chapter 6

Chapter 6

Problem Summary

Prob. # Concepts Covered Level of Difficulty

Notes

6.1 Decision Making Under Uncertainty -- maximax, maximin, minimax regret and principle of insufficient reason criteria

1

6.2 Expected Value Criterion, EVPI 16.3 Decision Making Under Uncertainty --

Maximax, Maximin, and Minimax Regret Criteria

1

6.4 Expected Value Criterion, EVPI 26.5 Bayesian Probability Revision, EVSI 46.6 Expected Value and Expected Utility Criteria 46.7 Utility, Expected Utility Criterion 46.8 Decision Tree Analysis 56.9 Constructing Payoff Tables, Minimax Regret and

Expected Value Criterion6

6.10 Game Theory 46.11 Decision Making Under Uncertainty-- Maximin

and Principle of Insufficient Reason Criteria1

6.12 Bayesian Probability Revision, EVSI, Efficiency 46.13 Maximax, Minimax Regret, and Expected Value

Criteria3

6.14 Maximax, Minimax Regret, and Principle of Insufficient Reason Criteria

2

6.15 Expected Value Criterion, EVSI 46.16 Decision Tree Analysis 56.17 Constructing Payoff Tables, Minimax Regret and

Expected Value Criteria, EVPI6

6.18 Utility, Expected Utility Criterion 56.19 Bayes’ Theorem 66.20 Calculation of State Probabilities, Expected

Value Criterion4

6.21 Decision Tree Analysis 76.22 Constructing Payoff Tables 46.23 Expected Value Criterion, EVPI 2 Part b. of this problem is looking for

the EVPI.6.24 Bayesian Probability Revision, EVSI, Efficiency 36.25 Minimax Regret, Maximin, and Expected Value

Criteria3

6.26 EVPI, EVSI, and Efficiency 46.27 Constructing Payoff and Regret Tables,

Maximax, Maximin, and Minimax Regret Criteria

2

6.28 Expected Value Criterion, EVPI, EVSI, 3

6-1

Efficiency6.29 Constructing Payoff Tables, Expected Value

Criterion5

6-2

6.30 Decision Tree Analysis 56.31 Expected Value Criterion 16.32 Game Theory 16.33 Decision Tree Construction and Solution 76.34 Expected Utility Criterion 46.35 Minimax Regret and Expected Value Criteria 1 In Problem the wording for Expected

Value Criterion is Expected Monetary Criterion

6.36 Game Theory, Expected Value Criterion 4 Note that the bids are in ounces but the mail quantity is in pounds.

6.37 Expected Value Criterion, EVPI, Risk Classification, Expected Utility Criterion

4

6.38 Expected Value Criterion, EVPI, Expected Utility Criterion

6

6.39 Decision Tree Analysis 66.40 Decision Tree Construction and Solution 76.41 Constructing Payoff Tables, Maximax and

Expected Value Criterion7

6.42 Game Theory 66.43 Decision Tree Construction and Solution 66.44 Constructing Payoff Tables, Expected Value

Criterion5

6.45 Decision Tree Construction and Solution, Bayesian Probability Revision

9 This is a long problem and could easily serve as a case study.

6.46 Constructing Payoff Tables, Minimax Regret and Expected Value Criterion, EVPI

6

6.47 Expected Utility Criterion 36.48 Game Theory, Expected Value Criterion 46.49 Expected Value Criterion, Bayesian Probability

Revisions7

6.50 Utility Theory, Expected Utility Criterion 5Case 6.1 Payoff Table, Decision Tree 8Case 6.2 Decision Tree, Utility 8Case 6.3 Decision Tree, Bayesian Probability Revision 8Case 6.4 Decision Tree Analysis 6

6-3

Problem Solutions

6.1 See File ch6.1.xls

a. Maximax Criterion -- 400

b. Maximin Criterion – 320

c. Minimax Regret Criterion -- 360

6-4

Maximax Maximin 2800 24803200 26003600 24004000 2200

7 8 9 10 MinmaxReg280 0 480 960 1520 1520320 200 0 560 1120 1120360 400 200 0 560 560400 600 400 200 0 600

6.2 See File ch6.2.xls

a. Expected Value Criterion -- 360. This criterion seems appropriate from the Bookstore's point of view if their objective is to maximize profit. If their objective is to serve student needs, this is probably not the appropriate criteria.

b. EVPI = $212. EVPI represents the expected gain in profit from knowing with certainty how many economics course sections will be ordered.

