# Quantitative Decisions: Statistics, Probability, and Game Theory

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1. Nick has plans to open some pizza restaurants, but he is not sure how many to open. He has prepared a payoff table to help analyze the situation.

States of Nature

Alternatives Good Market Fair Market Poor Market

Open 1 380,000 70,000 -400,000

Open 2 200,000 80,000 -200,000

Do Nothing 0 0 0

Nick believes there is a 40 percent chance that the market will be good, a 30 percent chance that it will be fair, and a 30 percent chance that it will be poor. A market research firm will analyze market conditions and will provide a perfect forecast (they provide a money back guarantee). What is the most that should be paid for this forecast?

(a) $ 44,000

(b) $ 53,000

(c) $123,000

(d) $176,000

(e) none of the above

2. Daily sales for a perishable food product are known to be 8, 9, 10, or 11 cases with probabilities 0.2, 0.3, 0.4, and 0.1, respectively. Cases not sold during the day are worthless, but cases can only be produced in the morning before the store opens. The cost of producing one of these is $4 while the selling price is $7. If you choose to produce 10 cases in the morning to sell, what is the probability that you will be able to meet today's demand?

(a) 0.20

(b) 0.30

(c) 0.40

(d) 0.10

(e) 0.90

3. Daily sales for a perishable food product are known to be 8, 9, 10, or 11 cases with probabilities 0.2, 0.3, 0.4, and 0.1, respectively. Cases not sold during the day are worthless, but cases can only be produced in the morning before the store opens. The cost of producing one of these is $4 while the selling price is $7. How many cases should you produce to maximize profit?

(a) 8

(b) 9

(c) 10

(d) 11

(e) none of the above

4. Consider the following two-person game, and determine the saddle point if it exists.

Y1 Y2

X1 4 6

X2 2 3

(a) 3

(b) 2

(c) 4

(d) 6

(e) there is no saddle point

5. Considering the following two-person game, what percentage of the time should X play strategy X2?

Y1 Y2

X1 6 3

X2 2 8

(a) 1/3

(b) 2/3

(c) 4/9

(d) 5/9

(e) none of the above

Given the following two-person game, which strategy can be eliminated by use of dominance?

Y1 Y2

X1 13 9

X2 6 8

X3 12 14

(a) X1

(b) X2

(c) X3

(d) Y1

(e) Y2

6. Enrollment in a particular class for the last four semesters has been 120, 126, 110, and 130. Suppose a one-semester moving average was used to forecast enrollment (this is sometimes referred to as a naive forecast). Thus, the forecast for the second semester would be 120, for the third semester it would be 126, and for the last semester it would be 110. What would the MSE be for this situation?

(a) 196.00

(b) 230.67

(c) 100.00

(d) 42.00

(e) none of the above

7. Given the MAD for the following forecast is 4.0 what is the forecast value in period 4?

Period Forecast Actual

1 15 11

2 20 13

3 25 21

4 23

(a) 24

(b) 30

(c) 23

(d) 33

(e) none of the above

#### Solution Preview

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1. Maximum amount that can be paid for the market research can be calculated using EVPI.

EVPI=EV w PI-Max EMV

EVwPI=0.4*380000+0.3*80000+0.3*(-400000)

EVwPI=176000

EVPI=176000-53000= $ 123,000

2. Probability of meeting today's demand with 10 cases=Probability of demand being 8 cases+Probability of demand being 9 cases+ probability of demand being 10 cases

Probability of meeting today's demand with 10 cases=0.2+0.3+0.4=0.9

Thus, answer is (e)

3. Following is the payoff table for demand and no of cases produced.

Payoff for the cells where Demand is less than no of cases produced:

=Demand*$7-No of cases produced*$4.

Payoff for ...

#### Solution Summary

This response answers a series of questions based on probability and game theory.