International Pictures is trying to decide how to distribute its new movie ‘Claws’. ‘Claws’ is the story of an animal husbandry experiment at the University of Southern Queensland that goes astray, with tragic results. An effort to breed meatier chickens somehow produces an intelligent, 200 kilogram chicken that escapes from the lab and terrorises the campus. In a surprise ending the chicken is befriended by coach Tim Galvano, who teaches it how to play Rugby and help his team win State, National and World Championships. Because of the movie’s controversial nature, it has the potential to be either a smash hit, a modest success, or a total bomb. International is trying to decide whether to release the picture for general distribution initially or to start out with a ‘limited first-run release’ at a few selected theaters, followed by general distribution after 3 months. The company has estimated the following probabilities and conditional profits for ‘Claws’:
PROFITS (Millions of $)
Level of
success Probability Limited General
release distribution
Smash .3 22 12
Modest .4 9 8
Bomb .3 –10 –2
International can run sneak previews of ‘Claws’ to get a better idea of the movies’ ultimate level of success. Preview audiences rate movies as either good or excellent. On the basis of past experiences, it was found that 90% of all smash successes were rated excellent (and 10% rated good), 75% of all modest successes were rated excellent (25% rated good) and 40% of all bombs were rated excellent (60% rated good). The cost of running sneak previews is not cheap. Currently, this stands at $1m.
What is the opportunity loss for a General Distribution for a Smash level of success?
What would the optimal action be for International before running the sneak preview?
1) Run a limited release with an expected payoff of $7.20m
2) Run a limited release with an expected payoff of $6.20m
3) Run a general distribution with an expected payoff of $7.20m
4)Run a general distribution with an expected payoff of $6.20m
What is the maximum amount of money that International would be prepared to pay for an absolutely reliable forecast of the movies’ level of success?
1) $9.6 m
2) $7.2 m
3) $6.2 m
4) $2.4 m
What would be the joint probability for a ‘bomb success’ and excellent preview given that in the past, it was found that 40% of all bomb successes were rated excellent?
What is the posterior probability of a bomb given the sneak preview indicates good?
What is the maximum amount that should be paid for the sneak preview [i.e. what would be the expected value of sample information (EVSI)]?
1) $1.04 million
2) $2.58 million
3) $7.20 million
4) $8.24 million
To calculate the opportunity loss for a General Distribution for a Smash level of success, we need to compare the expected payoff of a General Distribution for a Smash level of success with the expected payoff of a Limited Distribution for a Smash level of success.
Expected payoff of a General Distribution for a Smash level of success:
Probability(Smash) * Profits(Smash, General Distribution) = 0.3 * 12 = 3.6 million
Expected payoff of a Limited Distribution for a Smash level of success:
Probability(Smash) * Profits(Smash, Limited Distribution) = 0.3 * 22 = 6.6 million
The opportunity loss is the difference between the expected payoffs:
Opportunity loss = Expected payoff(Limited Distribution) - Expected payoff(General Distribution) = 6.6 million - 3.6 million = 3 million
Therefore, the opportunity loss for a General Distribution for a Smash level of success is 3 million dollars.
Before running the sneak preview, we need to determine the optimal action for International. To do this, we calculate the expected payoffs for both options: running a limited release and running a general distribution.
Expected payoff of a Limited Release:
Probability(Smash) * Expected payoff(Smash, Limited Distribution) + Probability(Modest) * Expected payoff(Modest, Limited Distribution) + Probability(Bomb) * Expected payoff(Bomb, Limited Distribution)
= 0.3 * 6.2 + 0.4 * 6.2 + 0.3 * 7.2 = 1.86 + 2.48 + 2.16 = 6.5 million
Expected payoff of a General Distribution:
Probability(Smash) * Expected payoff(Smash, General Distribution) + Probability(Modest) * Expected payoff(Modest, General Distribution) + Probability(Bomb) * Expected payoff(Bomb, General Distribution)
= 0.3 * 12 + 0.4 * 8 + 0.3 * -2 = 3.6 + 3.2 - 0.6 = 6.2 million
Based on these calculations, the optimal action for International before running the sneak preview would be to run a general distribution with an expected payoff of $6.2 million (option 4).
The maximum amount of money that International would be prepared to pay for an absolutely reliable forecast of the movie's level of success is the difference between the expected payoff of the optimal action and the expected payoff of the worst action.
Expected payoff of the worst action (running a limited release):
Probability(Smash) * Expected payoff(Smash, Limited Distribution) + Probability(Modest) * Expected payoff(Modest, Limited Distribution) + Probability(Bomb) * Expected payoff(Bomb, Limited Distribution)
= 0.3 * 7.2 + 0.4 * 6.2 + 0.3 * -2 = 2.16 + 2.48 - 0.6 = 4.04 million
Maximum amount International would pay = Expected payoff(optimal action) - Expected payoff(worst action) = 6.2 million - 4.04 million = 2.16 million
Therefore, the maximum amount that International would be prepared to pay for an absolutely reliable forecast of the movie's level of success is $2.16 million (option 4).
The joint probability for a 'bomb success' and excellent preview can be calculated using the information given. We know that in the past, it was found that 40% of all bomb successes were rated excellent.
Joint probability for 'bomb success' and excellent preview = Probability(Bomb) * Probability(Excellent given Bomb) = 0.3 * 0.4 = 0.12
Therefore, the joint probability for a 'bomb success' and excellent preview is 0.12.
To calculate the posterior probability of a bomb given the sneak preview indicates good, we use Bayes' theorem.
Posterior probability of a bomb = (Probability(Good given Bomb) * Probability(Bomb)) / (Probability(Good given Bomb) * Probability(Bomb) + Probability(Good given Not Bomb) * Probability(Not Bomb))
= (0.6 * 0.3) / (0.6 * 0.3 + 0.25 * 0.4 + 0.1 * 0.3) = 0.18 / (0.18 + 0.1 + 0.03) = 0.18 / 0.31 = 0.58
Therefore, the posterior probability of a bomb given the sneak preview indicates good is 0.58.
The maximum amount that should be paid for the sneak preview can be calculated using the concept of expected value of sample information (EVSI).
Expected value of sample information (EVSI) = Expected payoff with perfect information - Expected payoff without perfect information
Expected payoff with perfect information = Probability(Smash) * Expected payoff(Smash, Limited Distribution, perfect information) + Probability(Modest) * Expected payoff(Modest, Limited Distribution, perfect information) + Probability(Bomb) * Expected payoff(Bomb, Limited Distribution, perfect information)
= 0.3 * 6.2 + 0.4 * 6.2 + 0.3 * 7.2 = 1.86 + 2.48 + 2.16 = 6.5 million
Expected payoff without perfect information = Expected payoff of the optimal action = 6.2 million
EVSI = Expected payoff with perfect information - Expected payoff without perfect information = 6.5 million - 6.2 million = 0.3 million
Therefore, the maximum amount that International should be paid for the sneak preview (EVSI) is $0.3 million (option 1).