6-5

Expected value (EV)2800(.1)+2720(.3)+2640(.4)+2480(.2)2600(.1)+3200(.3)+3040(.4)+2880(.2)2400(.1)+3000(.3)+3600(.4)+3440(.2)=32682200(.1)+2800(.3)+3400(.4)+4000(.2)

EVWPI = 2800(.1)+3200(.3)+3600(.4)+4000(.2)= 3480EVPI = EVWPI – MaxEV =3480 – 3268 =212

6.3 See file ch6.3.xls

a. The optimistic decision involves the Maximax Criterion --- 3 commercials

b. The pessimistic decision involves the Maximin Criterion -- 0 commercials

c. Minimax Regret Criterion -- 2 commercials

6-6

6.4 See file ch6.4.xls

a. Expected Value Criterion -- 2 commercials

b. EVPI -- $170,000

6-7

6.5 See file ch6.5.xls

a. P(Game will be dull | Predict Game Interesting) = .083

6-8

6.5 continued

b. (i) If game prediction is interesting, buy 3 commercials, (ii) If game prediction is dull, buy 2 commercials.

c. EVSI = $25,000

6-9

6.6 See file Ch6.6.xls

a. Take 12 Leases

6-10

Ch 6.6 continued

b. Take 9 Leases

6-11

6.7 See file Ch6.7.xls

a. Concave utility function – management is risk averse.

b. Purchase the H-P server

6-12

6.8 See file Ch6.8.xls

Steve Johnson should repair the roof

6-13

6.9 See file Ch6.9.xls

a. Payoff Table

Number of Adoptions

0 1 2 3 4 5 6

1 2000 2000 2000 2000 2000 2000 2000

Plan 2 1000 1300 1600 1900 2200 2500 2800

3 0 700 1400 2100 2800 3500 4200

b. Minimax Regret Criterion -- Plan 2

c. Expected Value Criterion -- Plan 1

6-14

6.10 See file Ch6.10.xls

a. Advertise recreational facilities with probability .31, breakfast quality with probability .44, and room décor with probability .24.

b. A decrease of .91%.

6-15

6.11 See file Ch6.11.xls

a. The conservative approach is to use the Maximin decision strategy -- Unfurnished

b. Principle of Insufficient Reason (Equal Likelihood) -- Custom Decorated. For this problem this criterion provides the same choice as the Expected Value criterion.

6-16

6.12 See file Ch6.12.xls

a. (i) Predict Above Average Rise -- Custom Decorating, (ii) Predict Average Rise -- Unfurnished, (iii) Predict Below Average Rise -- Unfurnished

b. EVSI = $937.50 per lot

c. Efficiency = 58%

d. GNP is not a good indicator, other indicators that would specifically focus on the Atlanta housing market or economy (such as average per capita income) would be better.

6-17

6.13 See file Ch6.13.xls

a. An optimistic decision strategy employs the Maximax criterion -- Open a Mega store

b. The Minimax Regret decision strategy -- Open a Super store

c. The Expected Value decision strategy -- Open a Super store

6-18

6.14 See file Ch6.14.xls

a. Maximin criterion -- Order 1 car

b. Minimax Regret criterion – Order 2 cars

c. Principle of Insufficient Reason Criterion – Order 3 cars

6-19

6.15 See file Ch6.15.xls

a. P(Outstanding Review) = .15, King should order 3 cars

b. King should pay up to the EVPI = $3,950

6-20

6.16 See file Ch6.16.xls

The Dean should bid $300,000.

6-21

6.17 See file Ch6.17.xls

a. Payoff Table:

Demand

10,000 50,000 100,000

Do Nothing 0 0 0

Order 40,000 -130,000 130,000 80,000

Size 80,000 -250,000 110,000 360,000

120,000 -370,000 -10,000 440,000

b. Minimax Regret -- order 80,000

c. Expected Value -- order 80,000

d. The company should pay up to the EVPI = $58,889

6-22

6.18 See file Ch6.18.xls

a. Convex utility function – risk averse firm

b. Order 80,000

6-23

6.19 See file Ch6.19.xls

The posterior probability is .996

6-24

6.20 See file Ch6.20.xls

Open a 260 room hotel

6-25

6.21 See file Ch6.21.xls

Bill should not invest the $15,000.

