### C.4 The Value of Information

```Readings
Chapter 13
Decision Analysis
BA 452 Lesson C.4 The Value of Information
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Overview
Overview
BA 452 Lesson C.4 The Value of Information
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Overview
Expected Value of Perfect Information is the increase in the expected profit that
would result if one knew with certainty which state of nature would occur. It is
an upper bound on the value of information.
Bayes’ Rule revises prior probability estimates for the states of nature into
posterior probabilities. The rule uses conditional probabilities for the outcomes
or indicators of the sample or survey information.
Expected Value of Sample Information is the additional expected profit possible
through knowledge of sample information. It is typically less than the Expected
Value of Perfect Information.
BA 452 Lesson C.4 The Value of Information
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Expected Value of Perfect Information
Expected Value of Perfect
Information
BA 452 Lesson C.4 The Value of Information
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Expected Value of Perfect Information
Overview
Expected Value of Perfect Information is the increase in the
expected profit that would result if one knew with certainty
which state of nature would occur. It provides an upper
bound on the expected value of any sample or survey
information that better estimates the probability estimates
for the states of nature.
BA 452 Lesson C.4 The Value of Information
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Expected Value of Perfect Information




Frequently, information is available that can better
estimate the probabilities for the states of nature. (For
example, you can take the time to listen to the radio to
In the extreme, you can find the state of nature with
certainty. (For example, you can call the Pepperdine
hotline to see if PCH is open.)
The expected value of perfect information (EVPI) is the
increase in the expected profit that would result if one
knew with certainty which state of nature would occur.
The EVPI provides an upper bound on the expected
value of any sample or survey information that better
estimates the probability estimates for the states of
nature.
BA 452 Lesson C.4 The Value of Information
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Expected Value of Perfect Information

EVPI Calculation
• Step 1:
Determine the optimal return corresponding to
each state of nature.
• Step 2:
Compute the expected value of those optimal
returns.
• Step 3:
Subtract the EV of the optimal decision without
knowing the state of nature from the amount
determined in Step 2.
BA 452 Lesson C.4 The Value of Information
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Expected Value of Perfect Information

How much should Medieval Times be willing to pay to
determine the average number of customers per hour
(the state of nature)?
 The probabilities of states s1, s2, s3 were .4, .2, and
.4:
 Maximum expected profit was EV(C) = .4(\$6,000) +
.2(\$16,000) + .4(\$21,000) = \$14,000
Average Number of Customers Per Hour
s1 = 80 s2 = 100 s3 = 120
Model A
Model B
Model C
\$10,000
\$ 8,000
\$ 6,000
\$15,000
\$18,000
\$16,000
\$14,000
\$12,000
\$21,000
BA 452 Lesson C.4 The Value of Information
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Expected Value of Perfect Information





If you knew the state were s1, choose A and earn \$10,000.
If you knew the state were s2, choose B and earn \$18,000
If you knew the state were s3, choose C and earn \$21,000
EV = .4(10,000) + .2(18,000) + .4(21,000) = \$16,000
EV(perfect information) is \$2,000 than EV(no info).
Average Number of Customers Per Hour
s1 = 80 s2 = 100 s3 = 120
Model A
Model B
Model C
\$10,000
\$ 8,000
\$ 6,000
\$15,000
\$18,000
\$16,000
\$14,000
\$12,000
\$21,000
BA 452 Lesson C.4 The Value of Information
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Bayes’ Rule
Bayes’ Rule
BA 452 Lesson C.4 The Value of Information
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Bayes’ Rule
Overview
Bayes’ Rule revises prior probability estimates for the
states of nature into posterior probabilities. The rule uses
conditional probabilities for the outcomes or indicators of
the sample or survey information.
BA 452 Lesson C.4 The Value of Information
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Bayes’ Rule




Knowledge of sample (survey) information can better
estimate the probabilities for the states of nature.
