Chapter 6
Problem Summary
Prob. # / Concepts Covered / Level of Difficulty / Notes6.1 / Decision Making Under Uncertainty -- maximax, maximin, minimax regret and principle of insufficient reason criteria / 1
6.2 / Expected Value Criterion, EVPI / 1
6.3 / Decision Making Under Uncertainty -- Maximax, Maximin, and Minimax Regret Criteria / 1
6.4 / Expected Value Criterion, EVPI / 2
6.5 / Bayesian Probability Revision, EVSI / 4
6.6 / Expected Value and Expected Utility Criteria / 4
6.7 / Utility, Expected Utility Criterion / 4
6.8 / Decision Tree Analysis / 5
6.9 / Constructing Payoff Tables, Minimax Regret and Expected Value Criterion / 6
6.10 / Game Theory / 4
6.11 / Decision Making Under Uncertainty-- Maximin and Principle of Insufficient Reason Criteria / 1
6.12 / Bayesian Probability Revision, EVSI, Efficiency / 4
6.13 / Maximax, Minimax Regret, and Expected Value Criteria / 3
6.14 / Maximax, Minimax Regret, and Principle of Insufficient Reason Criteria / 2
6.15 / Expected Value Criterion, EVSI / 4
6.16 / Decision Tree Analysis / 5
6.17 / Constructing Payoff Tables, Minimax Regret and Expected Value Criteria, EVPI / 6
6.18 / Utility, Expected Utility Criterion / 5
6.19 / Bayes’ Theorem / 6
6.20 / Calculation of State Probabilities, Expected Value Criterion / 4
6.21 / Decision Tree Analysis / 7
6.22 / Constructing Payoff Tables / 4
6.23 / Expected Value Criterion, EVPI / 2 / Part b. of this problem is looking for the EVPI.
6.24 / Bayesian Probability Revision, EVSI, Efficiency / 3
6.25 / Minimax Regret, Maximin, and Expected Value Criteria / 3
6.26 / EVPI, EVSI, and Efficiency / 4
6.27 / Constructing Payoff and Regret Tables, Maximax, Maximin, and Minimax Regret Criteria / 2
6.28 / Expected Value Criterion, EVPI, EVSI, Efficiency / 3
6.29 / Constructing Payoff Tables, Expected Value Criterion / 5
6.30 / Decision Tree Analysis / 5
6.31 / Expected Value Criterion / 1
6.32 / Game Theory / 1
6.33 / Decision Tree Construction and Solution / 7
6.34 / Expected Utility Criterion / 4
6.35 / 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 / 6
6.40 / Decision Tree Construction and Solution / 7
6.41 / Constructing Payoff Tables, Maximax and Expected Value Criterion / 7
6.42 / Game Theory / 6
6.43 / Decision Tree Construction and Solution / 6
6.44 / Constructing Payoff Tables, Expected Value Criterion / 5
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 / 3
6.48 / Game Theory, Expected Value Criterion / 4
6.49 / Expected Value Criterion, Bayesian Probability Revisions / 7
6.50 / Utility Theory, Expected Utility Criterion / 5
Case 6.1 / Payoff Table, Decision Tree / 8
Case 6.2 / Decision Tree, Utility / 8
Case 6.3 / Decision Tree, Bayesian Probability Revision / 8
Case 6.4 / Decision Tree Analysis / 6
Problem Solutions
6.1 See File ch6.1.xls
a. Maximax Criterion -- 400
b. Maximin Criterion – 320
c. Minimax Regret Criterion -- 360
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.
6.3 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
6.4 See file ch6.4.xls
a. Expected Value Criterion -- 2 commercials
b. EVPI -- $170,000
6.5 See file ch6.5.xls
a. P(Game will be dull | Predict Game Interesting) = .083
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
6.6 See file Ch6.6.xls
a. Take 12 Leases
Ch 6.6 continued
b. Take 9 Leases
6.7 See file Ch6.7.xls
a. Concave utility function – management is risk averse.
b. Purchase the H-P server
6.8 See file Ch6.8.xls
Steve Johnson should repair the roof
6.9 See file Ch6.9.xls
a. Payoff Table
Number of Adoptions0 / 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
6.10 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%.
6.11 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.
6.12 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.
6.13 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
6.14 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
6.15 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
6.16 See file Ch6.16.xls
The Dean should bid $300,000.
6.17 See file Ch6.17.xls
a. Payoff Table:
Demand10,000 / 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
6.18 See file Ch6.18.xls
a. Convex utility function – risk averse firm
b. Order 80,000
6.19 See file Ch6.19.xls
The posterior probability is .996
6.20 See file Ch6.20.xls
Open a 260 room hotel
6.21 See file Ch6.21.xls
Bill should not invest the $15,000.
6.22 See file Ch6.22.xls
# Demanded
0 / 1 / 2 / 30 / 0 / -150 / -300 / -450
# 1 / -625 / 600 / 450 / 300
Ordered 2 / -1250 / -25 / 1200 / 1050
3 / -1875 / -650 / 575 / 1800
6.23 see file Ch6.23.xls
a. Order Two Sets
b. The manager should pay up to the EVPI = $587.50
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.
6.25 See file Ch6.25.xls
a. Minimax Regret criterion –Three screens
b. Maximin criterion – One screen
c. Expected Value criterion – Two screens
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%
6.27 See file 6.27.xls
a. Payoff Table
Sales
70,000 40,000 10,000
Introduce 150 50 -100
Do 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 $80
Do Not Introduce $170 70 0
d. Minimax Regret Criterion -- Introduce
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
6.28 continued
c. EVSI = $1,800, Efficiency = 8%
6.29 See file Ch6.29.xls
a. Payoff table in $1,000's
Demand0 / 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.
6.30 See file Ch6.30.xls
Stefan should rent 2 cabanas and charge $100 for each.
6.31 See file Ch6.31.xls
Zeus should select Plan II since it maximizes the expected value.
6.32 See file Ch6.32.xls
For both players the optimal strategy is to play rock, scissors, and paper randomly with equal likelihood.
6.33 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.
6.34 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.
6.34 continued
b. EU = .9952
c. Approximately $1,000 since EU = 0.9952 is close to the utility value of 0.995 for $1,000.
6.35 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) = 3/14
P(Sunny Days = 275) = 1/7
P(Sunny Days = 300) = 3/7
P(Sunny Days = 325) = 3/14
Expected Value Criterion -- Purchase 3 Tandem Bicycles
6.36 See file Ch6.36a.xls
a. Bid $.02
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
6.37 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.
6.38 See file Ch6.38.xls
a. Expected Value Criterion -- Spot + $.01
b. EVPI = $116
c. Convex utility function -- Sardon is risk loving
6.38 continued
d. Expected Utility Criterion -- Spot +$.01
6.39 See file Ch6.39.xls
United should bid $175 million for development work.
6.40. See file Ch6.40.xls
Company should import scooters and advertise only if tariff is not imposed.
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.
6.42 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%.
6.43 See file Ch6.43.xls
Midge should wait two months before buying her ticket.
6.44 See file Ch6.44.xls
Payoff Table
# Sold1 / 2 / 3
# Bought / 1 / 230 / 360 / 490
Prior to / 2 / 160 / 460 / 590
May 1 / 3 / 90 / 390 / 690
Adams should purchase 2 tractors prior to May 1.
6.45 See file Ch6.45.xls