6-26

6.22 See file Ch6.22.xls

# Demanded 0 1 2 3

0 0 -150 -300 -450

# 1 -625 600 450 300

Ordered 2 -1250 -25 1200 1050

3 -1875 -650 575 1800

6-27

6.23 see file Ch6.23.xls

a. Order Two Sets

b. The manager should pay up to the EVPI = $587.50

6-28

6.24 see file Ch6.24.xls

a. If survey shows at least one customer likely to buy -- order 2, otherwise order 1

b. It should pay up to the EVSI = $82.50

c. Efficiency = 14.04%

d. Survey could record the number of customers who are likely to purchase.

6-29

6.25 See file Ch6.25.xls

a. Minimax Regret criterion –Three screens

b. Maximin criterion – One screen

c. Expected Value criterion – Two screens

6-30

6.26 See file Ch6.26.xls

a. The manager should pay up to the EVPI = $1,030

b. Yes, EVSI = $335 is greater than the $50 fee.

c. Efficiency = 33%

6-31

6.27 See file 6.27.xls

a. Payoff Table

Sales70,000 40,000 10,000

Introduce 150 50 -100Do Not Intro. -20 -20 -20

b. A conservative strategy is the Maximin Criterion -- Do not introduce

c. Regret Table for Bee's Candy

Sales 70,000 40,000 10,000

Introduce 0 0 $80Do Not Introduce $170 70 0

d. Minimax Regret Criterion -- Introduce

6-32

6.28 See file Ch6.28.xls

a. Expected Value Criterion -- Introduce

b. If survey shows a favorable attitude Bees should introduce the lower calorie candy assortment

6-33

6.28 continued

c. EVSI = $1,800, Efficiency = 8%

6-34

6.29 See file Ch6.29.xls

a. Payoff table in $1,000's

Demand

0 1 2 3 4

1 -300 -75 -105 -135 -165

Number 2 -400 -175 50 20 10

Built 3 -300 -75 150 375 345

4 -400 -175 50 275 500

b. Build 3 computers and build 4 computers are undominated strategies.

c. Craig should build 3 computers.

6-35

6.30 See file Ch6.30.xls

Stefan should rent 2 cabanas and charge $100 for each.

6-36

6.31 See file Ch6.31.xls

Zeus should select Plan II since it maximizes the expected value.

6-37

6.32 See file Ch6.32.xls

For both players the optimal strategy is to play rock, scissors, and paper randomly with equal likelihood.

6-38

6.33 See file Ch6.33.xls

Roney should Purchase the house, submit plans for Plan B, and not contribute the $6,000. The overall expected value equals $15,200.

6-39

6.34 See file Ch6.34.xls

a. Since Steve is risk neutral the amount he should pay for the insurance will be equal to his expected loss. According to the Expected Value decision criterion, he should therefore pay $580.

6-40

6.34 continued

b. EU = .9952

c. Approximately $1,000 since EU = 0.9952 is close to the utility value of 0.995 for $1,000.

6-41

6.35 See file Ch6.35.xls

a. Purchasing 0, 2, or 3 tandem bicycles are undominated decisions.

b. Minimax Regret Criterion -- Purchase 2 Tandem Bicycles

c. Probabilities:

P(Sunny Days = 250) = P(Sunny Days = 325)P(Sunny Days = 300) = 2*P(Sunny Days = 325)P(Sunny Days = 300) = 3*P(Sunny Days = 275)

Hence, P(Sunny Days = 250) = 3/14P(Sunny Days = 275) = 1/7P(Sunny Days = 300) = 3/7P(Sunny Days = 325) = 3/14

Expected Value Criterion -- Purchase 3 Tandem Bicycles

6-42

6.36 See file Ch6.36a.xls

a. Bid $.02

6-43

6.36 continued See file Ch6.36b.xls

Payoff Table

.02 .05 .06

.02 100,000 220,000 220,000

.04 0 700,000 700,000

.06 0 0 580,000

b. Federal Parcel should bid $.04

6-44

6.37 See file Ch6.37.xls

a. Player 1 should play 1 with probability .25, 2 with probability .35, and 3 with probability .24 and 4 with probability .16. The expected value of the game is $.16.

b. The game could be made "fair" if Player 1 paid Player 2 $.16 each time the game is played.