Before getting this information, probability estimates for
the states of nature are called prior probabilities.
With knowledge of conditional probabilities for the
outcomes or indicators of the sample or survey
information, these prior probabilities can be revised by
employing Bayes' Theorem. (The accuracy of a sample
or survey is measured by its conditional probabilities.)
The revised probabilities are called posterior probabilities
or branch probabilities for decision trees.
BA 452 Lesson C.4 The Value of Information
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Bayes’ Rule


For example, suppose a certain drug test will correctly
identify a drug user as testing positive 99% of the time,
and will correctly identify a non-user as testing negative
99% of the time.
Suppose a corporation decides to test its employees for
opium use, and the corporation believes 0.5% of the
employees use the drug. We want to know the
probability that, given a positive drug test, an employee
is actually a drug user.
BA 452 Lesson C.4 The Value of Information
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Bayes’ Rule



Let "D" be the event of being a drug user; and "N“, being
a non-user. Let "+" be the event of a positive drug test.
The population is defined by P(D), or the prior probability
that the employee is a drug user. This is 0.005, since
0.5% of the employees are drug users. Hence, P(N), or
the probability that the employee is not a drug user, is 1P(D), or 0.995.
The accuracy of the drug test is defined by two numbers:
• P(+|D), or the probability that the test is positive,
given that the employee is a drug user. This is 0.99,
since the test will correctly identify a drug user as
testing positive 99% of the time.
• P(+|N), or the probability that the test is positive,
given the employee is not a user. This is 0.01, since
the test produces a false positive for 1% of non-users.
BA 452 Lesson C.4 The Value of Information
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Bayes’ Rule
Hence, follow a sequence of steps to compute the probability a person
is a drug user given a positive test:
 The probability a person is a drug user and tests positive is
P(D&+) = P(+|D) x P(D) = 0.99 x 0.005 = 0.00495 = 0.495%
 The probability a person is a non-drug user but tests positive is
P(N&+) = P(+|N) x P(N) = 0.01 x 0.995 = 0.00995 = 0.995%
 The probability a person tests positive
P(+) = P(D&+) + P(N&+) = 0.0149 = 1.49%.
 The probability a person is a drug user given a positive test
P(D|+) = P(D&+)/P(+) = 0.00495/0.0149 = .3322.
• Even though the test is 99% accurate, P(D|+) = .3322, only
33.22% of those testing positive are drug users!
BA 452 Lesson C.4 The Value of Information
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Expected Value of Sample Information
Expected Value of Sample
Information
BA 452 Lesson C.4 The Value of Information
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Expected Value of Sample Information
Overview
Expected Value of Sample Information is the additional
expected profit possible through knowledge of the sample
or survey information. It is less than the Expected Value of
Perfect Information when samples and surveys are
imperfect.
BA 452 Lesson C.4 The Value of Information
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Expected Value of Sample Information


In general, the expected value of sample information
(EVSI) is the additional expected profit possible through
knowledge of the sample or survey information.
EVSI calculation
• Step 1: Determine the optimal decision and its
expected profit for the possible outcomes of the
sample using the posterior probabilities for the states
of nature.
• Step 2: Compute the expected value of these optimal
profits.
• Step 3: Subtract the EV of the optimal decision
obtained without using the sample information from
the amount determined in Step 2.
BA 452 Lesson C.4 The Value of Information
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Expected Value
Question: Medieval Times Dinner Theater is considering
opening a new theater on Main Street. It has three
different models, each with a different seating capacity.
