333333333
0.25
0.75
Chapter 9 - Page 94
F G H
Chapter 9
DECISION ANALYSIS
b) The best course of action is to skip the test market, and immediately market the product
fully. The expected payoff is $1750.
0.66667
Do Well
Probabilities:
P("Well in Full Market")
0.5
P("Well in Test")
0.6
P("Well in Full Market | Well in Test"
0.6667
P("Well in Full Market | Poor in Test")
0.25
P("LSPAF")
0.2
$1,300
Fully Market
-$1,000
$2,000
$700
0.2
LSPAF Enters Market
0
0.33333
Do Poorly
-$500
1
Cost & Revenue Data:
Do Well, LSPAF $2,000
Do Well, No LSPAF $5,000
Do Poorly, LSPAF
$200
Do Poorly, No LSPAF
$500
Do Well in Test Market
$400
Do Poorly in Test Market
$40
Full Market Cost $1,000
Test Market Cost
$100
$1,300
$200
-$500
$700
Don't
$300
0.60
Do Well
$400
0
$300
0.66667
Do Well
$2,380
$4,300
Fully Market
-$1,000
$5,000
$2,800
0.8
No LSPAF
0.33333
Do Poorly
-$200
1
0
4300
$500
-200
$2,800
Don't
$300
0
$300
Test Market
-$100
0.25
Do Well
$1,604
$940
Fully Market
-$1,000
$2,000
-$410
0.2
LSPAF Enters Market
0.75
Do Poorly
-$860
2
0
$940
$200
-$860
-$60
Don't
-$60
0.40
Do Poorly
$40
0
-$60
0.25
Do Well
$440
$3,940
Fully Market
$5,000
$3,940
2
$1,750
Expected Profit
-1000
565
0.8
No LSPAF
-$560
1
0
0.75
Do Poorly
$500
-$560
$565
Don't
-$60
0
-$60
0.5
Do Well
$4,000
Fully Market
#####
#####
1750
$4,000
0.5
Do Poorly
No Test Market
-$500
1
$500
-$500
0 $1,750
Don't
0
0
0
Chapter 9 - Page 95
Chapter 9
DECISION ANALYSIS
c) If the probability that the LSPAFs enter the market before the test marketing would be
completed increases this would make the test market even less desirable, so it would
still not be worthwhile to do. However, if the probability decreases, this would make
the test market more desirable. It might reach the point where the test market is
worthwhile.
d)
Prob(LSPAF)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Expected
Payoff
$1,750
$1,906
$1,755
$1,750
$1,750
$1,750
$1,750
$1,750
$1,750
$1,750
$1,750
$1,750
Test
Market?
No
Yes
Yes
No
No
No
No
No
No
No
No
No
e) It is better to perform the test market if the probability that the LSPAFs enter the market
is 10% or less. It is better to skip the test market if the probability is greater than 10%.
Chapter 9 - Page 96
Chapter 9
DECISION ANALYSIS
9.3
a) The decision alternatives are to price the product high ($50), medium ($40), or low
($30), or don’t market the product at all. The possible states of nature are the demand
could be high (50,000), medium (30,000), or low (20,000), which in turn depends upon
the price, and whether the competition is severe, moderate, or weak. The various data
are summarized in the spreadsheet below.
