Question

Hale’s TV Productions is considering producing a pilot for a comedy series in the hope of selling it to a major television network. The network may decide to reject the series, but it may also decide to purchase the rights to the series for either one or two years. At this point in time, Hale may either produce the pilot and wait for the network’s decision or transfer the rights for the pilot and series to a competitor for $100,000. Hale’s decision alternatives and profits (in thousands of dollars) are as follows:

State of Nature

Decision Alternative

Reject, s1

1 Year, s2

2 Years, s3

Produce pilot, d1

-100

50

150

Sell to competitor, d2

100

100

100

The probabilities for the states of nature are P(s1) = 0.20, P(s2) = 0.30, and P(s3) = 0.50. For a consulting fee of $5000, an agency will review the plans for the comedy series and indicate the overall chances of a favorable network reaction to the series. Assume that the agency review will result in a favorable (F) or an unfavorable (U) review and that the following probabilities are relevant:

P(F) = 0.69

P(s1 | F) = 0.09

P(s1 | U) = 0.45

P(U) = 0.31

P(s2 | F) = 0.26

P(s2 | U) = 0.39



P(s3 | F) = 0.65

P(s3 | U) = 0.16



Construct a decision tree for this problem.

What is the recommended decision if the agency opinion is not used? What is the expected value?

Answer 1

To construct the decision tree, we can follow these steps:

1. Start with the initial decision nodes representing the two decision alternatives: "Produce pilot" (d1) and "Sell to competitor" (d2).

2. Assign the payoffs for each decision alternative under each state of nature.

3. Add chance nodes for each state of nature and connect them to the corresponding decision alternatives.

4. Assign the probabilities of each state of nature at the chance nodes.

5. Calculate the expected payoffs at each chance node by multiplying the payoffs with their respective probabilities and summing them up.

6. Determine the optimal decision by comparing the expected payoffs at the initial decision nodes.

Here is the decision tree for this problem:

| Produce pilot (d1)

| -100

|____________

/|\

/ | \

/ | \

/ | \

P(F) = 0.69 / | \ P(U) = 0.31

/ | \

/ | \

/ | \

/ | \

s1 / | \ s2

/ | \

/ | \

/ | \

/ | \

/ | \

50 | F U F | 100

| |

| |

| |

| s3 | s3

| |

150| F | 100

|_______________________________|

If the agency opinion is not used, the recommended decision would be to produce the pilot (d1) since it has a higher expected value compared to selling to the competitor (d2).

To calculate the expected value:

Expected value (d1) = (-100 * P(s1 | F) * P(F)) + (50 * P(s2 | F) * P(F)) + (150 * P(s3 | F) * P(F))

= (-100 * 0.09 * 0.69) + (50 * 0.26 * 0.69) + (150 * 0.65 * 0.69)

= -6.93 + 8.97 + 66.88

= 68.92

Expected value (d2) = (100 * P(s1 | U) * P(U)) + (100 * P(s2 | U) * P(U)) + (100 * P(s3 | U) * P(U))

= (100 * 0.45 * 0.31) + (100 * 0.39 * 0.31) + (100 * 0.16 * 0.31)

= 13.95 + 12.09 + 4.96

= 30

Comparing the expected values, the recommended decision is to produce the pilot (d1) with an expected value of 68.92.