Medieval Times estimates that the average number of
customers per show will be 800, 1000, or 1200. Here is the
payoff table for the three models:
Average Number of Customers Per Hour
s1 = 800 s2 = 1000 s3 = 1200
Model A
Model B
Model C
\$10,000
\$ 8,000
\$ 6,000
\$15,000
\$18,000
\$16,000
\$14,000
\$12,000
\$21,000
BA 452 Lesson C.4 The Value of Information
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Expected Value of Sample Information
Medieval Times Dinner Theater must decide whether or
not to spend \$1,000 for a marketing survey from
Stanton Marketing. The two possible results of the
survey are "favorable" or "unfavorable". The conditional
probabilities are:
P(favorable | 800 customers per show) = .2
P(favorable | 1000 customers per show) = .5
P(favorable | 1200 customers per show) = .9
Should Medieval Times spend \$1,000 for the survey
from Stanton Marketing if the prior probabilities of states
s1, s2, and s3 are .4, .2, and .4?
BA 452 Lesson C.4 The Value of Information
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Expected Value of Sample Information
Answer: Compare \$1000 to EVSI. Compute EVSI in three
steps.
• Step 1: Determine the optimal decision and its
expected profit for the possible outcomes of the
sample using the posterior probabilities for the states
of nature.
• Step 2: Compute the expected value of these optimal
profits.
• Step 3: Subtract the EV of the optimal decision
obtained without using the sample information from
the amount determined in Step 2.
BA 452 Lesson C.4 The Value of Information
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Expected Value of Sample Information
Favorable survey result
State Prior
800
.4
1000
.2
1200
.4
Conditional Joint Posterior
.2
.08
.148
.5
.10
.185
.9
.36
.667
Total .54
1.000
P(favorable) = .54
For the first state (80 customers),
 Joint probability = 0.08 = 0.4 x 0.2 = Prior x Conditional
 Marginal P(favorable) = 0.54 = 0.08+0.10+0.36 = S Joints
 Posterior probability = 0.148 = 0.08/0.54 = Joint/Marginal
BA 452 Lesson C.4 The Value of Information
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Expected Value of Sample Information
Unfavorable survey result
State Prior
800
.4
1000
.2
1200
.4
Conditional Joint Posterior
.8
.32
.696
.5
.10
.217
.1
.04
.087
Total .46
1.000
P(unfavorable) = .46
For the first state (80 customers),
 Joint probability = 0.32 = 0.4 x 0.8 = Prior x Conditional
 Marginal P(unfavorable) = 0.46 = 0.32+0.10+0.04 = S
Joints
 Posterior probability = 0.696 = 0.32/0.46 = Joint/Marginal
BA 452 Lesson C.4 The Value of Information
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Expected Value of Sample Information

Top half (favorable survey result)
s1 (.148)
d1
2
I1
d2
4
5
d3
s2 (.185)
\$15,000
s3 (.667)
\$14,000
s1 (.148)
\$8,000
s2 (.185)
\$18,000
s3 (.667)
s1 (.148)
(.54)
6
\$10,000
s2 (.185)
s3 (.667)
1
\$12,000
\$6,000
\$16,000
\$21,000
BA 452 Lesson C.4 The Value of Information
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Expected Value of Sample Information

Bottom half (unfavorable survey result)
1
I2
(.46)
d1
7
s1 (.696)
\$10,000
s2 (.217)
\$15,000
s3 (.087)
\$14,000
d2
8
3
s1 (.696)
\$8,000
s2 (.217)
s3 (.087)
\$18,000
\$12,000
\$6,000
d3
9
s1 (.696)
s2 (.217)
s3 (.087) \$16,000
\$21,000
BA 452 Lesson C.4 The Value of Information
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Expected Value of Sample Information
d1
\$17,855
I1
(.54)
2
d2
4
EMV = .148(10,000) + .185(15,000)
+ .667(14,000) = \$13,593
5
EMV = .148 (8,000) + .185(18,000)
+ .667(12,000) = \$12,518
6
EMV = .148(6,000) + .185(16,000)
+.667(21,000) = \$17,855
7
EMV = .696(10,000) + .217(15,000)
+.087(14,000)= \$11,433
8
EMV = .696(8,000) + .217(18,000)
+ .087(12,000) = \$10,554
9
EMV = .696(6,000) + .217(16,000)
+.087(21,000) = \$9,475
d3
1
d1
I2
(.46)
3
\$11,433
d2
d3
BA 452 Lesson C.4 The Value of Information
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Expected Value of Sample Information
If the outcome of the survey is "favorable”, choose Model
C. If it is “unfavorable”, choose Model A.
EVSI = .54(\$17,855) + .46(\$11,433) - \$14,000 = \$900.88
Since that is less than the cost of the survey, the survey
should not be purchased.