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
High
Medium
Low
B
Price
$50
$40
$30
Sales
(thousands)
High
50
Medium
30
Low
20
C
D
Prior Probability
E
Severe
0.2
F
Moderate
0.7
G
Weak
0.1
High Price
Sales High
Sales Medium
Sales Low
Severe
0.20
0.25
0.55
Moderate
0.25
0.30
0.45
Medium Price
Sales High
Sales Medium
Sales Low
Severe
0.25
0.35
0.40
Low Price
Sales High
Sales Medium
Sales Low
Severe
0.35
0.40
0.25
H
Prior
Probability
=SUMPRODUCT($E$2:$G$2,E6:G6)
=SUMPRODUCT($E$2:$G$2,E7:G7)
=SUMPRODUCT($E$2:$G$2,E8:G8)
I
Revenue
($thousands)
=$B$2*B8
=$B$2*B9
=$B$2*B10
=SUMPRODUCT($E$2:$G$2,E11:G11)
=SUMPRODUCT($E$2:$G$2,E12:G12)
=SUMPRODUCT($E$2:$G$2,E13:G13)
=$B$3*B8
=$B$3*B9
=$B$3*B10
=SUMPRODUCT($E$2:$G$2,E16:G16)
=SUMPRODUCT($E$2:$G$2,E17:G17)
=SUMPRODUCT($E$2:$G$2,E18:G18)
=$B$4*B8
=$B$4*B9
=$B$4*B10
H
I
Weak
0.30
0.35
0.35
Prior
Probability
0.245
0.295
0.46
Revenue
($thousands)
2,500
1,500
1,000
Moderate
0.30
0.40
0.30
Weak
0.40
0.50
0.10
0.3
0.4
0.3
2,000
1,200
800
Moderate
0.40
0.50
0.10
Weak
0.50
0.45
0.05
0.4
0.475
0.125
1,500
900
600
The payoff table can be generated based on the results in column I.
Chapter 9 - Page 97
Chapter 9
DECISION ANALYSIS
The decision tree for this problem follows (over three pages):
0.20
Sales High
2,500
2,500
0.2
Severe Competition
2,500
0.25
Sales Medium
1,500
0
1425
1,500
1,500
0.55
Sales Low
1,000
1,000
1,000
0.25
Sales High
2,500
2,500
0.7
Moderate Competition
Price High
2,500
0.30
Sales Medium
1,500
0
1515
0
1525
1,500
1,500
0.45
Sales Low
1,000
1,000
1,000
0.30
Sales High
2,500
2,500
0.1
Weak Competition
2,500
0.35
Sales Medium
1,500
0
1625
1,500
1,500
0.35
Sales Low
1,000
1,000
Chapter 9 - Page 98
1,000
Chapter 9
DECISION ANALYSIS
0.25
Sales High
2,000
2,000
0.2
Severe Competition
2,000
0.35
Sales Medium
1,200
0
1240
1,200
1,200
0.40
Sales Low
800
800
800
0.30
Sales High
2,000
2,000
0.7
Moderate Competition
Price Medium
2,000
0.40
Sales Medium
1
1515
1,200
0
1320
0
1320
1,200
1,200
0.30
Sales Low
800
800
800
0.40
Sales High
2,000
2,000
0.1
Weak Competition
2,000
0.50
Sales Medium
1,200
0
1480
1,200
1,200
0.10
Sales Low
800
800
Chapter 9 - Page 99
800
Chapter 9
DECISION ANALYSIS
0.35
Sales High
1,500
1,500
1,500
0.2
Severe Competition
0.40
Sales Medium
900
0
1035
900
900
0.25
Sales Low
600
600
600
0.40
Sales High
1,500
1,500
0.7
Moderate Competition
Price Low
1,500
0.50
Sales Medium
900
0
1102.5
0
1110
900
900
0.10
Sales Low
600
600
600
0.50
Sales High
1,500
1,500
0.1
Weak Competition
1,500
0.45
Sales Medium
900
0
1185
900
900
0.05
Sales Low
600
600
Chapter 9 - Page 100
600
Chapter 9
DECISION ANALYSIS
b) The most likely state depends upon the price. As calculated in the spreadsheet below, if
the price is set high ($50), the most likely outcome is low sales (20,000), with $1
million in revenue. If the price is set medium ($40), the most likely outcome is medium
sales (30,000), with $1.2 million in revenue. If the price is set low ($30), the most likely
outcome is medium sales (30,000), with $0.9 million in revenue.
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
High
Medium
Low
B
Price
$50
$40
$30
Sales
(thousands)
High
50
Medium
30
Low
20
C
D
Prior Probability
E
Severe
0.2
F
Moderate
0.7
G
Weak
0.1
H
I
High Price
Sales High
Sales Medium
Sales Low
Severe
0.20
0.25
0.55
Moderate
0.25
0.30
0.45
Weak
0.30
0.35
0.35
Prior
Probability
0.245
0.295
0.46
Revenue
($thousands)
2,500
1,500
1,000
Medium Price
Sales High
Sales Medium
Sales Low
Severe
0.25
0.35
0.40
Moderate
0.30
0.40
0.30
Weak
0.40
0.50
0.10
0.3
0.4
0.3
2,000
1,200
800
Low Price
Sales High
Sales Medium
Sales Low
Severe
0.35
0.40
0.25
Moderate
0.40
0.50
0.10
Weak
0.50
0.45
0.05
0.4
0.475
0.125
1,500
900
600
Thus, to maximize revenue under the maximum likelihood criterion, Charlotte should
charge the medium price ($40), for a most likely outcome of $1.2 million in revenue.
