MODELING UNCERTAINTY Part 1 Agenda Monty Hall problem

MODELING UNCERTAINTY Part 1

Agenda • Monty Hall problem • Decision trees • Assignment 2 announcement • Market research problem taken from the lectures of Prof. Paat Rusmevichientong delivered for course DSO 570, USC Spring 2017

Market Research Case • Please split into groups of 4 or 5 and attempt the following case • You have 20 minutes to solve this case as a group following which we will discuss the result • First team to submit the correct answer gets a prize (probably bonus marks in exam)

Decision Trees • A national chain of quick service restaurants has developed a new specialty sandwich. Initially, the company faces two possible decisions: • Invest $200 million to introduce the sandwich nationally • Invest $20 million to evaluate it in a regional test market • If it introduces the sandwich nationally, the chain might find either a high or low response to the idea: • In a high response case, the chain will make $700 million in revenue • In a low response case, the chain will make $150 million in revenue. • There is a 62. 5% chance that the national response is high, and a 37. 5% chance that it’s low.

Decision Trees • On the other hand, if the company starts with a regional marketing strategy, it might find a low response or a high response at the regional level. • However, the region response may or may not reflect the national market potential. • In any case, after the regional evaluation, the chain needs to decide whether to 1) remain regional, or 2) market nationally. • Note: It’s possible to allow the option to drop the product, but we will not pursue that here. • Payoff for Remaining Regional: • If the regional response is high, and the company decides to remain regional, the revenue will be $200 million. • If the regional response is low, and the company decides to remain regional, the revenue will be $100 million. • There is no further cost of launching regionally if a regional test is conducted.

Decision Trees • Based on launches of other similar products in the past, the chain estimates • If the national response is low, then it is likely to see a low regional response about 72% of the time • On the other hand, if the national response is high, then the chain is likely to see a high regional response 95. 2% of the time • In this case, the product is likely to have a universal appeal • Question: What is the optimal strategy in terms of expected monetary value?

Squares = Decisions Circles = Events

700 MM – 200 MM = 500 MM 150 MM – 200 MM = -50 MM 200 MM – 20 MM = 180 MM 700 MM – 20 MM = 480 MM 150 MM – 20 MM = -70 MM 100 MM – 20 MM = 80 MM 700 MM – 20 MM = 480 MM 150 MM – 20 MM = -70 MM

? ? 700 MM – 200 MM = 500 MM 150 MM – 200 MM = -50 MM 200 MM – 20 MM = 180 MM ? ? 700 MM – 20 MM = 480 MM 150 MM – 20 MM = -70 MM 100 MM – 20 MM = 80 MM ? ? 700 MM – 20 MM = 480 MM 150 MM – 20 MM = -70 MM

62. 5% 37. 5% 700 MM – 200 MM = 500 MM 150 MM – 200 MM = -50 MM 200 MM – 20 MM = 180 MM ? ? 700 MM – 20 MM = 480 MM 150 MM – 20 MM = -70 MM 100 MM – 20 MM = 80 MM ? ? 700 MM – 20 MM = 480 MM 150 MM – 20 MM = -70 MM

Decision Tree • For the regional response we do not know the probabilities directly • We know probability of a high and a low national response • And we know: • If the national response is low, then it is likely to see a low regional response about 72% of the time • On the other hand, if the national response is high, then the chain is likely to see a high regional response 95. 2% of the time • If we consider event A as regional response and B as national response then we know P(B) and P(A|B). We need P(A) and P(B|A)

Bayes Rule National Response Regional Response High Low Total High 62. 5% * 95. 2% = 59. 5% 37. 5% * 28% = 10. 5% 59. 5% + 10. 5% = 70% Low 62. 5% * 4. 8% = 3% 37. 5% * 72% = 27% 3% + 27% = 30% Total 62. 5% 37. 5% 100% P(High National | High Regional) = 59. 5/70 = 85% --- Hence P(Low National | High Regional) = 15% P(High National | Low Regional) = 3/30 = 10% --- Hence P(Low National | High Regional) = 90%

62. 5% 37. 5% 700 MM – 200 MM = 500 MM 150 MM – 200 MM = -50 MM 200 MM – 20 MM = 180 MM 70% 85% 15% 700 MM – 20 MM = 480 MM 150 MM – 20 MM = -70 MM 100 MM – 20 MM = 80 MM 30% 10% 90% 700 MM – 20 MM = 480 MM 150 MM – 20 MM = -70 MM

62. 5% 37. 5% 700 MM – 200 MM = 500 MM 150 MM – 200 MM = -50 MM E(X) = 180 MM 70% 200 MM – 20 MM = 180 MM 85% E(X) = 397. 5 MM 15% 700 MM – 20 MM = 480 MM 150 MM – 20 MM = -70 MM 100 MM – 20 MM = 80 MM E(X) = 80 MM 30% 10% E(X) = -15 MM 90% 700 MM – 20 MM = 480 MM 150 MM – 20 MM = -70 MM

62. 5% (0. 625 * 500) + (0. 375 * -50) = 293. 75 MM 37. 5% 700 MM – 200 MM = 500 MM 150 MM – 200 MM = -50 MM E(X) = 180 MM 70% 200 MM – 20 MM = 180 MM 85% E(X) = 397. 5 MM 15% (0. 7 * 397. 5) + (0. 3 * 80) = 302. 25 MM 700 MM – 20 MM = 480 MM 150 MM – 20 MM = -70 MM 100 MM – 20 MM = 80 MM E(X) = 80 MM 30% 10% E(X) = -15 MM 90% 700 MM – 20 MM = 480 MM 150 MM – 20 MM = -70 MM

Decision Tree • Our optimal strategy, based on expected monetary value, is to test regionally first, and then if there is regional success we should launch nationally, else we should stay regional • Question: If testing regionally cost 30 MM instead of 20 MM, should we still test regionally? • Question: What is the maximum we should be willing to pay for a regional test? • Question: In a hypothetical situation where the regional test is 100% accurate and yields a perfect decision on going national or staying regional, how much should we be willing to pay for the test?

Decision Trees • To solve a decision tree problem, perform the following steps: Draw the decision tree Calculate terminal node values Calculate event probabilities Choose value maximizing decision at each decision node Calculate expected value over all decisions nodes assuming you will take value maximizing decisions • Finally, select the decision route with the highest expected value • • •