BA 452 Lesson C.4 The Value of Information
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Review Questions
Review Questions
 You should try to answer some of the following
questions before the next class.
request to discuss their answers to begin the next class.
 Your upcoming Final Exam will contain some similar
questions, so you should eventually consider every
review question before taking your exams.
BA 452 Lesson C.4 The Value of Information
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Review 1: Expected Value of Sample Information
Review 1: Expected Value of
Sample Information
BA 452 Lesson C.4 The Value of Information
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Review 1: Expected Value of Sample Information
The Gorman Manufacturing Company must decide whether
to make a component part at its Milan, Michigan, plan or
buy the part from a supplier. The profit depends on the
demand for the product. The following table shows the
profit (in thousands of dollars):
States of Demand
s1 (low) s2 (med) s3 (high)
Manufacture,
Purchase,
d1
d2
-20
10
40
45
BA 452 Lesson C.4 The Value of Information
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70
30
Review 1: Expected Value of Sample Information
Manufacture,
Purchase,
d1
d2
States of Demand
s1 (low) s2 (med) s3 (high)
-20
40
100
10
45
70
Compute the optimal decision (which maximizes
expected value) given priors
P(s1) = 0.35, P(s2) = 0.35, P(s3) = 0.30.
EV(d1) = 0.35 x (-20) + 0.35 x (40) + 0.30 x (100) =
37.
EV(d2) = 0.35 x (10) + 0.35 x (45) + 0.30 x (70) =
40.25.
So, choose Purchase, d2.
BA 452 Lesson C.4 The Value of Information
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Review 1: Expected Value of Sample Information
Manufacture,
Purchase,
d1
d2
States of Demand
s1 (low) s2 (med) s3 (high)
-20
40
100
10
45
70
Use EVPI (expected value of perfect information) to
determine whether Gorman should try to better
estimate demand.
If you had perfect information, then in state s1, pick
d2 for payoff 10; in state s2, pick d2 for payoff 45;
and in state s3, pick d1 for payoff 100. Hence, the
expected payoff is 0.35 x (10) + 0.35 x (45) + 0.30 x
(100) = 49.25, which is 9 more than the expected
payoff EV(d2) = 40.25 from the optimum given the
priors. EVPI thus = 9 (that is, 9 thousand), which is
positive so information is valuable.
BA 452 Lesson C.4 The Value of Information
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Review 1: Expected Value of Sample Information
Manufacture,
Purchase,
d1
d2
States of Demand
s1 (low) s2 (med) s3 (high)
-20
40
100
10
45
70
What is Gorman’s optimal strategy (which
market study that concludes a F = Favorable or U =
Unfavorable outcome with the following conditional
probabilities?
P(favorable | low state s1) = 0.1
P(favorable | med state s2) = 0.4
P(favorable | high state s3) = 0.6
BA 452 Lesson C.4 The Value of Information
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Review 1: Expected Value of Sample Information
Manufacture,
Purchase,
d1
d2
States of Demand
s1 (low) s2 (med) s3 (high)
-20
40
100
10
45
70
Favorable survey result
State
s1
s2
s3



Prior P(si) Cond. P(F|si) Joint P(si&F) Post P(si|F)
0.35
0.1
0.035
0.0986
0.35
0.4
0.140
0.3944
0.30
0.6
0.180
0.5070
P(favorable) = 0.355
EV(d1) = 0.0986 x (-20) + 0.3944 x (40) + 0.5070 x (100) = 64.51.
