Solution to Witkowski TV Productions[1]

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:

States of Nature

s1 = Reject / s2 = 1 Year / s3 = 2 Years
Produce Pilot
/ d1 / -100 / 50 / 150
Sell to Competitor / d2 / 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:

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 / Probabilities / Net Payout if Pilot is Produced / Net Payout if Sold to Competitor / Optimal Decision
Reject / 0.2 / -100 / 100 / Sell to Competitor
1 Year / 0.3 / 50 / 100 / Sell to Competitor
2 Years / 0.5 / 150 / 100 / 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:

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.

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.

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
Reject / /
1 year / /
2 years / /

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:

Probabilities
States / Prior / Conditional / Joint / Posterior

Reject

/ 0.20 / 0.30 / 0.06 / 0.087
1 Year / 0.30 / 0.60 / 0.18 / 0.261
2-Year / 0.50 / 0.90 / 0.45 / 0.652
Total

The total probability of a favorable O’Donnell report is 69%.

Probabilities
States / Prior / Conditional / Joint / Posterior

Reject

/ 0.20 / 0.70 / 0.14 / 0.452
1 Year / 0.30 / 0.40 / 0.12 / 0.387
2-Year / 0.50 / 0.10 / 0.05 / 0.161
Total

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
s1 = Reject / s2 = 1 Year / s3 = 2 Years
No Report /
d1 = Produce Pilot
/ -100 / 50 / 150
d2 = Sell to Competitor / 100 / 100 / 100
Get Report / I1 = Favorable Report /
d1 = Produce Pilot
/ -102.5 / 47.5 / 147.5
d2 = Sell to Competitor / 97.5 / 97.5 / 97.5
I2 = Unfavorable Report /
d1 = Produce Pilot
/ -102.5 / 47.5 / 147.5
d2 = Sell to Competitor / 97.5 / 97.5 / 97.5

In the event of a favorable O’Donnell report, the expected value of producing the pilot is:

In the event of a favorable O’Donnell report, the expected value of selling to the competitor is:

In the event of an unfavorable O’Donnell report, the expected value of producing the pilot is:

In the event of an unfavorable O’Donnell report, the expected value of selling to the competitor is:

Looking at the table here, we can see that for each possible outcome of the O’Donnell project there is an optimal strategy:

Decision Alternative / Expected Value
Favorable O’Donnell Report / Produce Pilot / 99.65 / ← Optimal
Sell to Competitor / 97.50
Unfavorable O’Donnell Report / Produce Pilot / -4.20
Sell to Competitor / 97.50 / ← Optimal

The overall expected value with sample information (EVwSI) is:

(Note that we are assuming here that we will always adopt the optimal strategy in light of whatever information O’Donnell provides.)

Developing an Optimal Decision Strategy

(b)Show a decision tree to represent the revised problem.

(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.

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

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

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.

Efficiency of Sample Information

The efficiency of sample information is calculated using this formula:

E /

(e)What is the efficiency of the O’Donnell’s sample information?

E /

In other words, the market research project gives us information with less than 6% of the utility of having perfect information.

B60.23501Prof. Juran

[1] David Juran.