6-45

6.38 See file Ch6.38.xls

a. Expected Value Criterion -- Spot + $.01

b. EVPI = $116

c. Convex utility function -- Sardon is risk loving

6-46

6.38 continued

d. Expected Utility Criterion -- Spot +$.01

6-47

6.39 See file Ch6.39.xls

United should bid $175 million for development work.

6-48

6.40. See file Ch6.40.xls

Company should import scooters and advertise only if tariff is not imposed.

6-49

6.41 See file Ch6.41.xls

a. Payoff Table (in Costs)

Driving Distance20,000 24,000 28,000 32,000 36,000

Plan A 6760 7152 7544 7936 8328

Plan B 8290 8498 8706 8914 9122

Plan C 6280 6840 7400 7960 8520

b. An optimistic approach involves using the Minimin criterion (equivalent to Maximax profit), John's optimal plan is C.

c. Using the Expected Value criterion John's optimal plan is C.

6-50

6.42 See file Ch6.42.xls

a. As its principal promotional strategy, Merck should advertise in medical journals with a .46 probability, advertise in consumer magazines with a .40 probability, and offer consumer rebates with a .14 probability.

b. As its principal promotional strategy, Upjohn should do sales calls to doctors with a .25 probability, advertise in medical journals with a .22 probability, and advertise in consumer magazines with a .53 probability.

c. Merck has an expected gain in market share of .52%.

6-51

6.43 See file Ch6.43.xls

Midge should wait two months before buying her ticket.

6-52

6.44 See file Ch6.44.xls

Payoff Table# Sold

1 2 3# Bought 1 230 360 490Prior to 2 160 460 590May 1 3 90 390 690

Adams should purchase 2 tractors prior to May 1.

6-53

6.45 See file Ch6.45.xls

John should bid $1.2 million without having the survey done.

6-54

6.46 See file Ch6.46.xls

a. Payoff Table (profits in $1,000's)Demand

1 2 3 4

1 -8 -12 -16 -20

Number 2 -9.5 .5 -3.5 -7.5

Produced 3 -10 0 10 6

4 -9.5 .5 10.5 20.5

b. Producing 2 and 3 robots are dominated decisions.

c. Principle of Insufficient Reason (Equal Likelihood) -- Produce 4 robots.

d. Minimax Regret Criterion -- Produce 4 Robots

e. P(Demand = 1) = .40, P(Demand = 2) = .30, P(Demand = 3) = .20, P(Demand = 4) = .10, Expected Value Criterion -- Produce 4 Robots

f. Ultima should pay up to the EVPI = $600 for the survey.

6-55

6.47 See file Ch6.47b.xls

a. Concave utility function -- risk averse

b. Hire consultant. If consultant predicts approval purchase the option. If consultant predicts denial do nothing.

6-56

6.48 See file Ch6.48b.xls and Ch6.48c.xls

a. Alaska would always lose market share to United in this case.

b. United should select the 9 a.m. time slot with probability .53 and the 10 a.m. time slot with probability .47. Expected market share would be 73.33%

6-57

6.48 continued

c. United should use the 9 a.m. slot.

d. The first approach assumed Alaska is also actively engaged in making decisions. It is the more realistic approach.

6-58

6.49 See file Ch6.49.xls

a. No maintenance

6-59

6.49 continued

6-60

b. Major maintenance is called for

6-61

6.50 See file Ch6.50.xls

a. Concave utility function. Colton is risk averse.

b. Major maintenance is called for.

6-62

Case 6.1

Swan Valley’s optimal strategy is not to use either plan, but to contract to purchase 50 tons of apricots under the current pricing plan. Since Plan 2 brings less expected profit than Plan 1 which is not favored, Plan 2 cannot be optimal.

6-63

Case 6.2

Pharmgen’s optimal strategy is to sell a ½ interest in the drug to Wyler Laboratories.

6-64

Case 6.3

Pickens should do the seismic test. If the test predicts oil the company should sell a 50% interest in the lease. If the test does not predict oil the company the company should not purchase the lease.

6-65

Case 6.4

Northwestern should make a pledge of the land and do a partial clearcut.

6-66