Chapter 9 - Page 101
Chapter 9
DECISION ANALYSIS
c) As shown in the decision tree for part a (recall that decision trees assume Bayes’
decision rule), Charlotte should charge the high price ($50), since this maximizes the
expected revenue ($1.515 million). Alternatively, the expected revenues for each
possible decision can be calculated directly as shown in the following spreadsheet.
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
High
Medium
Low
B
Price
$50
$40
$30
Sales
(thousands)
High
50
Medium
30
Low
20
C
D
Prior Probability
E
Severe
0.2
F
Moderate
0.7
H
I
Revenue
($thousands)
2,500
1,500
1,000
J
Expected
Revenue
($thousands)
High Price
Sales High
Sales Medium
Sales Low
Severe
0.20
0.25
0.55
Moderate
0.25
0.30
0.45
Weak
0.30
0.35
0.35
Prior
Probability
0.245
0.295
0.46
Medium Price
Sales High
Sales Medium
Sales Low
Severe
0.25
0.35
0.40
Moderate
0.30
0.40
0.30
Weak
0.40
0.50
0.10
0.3
0.4
0.3
2,000
1,200
800
1,320
Low Price
Sales High
Sales Medium
Sales Low
Severe
0.35
0.40
0.25
Moderate
0.40
0.50
0.10
Weak
0.50
0.45
0.05
0.4
0.475
0.125
1,500
900
600
1,102.5
H
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
G
Weak
0.1
I
1,515
J
Expected
Revenue
($thousands)
Prior
Probability
=SUMPRODUCT($E$2:$G$2,E6:G6)
=SUMPRODUCT($E$2:$G$2,E7:G7)
=SUMPRODUCT($E$2:$G$2,E8:G8)
Revenue
($thousands)
=$B$2*B8
=$B$2*B9
=SUMPRODUCT(H6:H8,I6:I8)
=$B$2*B10
=SUMPRODUCT($E$2:$G$2,E11:G11)
=SUMPRODUCT($E$2:$G$2,E12:G12)
=SUMPRODUCT($E$2:$G$2,E13:G13)
=$B$3*B8
=$B$3*B9
=$B$3*B10
=SUMPRODUCT(H11:H13,I11:I13)
=SUMPRODUCT($E$2:$G$2,E16:G16)
=SUMPRODUCT($E$2:$G$2,E17:G17)
=SUMPRODUCT($E$2:$G$2,E18:G18)
=$B$4*B8
=$B$4*B9
=$B$4*B10
=SUMPRODUCT(H16:H18,I16:I18)
Chapter 9 - Page 102
Chapter 9
DECISION ANALYSIS
d) With more information from the marketing research company, the posterior
probabilities for the state of competition can be found using the template for posterior
probabilities as follows.
B
2 Data:
3
State of
4
Nature
5
Severe
6
Moderate
7
Weak
8
9
10
11 Posterior
12 Probabilities:
13
Finding
14
Predict Severe
15 Predict Moderate
16
Predict Weak
17
18
C
D
Prior
Probability
0.2
0.7
0.1
Predict Severe
0.8
0.15
0.03
P(Finding)
0.268
0.597
0.135
Severe
0.597
0.050
0.074
E
F
P(Finding | State)
Finding
Predict Moderate
Predict Weak
0.15
0.05
0.8
0.05
0.07
0.9
P(State | Finding)
State of Nature
Moderate
0.392
0.938
0.259
G
H
Weak
0.011
0.012
0.667
To keep the decision tree from becoming too unwieldy, we will break it into parts. The
first three parts consider the situation after each possible prediction by the marketing
research company. The decision tree from part a is reused with the only change being
the prior probabilities of severe, moderate and weak competition used in part a are
replaced by the appropriate posterior probabilities calculated above, depending upon the
prediction of the marketing research company. For example, if the marketing research
company predicts the competition will be severe, the probability of severe, moderate,
and weak competition are 0.597, 0.392, and 0.011, respectively.
