Solution to Witkowski TV Productions1
Witkowski TV Productions is considering a pilot for a comedy series for a major
television network. The network may reject the pilot and the series, or it may
purchase the program for one or two years. Witkowski may decide to produce
the pilot or transfer the rights for the series to a competitor for $100,000.
Witkowski’s profits are summarized in the following profit ($1000s) payoff table:
Produce Pilot
Sell to Competitor
d1
d2
States of Nature
s1 = Reject
s2 = 1 Year s3 = 2 Years
-100
50
150
100
100
100
(a) If the probability estimates for the states of nature are P(Reject) = 0.20, P(1
Year) = 0.30, and P(2 Years) = 0.50, what should Witkowski do?
On the basis of expected value, the best thing to do is to sell to the competitor:
Reject
20.0%
-100
Produce Pilot
FALSE
0
Expected Value
70
1 Year
30.0%
50
2 Years
50.0%
150
Witkowski
Expected Value
100
Reject
20.0%
100
Sell to Competitor
TRUE
0
Expected Value
100
1 Year
30.0%
2 Years
50.0%
100
100
1
David Juran.
Decision Analysis with Perfect Information
A more advanced concept in decision analysis involves considering how the
optimal choice is affected by knowing in advance what the future state of nature
will be.
In this TV Production problem there is clearly a “best” choice of decision
alternative for each of the possible states of nature. Here, the optimal decision for
each state of nature is identified with a non-shaded cell:
State of
Nature
Reject
1 Year
2 Years
Probabilities
0.2
0.3
0.5
Net Payout if Pilot is
Produced
-100
50
150
Net Payout if Sold to
Competitor
100
100
100
Optimal Decision
Sell to Competitor
Sell to Competitor
Produce Pilot
We calculate the expected value with perfect information by summing up the
probability-weighted best payoffs for each state of nature. For this example:
EVwPI
 0.20 * 100  0.30 * 100  0.50 * 150
 20  30  75
 125
This result can be interpreted as follows: If we know ahead of time that the true
state of nature will be revealed before we make the decision, then the expected
value of the problem is 125 instead of 100. Therefore, perfect information (if it
were available) would be worth up to 125 - 100 = 25 thousand dollars to
Witkowski. This is referred to as expected value of perfect information.
This number may not appear to have much practical significance, but it does
provide some basis for considering whether to collect additional information
before making the decision. If the expected value of perfect information is small,
then there is little to be gained from additional research, no matter what the
results of the new information might be. However, if the expected value of
perfect information is large, then there is an opportunity to create value by
conducting a research project before making the decision.
B60.2350
2
Prof. Juran
Decision Analysis with Sample Information
For a consulting fee of $2,500, the O’Donnell agency will review the plans for the
comedy series and indicate the overall chance of a favorable network reaction.
PI 1 s 1   0.30
PI 2 s 1   0.70
PI 1 s 3   0.90
PI 2 s 3   0.10
PI 1 s 2   0.60
PI 2 s 2   0.40
Perfect information may be impossible to obtain, but we can often get sample
information about the future states of nature, for example by performing a
market research project. New information from the results of such a project
might make us more confident in choosing one of the decision alternatives. The
following analysis is aimed at placing a monetary value on this improvement in
confidence.
In this problem, Witkowski could hire O’Donnell to review the plans for the
comedy series and indicate the overall chance of a favorable network reaction.
Here are the conditional probabilities of each state of nature, given each possible
outcome from O’Donnell (based on historical outcomes):
O’Donnell Results
I1 = Favorable I2 = Unfavorable
PI 1 s1   0.30
PI 2 s1   0.70
Reject
PI 2 s 2   0.40
1 year PI 1 s 2   0.60
PI 1 s3   0.90
2 years
PI 2 s3   0.10
Using Bayes’ Law, we can use these conditional probabilities to calculate
posterior probabilities (probabilities for each state of nature given each possible
outcome of the O’Donnell report):
Probability calculations:
States
Prior
sj
P sj
P I1 s j
Reject
1 Year
2-Year
0.20
0.30
0.50
0.30
0.60
0.90
 
Probabilities
Joint
Conditional


  

P s j  I1  P s j P I1 s j

0.06
0.18
0.45
Total PI 1   0.69
Posterior


P s j I1 

P s j  I1
P I 1 

0.087
0.261
0.652
The total probability of a favorable O’Donnell report is 69%.
B60.2350
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Prof. Juran
States
Prior
sj
P sj
P I2 sj
Reject
1 Year
2-Year
0.20
0.30
0.50
0.70
0.40
0.10
Probabilities
Joint
Conditional
 


  