EV(d2) = 0.0986 x (10) + 0.3944 x (45) + 0.5070 x (70) = 54.23.
So, if the survey is Favorable, choose Manufacture, d1.
BA 452 Lesson C.4 The Value of Information
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Review 1: Expected Value of Sample Information
Manufacture,
Purchase,
d1
d2
States of Demand
s1 (low) s2 (med) s3 (high)
-20
40
100
10
45
70
Unfavorable survey result
State
s1
s2
s3



Prior P(si) Cond. P(U|si) Joint P(si&U)
0.35
0.9
0.315
0.35
0.6
0.210
0.30
0.4
0.120
P(Unfavorable) = 0.645
Post P(si|U)
0.4884
0.3256
0.1860
EV(d1) = 0.4884 x (-20) + 0.3256 x (40) + 0.1860 x (100) = 21.86.
EV(d2) = 0.4884 x (10) + 0.3256 x (45) + 0.1860 x (70) = 32.56.
So, if the survey is Unfavorable, choose Purchase, d2.
BA 452 Lesson C.4 The Value of Information
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Review 1: Expected Value of Sample Information
Manufacture,
Purchase,





d1
d2
States of Demand
s1 (low) s2 (med) s3 (high)
-20
40
100
10
45
70
The survey is Favorable with probability P(Favorable) =
0.355. And contingent on Favorable, the expected
payoff is 64.51.
The survey is Unfavorable with probability
P(Unfavorable) = 0.645. And contingent on
Unfavorable, the expected payoff is 32.56.
Therefore, the un-contingent expected payoff is
0.355 x 64.51 + 0.645 x 32.56 = 43.90.
The expected payoff with no information (priors) is 40.25.
Therefore, the EVSI = 43.90-40.25 = 3.65 (\$3,650).
BA 452 Lesson C.4 The Value of Information
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Review 2: Expected Value of Sample Information
Review 2: Expected Value of
Sample Information
BA 452 Lesson C.4 The Value of Information
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Review Problems




Dollar Department Stores has received an offer from
Harris Diamonds to purchase Dollar’s store on Grove
Street for \$120,000. Dollar is an expected-value
maximizer. Dollar has determined probability estimates
of the store's future profitability, based on economic
outcomes, as:
• P(\$80,000) = .2, P(\$100,000) = .3, P(\$120,000) = .1, and
P(\$140,000) = .4.
Should Dollar sell the store on Grove Street?
What is the EVPI?
Dollar can have an economic forecast performed. The forecast
conditions. Probabilities of the indicators conditional on future
profitability are P(G|\$80,000) = .1; P(G|\$100,000) = .2;
P(G|\$120,000) = .6; P(G|\$140,000) = .3. Should Dollar purchase
that forecast for \$10,000? For \$1,000?
BA 452 Lesson C.4 The Value of Information
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Review Problems

This problem can be solved like the previous problem by
first converting the profit data and selling price into a
payoff table:
• P(s1 = \$80,000) = .2, P(s2 = \$100,000) = .3,
P(s3 = \$120,000) = .1, and P(s4 = \$140,000) = .4
• Offer to sell for \$120,000.
s1
Do not sell, d1
Sell,
d2
States of Demand
s2
s3
80,000 100,000
120,000 120,000
s4
120,000
120,000
140,000
120,000
BA 452 Lesson C.4 The Value of Information
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Review Problems







Consider priors
• P(\$80,000) = .2, P(\$100,000) = .3, P(\$120,000) = .1,
and P(\$140,000) = .4.
Should Dollar sell the store on Grove Street?
Expected profit (in thousands of \$)
= .2 x 80 + .3 x 100 + .1 x 120 + .4 x 140 = 114, so
selling for \$120,000 increases value from \$114,000.
Here is a more elaborate way to say the same thing:
EV(d1=Not sell) = .2x80 + .3x100 + .1x120 + .4x140 =
114.