Chapter 9 - Page 103
Chapter 9
DECISION ANALYSIS
The optimal decision if the marketing research company predicts severe is to price high
($50), with expected revenue of $1.466 million.
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
High
Medium
Low
B
Price
$50
$40
$30
C
Sales
(thousands)
High
50
Medium
30
Low
20
D
Prior Probability
E
Severe
0.597
F
Moderate
0.392
G
Weak
0.011
H
I
High Price
Sales High
Sales Medium
Sales Low
Severe
0.20
0.25
0.55
Moderate
0.25
0.30
0.45
Weak
0.30
0.35
0.35
Prior
Probability
0.220709
0.270709
0.5085821
Revenue
($thousands)
2,500
1,500
1,000
Medium Price
Sales High
Sales Medium
Sales Low
Severe
0.25
0.35
0.40
Moderate
0.30
0.40
0.30
Weak
0.40
0.50
0.10
0.2712687
0.3712687
0.3574627
2,000
1,200
800
Low Price
Sales High
Sales Medium
Sales Low
Severe
0.35
0.40
0.25
Moderate
0.40
0.50
0.10
Weak
0.50
0.45
0.05
0.3712687
0.4397388
0.1889925
1,500
900
600
Optimal Decision Price High
Expected Revenue
1466.42
($thousands)
The optimal decision if the marketing research company predicts moderate competition
is to price high ($50), with expected revenue of $1.521 million.
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
High
Medium
Low
B
Price
$50
$40
$30
Sales
(thousands)
High
50
Medium
30
Low
20
C
D
Prior Probability
E
Severe
0.050
F
Moderate
0.938
G
Weak
0.012
High Price
Sales High
Sales Medium
Sales Low
Severe
0.20
0.25
0.55
Moderate
0.25
0.30
0.45
Medium Price
Sales High
Sales Medium
Sales Low
Severe
0.25
0.35
0.40
Low Price
Sales High
Sales Medium
Sales Low
Severe
0.35
0.40
0.25
Optimal Decision Price High
Expected Revenue
1521.15
($thousands)
Chapter 9 - Page 104
H
I
Weak
0.30
0.35
0.35
Prior
Probability
0.2480737
0.2980737
0.4538526
Revenue
($thousands)
2,500
1,500
1,000
Moderate
0.30
0.40
0.30
Weak
0.40
0.50
0.10
0.29866
0.39866
0.3026801
2,000
1,200
800
Moderate
0.40
0.50
0.10
Weak
0.50
0.45
0.05
0.39866
0.4943886
0.1069514
1,500
900
600
Chapter 9
DECISION ANALYSIS
The optimal decision if the marketing research company predicts weak competition is
to price high ($50), with expected revenue of $1.521 million.
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
High
Medium
Low
B
Price
$50
$40
$30
C
Sales
(thousands)
High
50
Medium
30
Low
20
D
Prior Probability
E
Severe
0.074
F
Moderate
0.259
G
Weak
0.667
H
I
High Price
Sales High
Sales Medium
Sales Low
Severe
0.20
0.25
0.55
Moderate
0.25
0.30
0.45
Weak
0.30
0.35
0.35
Prior
Probability
0.2796296
0.3296296
0.3907407
Revenue
($thousands)
2,500
1,500
1,000
Medium Price
Sales High
Sales Medium
Sales Low
Severe
0.25
0.35
0.40
Moderate
0.30
0.40
0.30
Weak
0.40
0.50
0.10
0.362963
0.462963
0.1740741
2,000
1,200
800
Low Price
Sales High
Sales Medium
Sales Low
Severe
0.35
0.40
0.25
Moderate
0.40
0.50
0.10
Weak
0.50
0.45
0.05
0.462963
0.4592593
0.0777778
1,500
900
600
Optimal Decision Price High
Expected Revenue
1584.26
($thousands)
Then, incorporating the expected payoff with each possible prediction by the marketing
company, along with the expected revenue without information from part a, we
combine the whole problem into the following decision tree.