P sj  I2  P sj P I2 sj

Posterior


P sj I2 
0.14
0.12
0.05

P sj  I2
PI 2 

0.452
0.387
0.161
Total PI 2   0.31
The total probability of an unfavorable O’Donnell report is 31%.
Now we can calculate an expected value for each decision alternative for each
possible outcome of the O’Donnell project, and we can calculate an overall
expected value.
A revised payoff table:
States of Nature
No
Report
Get
Report
s1 = Reject
s2 = 1 Year
s3 = 2 Years
d1 = Produce Pilot
-100
50
150
d2 = Sell to Competitor
100
100
100
-102.5
47.5
147.5
97.5
97.5
97.5
-102.5
47.5
147.5
97.5
97.5
97.5
d1 = Produce Pilot
I1 = Favorable Report
d2 = Sell to Competitor
I2 = Unfavorable Report
d1 = Produce Pilot
d2 = Sell to Competitor
In the event of a favorable O’Donnell report, the expected value of producing the
pilot is:
EV d 1 I 1 
 v11 Ps 1 I 1   v12 Ps 2 I 1   v13 Ps 3 I 1 
  102.5 0.087   47.5 0.261  147.5 0.652 
 99.65
In the event of a favorable O’Donnell report, the expected value of selling to the
competitor is:
EV d 2 I 1 
 v 21 Ps 1 I 1   v 22 Ps 2 I 1   v 23 Ps 3 I 1 
 97.5 0.087   97.5 0.261  97.5 0.652 
 97.50
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Prof. Juran
In the event of an unfavorable O’Donnell report, the expected value of producing
the pilot is:
EV d 1 I 2 
 v11 Ps 1 I 2   v12 Ps 2 I 2   v13 Ps 3 I 2 
  102.5 0.452   47.5 0.387   147.5 0.161
 4.20
In the event of an unfavorable O’Donnell report, the expected value of selling to
the competitor is:
EV d 2 I 2 
 v 21 Ps 1 I 2   v 22 Ps 2 I 2   v 23 Ps 3 I 2 
 97.5 0.452   97.5 0.387   97.5 0.161
 97.5
Looking at the table here, we can see that for each possible outcome of the
O’Donnell project there is an optimal strategy:
Favorable O’Donnell Report
Unfavorable O’Donnell Report
Decision Alternative
Expected Value
Produce Pilot
99.65
Sell to Competitor
97.50
Produce Pilot
-4.20
Sell to Competitor
97.50
← Optimal
← Optimal
The overall expected value with sample information (EVwSI) is:
EV I 1 PI 1   EV I 2 PI 2 
 99.65 * 0.69   97.5 * 0.31  98.98
(Note that we are assuming here that we will always adopt the optimal strategy
in light of whatever information O’Donnell provides.)
B60.2350
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Prof. Juran
Developing an Optimal Decision Strategy
(b) Show a decision tree to represent the revised problem.
8.7%
Reject
-102.5
Produce Pilot
TRUE
Expected Value
0.0
99.7
1 Year
26.1%
2 Years
65.2%
47.5
147.5
Favorable
0.7
Expected Value
0.0
99.7
8.7%
Reject
97.5
Sell to Competitor
FALSE
Expected Value
0.0
97.5
1 Year
26.1%
2 Years
65.2%
97.5
97.5
Get O'Donnell Report
FALSE
0.0
Expected Value
99.0
45.2%
Reject
-102.5
Produce Pilot
FALSE
Expected Value
0.0
-4.1
1 Year
38.7%
2 Years
16.1%
47.5
147.5
Unfavorable
0.3
Expected Value
0.0
97.5
45.2%
Reject
97.5
Sell to Competitor
TRUE
Expected Value
0.0
97.5
1 Year
38.7%
2 Years
16.1%
97.5
97.5
Expected Value
Witkowski
100.0
Do Not Get O'Donnell Report
TRUE
0.0
Expected Value
100.0
20.0%
Reject
-100.0
Produce Pilot
FALSE
Expected Value
0.0
70.0
1 Year
30.0%
2 Years
50.0%
50.0
150.0
1.0
Expected Value
0.0
100.0
20.0%
Reject
100.0
Sell to Competitor
TRUE
Expected Value
0.0
100.0
1 Year
30.0%
2 Years
50.0%
100.0
100.0
(c) What should Witkowski’s strategy be? What is the expected value of this
strategy?
The best thing to do is to forget about O’Donnell and sell the rights for $100,000.
Note that this conclusion is not affected by whether or not Witkowski buys the
report from O’Donnell.
B60.2350
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Prof. Juran
Expected Value of Sample Information
The expected value of sample information is calculated using this formula:
EVSI = EVwSI - EVwoSI
where
EVSI
= expected value of sample information
EVwSI
= expected value with sample information about the states of nature
EVwoSI
= expected value without sample information about the states of nature
In our example, the expected value of sample information is:
EVSI = EVwSI - EVwoSI
 98.98  100
 1.02
That means that if we pay O’Donnell the $2,500 fee, our overall expected value
drops by $1,020. This implies that the O’Donnell report is worth
$2 ,500  $1,020
 $1,480
We would be willing to pay up to (but no more than) $1,480 for the O’Donnell
report.
(d) What is the expected value of the O’Donnell agency’s sample information? Is
the information worth the $2,500 fee?
This is one way to address the question, “How much should Witkowski be
prepared to pay for the research study?” Clearly, it is not worth anything if we
have to pay $2,500 for it; at that price it actually has a negative expected value.
B60.2350
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Prof. Juran
Efficiency of Sample Information
The efficiency of sample information is calculated using this formula:
E

EVSI
EVPI
(e) What is the efficiency of the O’Donnell’s sample information?
E

EVSI
EVPI

$1, 480
$25,000
 0.0592
In other words, the market research project gives us information with less than
6% of the utility of having perfect information.
B60.2350
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Prof. Juran