Answer:
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.