Proceed with no info
1515
1515
1515
0.268
Predict Severe
1
1456.42
1515
1466.41791
Employ Marketing Research
1456.41791
0.597
Predict Moderate
1511.15
-10
1505
1521.1474
1511.147404
0.135
Predict Weak
1574.26
1584.25926
1574.259259
Charlotte should not purchase the services of the market research company. The
information is not worth anything since it does not affect the decision. Regardless of the
prediction, the optimal policy is to set the price at $50.
Chapter 9 - Page 105
Chapter 9
DECISION ANALYSIS
9.4
a) The available data are summarized in the following spreadsheet.
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
B
Costs
($million)
0.3
0.8
0.2
Research
Development
Marketing
C
D
Probability
of Success
0.8
0.65
Revenues from R&D
($million)
Sell Product Rights
1
Sell Research Results
0.2
Sell Current DSS
2
Sales
Revenue
($million)
8
4
2.2
High
Medium
Low
Probability
0.3
0.5
0.2
b) The basic decision tree is shown below.
No Success (Sell Current DSS)
Do Research
Don't Develop (Sell Current DSS & Research Results)
Success
No Success (Sell Current DSS & Research Results)
Develop
Don't Market (Sell Current DSS & Rights)
Success
High
Market
Medium
Low
Don't do Research (Sell Current DSS)
Chapter 9 - Page 106
Chapter 9
DECISION ANALYSIS
c) The decision tree displays all the expected payoffs and probabilities.
0.2
No Success (Sell Current DSS)
1.7
2
1.7
Do Research
Don't Develop (Sell Current DSS & Research Results)
1.9
-0.3
2.489
2.2
1.9
0.8
Success
0.35
No Success (Sell Current DSS & Research Results)
2
0
1.1
2.69
2.2
1.1
Develop
-0.8
Don't Market (Sell Current DSS & Product Rights)
1.9
3
1.9
2.69
0.65
Success
0.3
High
1
2
2.489
0
6.7
3.54
8
6.7
0.5
Medium
Market
2.7
-0.2
3.54
4
2.7
0.2
Low
0.9
2.2
0.9
Don't do Research (Sell Current DSS)
2
2
2
d) The best course of action is to do the research project. The expected payoff equals
$2.489 million.
Chapter 9 - Page 107
Chapter 9
DECISION ANALYSIS
e) The decision tree with perfect information on research is displayed. The expected value
in this case equals $2.549 million. The difference between the expected values with and
without information equals $60,000, which is the value of perfect information on
research.
No Research (Sell Current DSS)
2
2
2
0.8
Research will be Success
2
0
2.686
Don't Develop (Sell Current DSS & Research Results)
1.9
2.2
1.9
0.35
No Success (Sell Current DSS & Research Results)
Do Research
2
-0.3
1.1
2.686
2.2
1.1
Develop
Don't Market (Sell Current DSS & Product Rights)
1.9
-0.8
2.686
3
1.9
0.65
Success
0.3
High
2
2.5488
0
6.7
3.54
8
6.7
0.5
Medium
Market
2.7
-0.2
3.54
4
2.7
0.2
Low
0.9
2.2
0.9
0.2
Research won't be Successful (Sell Current DSS)
2
2
2
Chapter 9 - Page 108
Chapter 9
DECISION ANALYSIS
f) The decision tree with perfect information on development is displayed. The expected
value in this case equals $2.762 million. The difference between the expected values
with and without information equals $273,000, which is the value of perfect
information on development.
No Research (Sell Current DSS)
2
2
2
0.65
Development Successful
0.2
No Success (Sell Current DSS)
2
0
1.7
3.172
2
1.7
Do Research
Don't Develop (Sell Current DSS & Research Results)
1.9
-0.3
3.172
2.2
1.9
0.8
Success
Don't Market (Sell Current DSS & Product Rights)
1.9
3
1.9
2
0
3.54
0.3
High
Develop
2
2.7618
-0.8
6.7
3.54
8
6.7
0.5
Medium
Market
2.7
-0.2
3.54
4
2.7
0.2
Low
0.9
2.2
0.9
0.35
Development would not be Successful (Sell Current DSS)
2
2
2
Chapter 9 - Page 109
Chapter 9
DECISION ANALYSIS
g-i) The decision tree with expected utilities is displayed. The expected utilities are
calculated in the following way: for each of the outcome branches of the decision tree
(e.g. profit of $1.7 million) the corresponding utility is calculated (e.g. 9.4737). Once
this is done, the expected utilities are automatically calculated by TreePlan. The best
course of action is not to do the research (expected utility of 10.145 vs. 9.846 in the
case of “Do research”).