EV(d2=Sell) = .2x120 + .3x120 + .1x120 + .4x120 = 120.
So, choose Sell, d2.
BA 452 Lesson C.4 The Value of Information
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Review Problems





Consider priors
• P(\$80,000) = .2, P(\$100,000) = .3, P(\$120,000) = .1,
and P(\$140,000) = .4.
What is the EVPI? Expected value of perfect information
is what you can expect if you know the store’s future
profitability before you have to decide whether to sell.
If future profit is either \$80,000 or \$100,000, then you sell for
\$120,000 (choose d2); if future profit is \$120,000, then it does not
matter whether you sell or not (choose d1 or d2); and if future profit is
\$140,000, then you do not sell (choose d1).
So, with perfect information, you earn \$120,000, except in the
probability P(\$140,000) = .4 state 4 event, when you earn \$140,000.
So you can expect .6 x \$120,000 + .4 x \$140,000 = \$128,000, which
is \$8,000 more than selling for \$120,000.
Therefore, EVPI = \$8,000.
BA 452 Lesson C.4 The Value of Information
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Review Problems




Consider priors
• P(\$80,000) = .2, P(\$100,000) = .3, P(\$120,000) = .1,
and P(\$140,000) = .4.
EVPI = \$8,000.
Should Dollar purchase the forecast for \$10,000?
No, because the value of the forecast is at most EVPI,
which is less than \$10,000.
BA 452 Lesson C.4 The Value of Information
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Review Problems










Should Dollar purchase the forecast for \$1,000?
First, compute posterior probabilities under the two possible forecasts: G =
If G, then
Priors
Conditional
Joint
Posterior
P(s1) = .2
P(G|s1) = .1
P(G&s1) = .02
P(s1|G) = .02/.26 = .077
P(s2) = .3
P(G|s2) = .2
P(G&s2) = .06
P(s2|G) = .06/.26 = .231
P(s3) = .1
P(G|s3) = .6
P(G&s3) = .06
P(s3|G) = .06/.26 = .231
P(s4) = .4
P(G|s4) = .3
P(G&s4) = .12
P(s4|G) = .12/.26 = .462
P(G) = .26
Expected Value of Store Given G =
.077x\$80,000 + .231x\$100,000+.231x\$120,000+.462x\$140,000=\$121,660
Since that value is greater than \$120,000, you do not sell the store if the
report is G = Good business conditions.
BA 452 Lesson C.4 The Value of Information
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Review Problems









Should Dollar purchase the forecast for \$1,000?
If B, then
Priors
Conditional
Joint
Posterior
P(s1) = .2
P(B|s1) = .9
P(B&s1) = .18
P(s1|B) = .18/.74 = .243
P(s2) = .3
P(B|s2) = .8
P(B&s2) = .24
P(s2|B) = .24/.74 = .324
P(s3) = .1
P(B|s3) = .4
P(B&s3) = .04
P(s3|B) = .04/.74 = .054
P(s4) = .4
P(B|s4) = .7
P(B&s4) = .28
P(s4|B) = .28/.74 = .378
P(B) = .74
Expected Value of Store Given B =
.243x\$80,000 + .324x\$100,000+.054x\$120,000+.378x\$140,000=\$111,240
Since that value is less than \$120,000, you do sell the store if the report is B
BA 452 Lesson C.4 The Value of Information
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Review Problems





Putting it all together,
P(G) = .26 of a Good report and expected profit
=\$121,660.
P(B) = .74 of a Bad report and expected profit \$120,000.
Overall, expected profit is .26x\$121,660 + .74\$120,000 =
\$120,431.
Therefore, EVSI = \$120,431-\$120,000 = \$431, and you
should not pay \$1,000 for the economic forecast.
BA 452 Lesson C.4 The Value of Information
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BA 452
Quantitative Analysis
End of Lesson C.4
BA 452 Lesson C.4 The Value of Information
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