A television station would like to decide between extending a current television show for another season or develop a completely new show for that time slot.

It will cost $6M to develop the new show. The new show may be very successful, moderately successful, or not successful with associated advertising revenues of $25M, $15M, or $5M, respectively.

If the station chooses to extend the current show, it will cost $2M. The extended show may be very successful, moderately successful, or not successful with associated advertising revenues of $15M, $10M, or $5M, respectively.

Which of the following are true regarding the payoffs on the terminal nodes of the associated decision tree?

Question

A television station would like to decide between extending a current television show for another season or develop a completely new show for that time slot.

It will cost $6M to develop the new show.  The new show may be very successful, moderately successful, or not successful with associated advertising revenues of $25M, $15M, or $5M, respectively.

If the station chooses to extend the current show, it will cost $2M.  The extended show may be very successful, moderately successful, or not successful with associated advertising revenues of $15M, $10M, or $5M, respectively.

Which of the following are true regarding the payoffs on the terminal nodes of the associated decision tree?

BlackTom AI · Accepted Answer

Question restatement: A TV station must choose between extending the current show or developing a new one. Costs: new show development is $6M with revenues of $25M (very successful), $15M (moderately), or $5M (not successful). Extending the current show costs $2M with revenues of $15M (very successful), $10M (moderately), or $5M (not successful). We’re asked about the payoffs on the terminal nodes of the decision tree.

Option 1: If the new show is developed and it is very successful, the payoff is $19M.
- Analysis: Payoff = revenue − cost = 25 − 6 = 19. This is correct. The calculation follows the standard payoff definition used in decision trees, where payoff equals net profit.

Option 2: If the new show is developed and it is moderately successful, the payoff is $8M.
- Analysis: Payoff = 15 − 6 = 9, not 8. This is incorrect because the revenue is $15M and the development cost is $6M, yielding $9M, not $8M.

Option 3: If the new show is developed and it is not successful, the payoff is $1M.
- Analysis: Payoff = 5 − 6 = −$1M, i.e., a loss of $1M, not a positive $1M. This is incorrect because the development cost exceeds the revenue in the not-successful scenario, producing a negative payoff.

Option 4: If the existing show is extended and it is moderately successful, the payoff is $8M.
- Analysis: Payoff = 10 − 2 = 8. This is correct because the extended show yields $10M in revenue and costs $2M to extend, giving an $8M payoff.

Option 5: Exactly two answers are correct.
- Analysis: At first glance, there are two numerically correct payoff statements (Options 1 and 4). However, declaring that exactly two answers are correct would have to be evaluated against the entire set of options. If Options 1 and 4 are the only true payoffs, then the statement itself would claim a count of two correct answers, which would be true only if no other option is correct. In this structure, including Option 5 as a true statement would imply a different count, creating a self-referential issue. Consequently, it is best treated as an evaluation trap: its truth value depends on the count of other true statements, leading to potential inconsistency. In practice, it is safer to mark this as not a straightforward factual payoff and treat it as a self-referential option that cannot be definitively classified without additional rules.

Option 6: None of the answers are correct.
- Analysis: This is false because Option 1 and Option 4 are correct payoffs, so it cannot be true that none are correct.

Summary of option-by-option reasoning:
- Option 1: Correct payoff calculation (19M).
- Option 2: Incorrect; should be 9M, not 8M.
- Option 3: Incorrect; should be −1M, not +1M.
- Option 4: Correct payoff calculation (8M).
- Option 5: A tricky self-reference; given Options 1 and 4 are correct, asserting that exactly two answers are correct would imply a specific count, but including this option as a true statement creates a circular logic. In standard analysis, it is treated as not reliably true based on the payoff data alone.
- Option 6: Incorrect; there are correct answers among the options.

Additional notes on interpretation:
- Payoffs in a decision tree are typically net profits (revenue minus development/extension costs) at the terminal nodes. This is why the calculations for Options 1 and 4 reflect net gains of $19M and $8M, respectively.
- The self-referential nature of Option 5 complicates its evaluation; unless the problem explicitly defines how to resolve such self-referential statements, it is safest to flag it as not reliably true within the payoff context.

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