0.2
No Success (Sell Current DSS)
2
9.4737 Utility
1.7 $million
9.47
Do Research
-0.3
Don't Develop (Sell Current DSS & Research Results)
9.846
2.2
9.9394 Utility
1.9 $million
9.94
0.8
Success
0.35
No Success (Sell Current DSS & Research Results)
1
0
9.94
2.2
7.506 Utility
1.1 $million
7.51
Develop
-0.8
Don't Market (Sell Current DSS & Product Rights)
9.9394 Utility
3 9.94
1.9 $million
9.55
0.65
Success
0.3
High
2
2
10.14
0
10.7
12.46
12.46 Utility
6.7 $million
4 11.189
11.189 Utility
2.7 $million
8
0.5
Medium
Market
-0.2
10.7
0.2
Low
2.2 6.5991
6.5991 Utility
0.9 $million
Don't do Research (Sell Current DSS)
2
10.145 Utility
2 $million
10.145
Chapter 9 - Page 110
Chapter 9
DECISION ANALYSIS
j) The expected utility of doing research, even if we know it will be successful (with
perfect information) equals 9.9394 which is still less than the expected utility of the
alternative “No research” (10.145). Therefore, the best course of action is not to do the
research no matter what, implying a value of zero for perfect information on the
outcome of the research effort.
No Research (Sell Current DSS)
10.145 Utility
2 $million
2 10.145
0.8
Research will be Success
1
0 10.14469
Don't Develop (Sell Current DSS & Research Results)
2.2
9.9394 Utility
1.9 $million
9.9394
0.35
No Success (Sell Current DSS & Research Results)
Do Research
1
-0.3 9.9394
2.2
7.506 Utility
1.1 $million
7.506
Develop
Don't Market (Sell Current DSS & Product Rights)
9.9394 Utility
3 9.9394
1.9 $million
-0.8 9.55102
0.65
Success
0.3
High
2
10.145
0
10.65
8
12.46 Utility
6.7 $million
11.19
11.189 Utility
2.7 $million
6.599
6.5991 Utility
0.9 $million
0.5
Medium
Market
-0.2
12.46
10.652
4
0.2
Low
2.2
0.2
Research won't be Successful (Sell Current DSS)
10.145 Utility
2 $million
2 10.14469
Chapter 9 - Page 111
Chapter 9
DECISION ANALYSIS
k) The expected utility for perfect information on development equals 10.321 which is
more than the expected utility for the case with no information (10.145). The value of
perfect information on development is calculated as the difference between the inverses
of these two utility values (U-1[10.321] - U-1[10.145] = 20.933 – 20 = 0.933. The value
of perfect information on the outcome of the development effort equals $93,300.
No Research (Sell Current DSS)
10.1447 Utility
2 $million
2 10.145
0.65
Development Successful
0.2
No Success (Sell Current DSS)
2
0
10.4164711
9.47375 Utility
1.7 $million
2 9.4737
Do Research
Don't Develop (Sell Current DSS & Research Results)
-0.3 10.416
9.9394 Utility
1.9 $million
2.2 9.9394
0.8
Success
Don't Market (Sell Current DSS & Product Rights)
9.9394 Utility
3 9.9394
1.9 $million
2
0 10.652
0.3
High
Develop
2
10.321
-0.8 10.652
8 12.5
Market
-0.2 10.652
12.4599 Utility
6.7 $million
0.5
Medium
4 11.2
11.1887 Utility
2.7 $million
0.2
Low
2.2
6.6
6.59908 Utility
0.9 $million
0.35
Development would not be Successful (Sell Current DSS)
2
10.1447 Utility
2 $million
10.1446871
Chapter 9 - Page 112