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Review Questions
12.1-1 The decision alternatives are to drill for oil or to sell the land.
12.1-2 The consulting geologist believes that there is 1 chance in 4 of oil on the tract of land.
12.1-3 Max does not put much faith in the assessment.
12.1-4 A detailed seismic survey of the land could be done to obtain more information.
12.1-5 The possible states of nature are the possible outcomes of the random factors that affect the payoff that would be obtained from a decision alternative.
12.1-6 Prior probabilities are the estimated probabilities of the states of nature prior to obtaining additional information through a test or survey.
12.1-7 The payoffs are quantitative measures of the outcomes from a decision alternative and a state of nature. Payoffs are generally expressed in monetary terms.
12.2-1 The maximax criterion identifies the maximum payoff for each decision alternative and chooses the decision alternative with the maximum of these maximum payoffs. The maximax criterion is for the eternal optimist.
12.2-2 The maximax citerion completely ignores the prior probabilities and ignores all payoffs except for the largest one.
12.2-3 The maximin criterion identifies the minimum payoff for each decision alternative and chooses the decision alternative with the maximum of these minimum payoffs. The maximin criterion is for the total pessimist.
12.2-4 The maximin criterion ignores the prior probabilities and ignores all payoffs except the maximin payoff.
12.2-5 The maximum likelihood criterion focuses on the most likely state of nature, the one with the largest prior probability.
12.2-6 Criticisms of the maximum likelihood criterion include: 1) this criterion chooses an alternative without considering its payoffs for states of nature other than the most likely one, 2) for alternatives that are not chosen, this criterion ignores their payoffs for states of nature other than the most likely one, 3) if the differences in the payoffs for the most likely state of nature are much less than for another somewhat likely state of nature, then it might make sense to focus on this latter state of nature instead, and 4) if there are many states of nature and they are nearly equally likely, then the probability that the most likely state of nature will be the true one is fairly low.
12.2-7 Bayes’ decision rule says to choose the alternative with the largest expected payoff.
12.2-8 The expected payoff is calculated by multiplying each payoff by the prior probability of the corresponding state of nature and then summing these products.
12.2-9 Criticisms of Bayes’ decision rule include: 1) there usually is considerable uncertainty involved in assigning values to prior probabilities, 2) prior probabilities inherently are at least largely subjective in nature, whereas sound decision making should be based on objective data and procedures, and 3) by focusing on average outcomes, expected payoffs ignore the effect that the amount of variability in the possible outcomes should have on the decision making.
12.3-1 A decision tree is a graphical display of the progression of decisions and random events to be considered.
12.3-2 A decision node indicates that a decision needs to be made at that point in the process. An event node indicates that a random event occurs at that point.
12.3-3 Decision nodes are represented by squares while circles represent event nodes.
12.4-1 Sensitivity analysis might be helpful to study the effect if some of the numbers included in the model are not correct.
12.4-2 It assures that each piece of data is in only one place and it makes it easy for anyone to interpret the model, even if they don’t understand TreePlan or decision trees.
12.4-3 A data table displays results of selected output cells for various trial values of a data cell.
12.4-4 If there is less than a 23.75% chance of oil, they should sell. If it’s more, they should drill.
12.5-1 Perfect information means knowing for sure which state of nature is the true state of nature.
12.5-2 The expected payoff with perfect information is calculated by multiplying the maximum payoff for each alternative by the prior probability of the corresponding state of nature.
12.5-3 The decision tree should be started with a chance node whose branches are the various states of nature.
12.5-4 EVPI = EP (with perfect information) – EP (without more information)
12.5-5 If the cost of obtaining more information is more than the expected value of perfect information then it is not worthwhile to obtain more information.
12.5-6 If the cost of obtaining more information is less than the expected value of perfect information then it might be worthwhile to obtain more information.
12.5-7 In the Goferbroke problem the EVPI >C so it might be worthwhile to do the seismic survey.
12.6-1 Posterior probabilities are revised probabilities of the states of nature after doing a test or survey to improve the prior probabilities.
12.6-2 The possible findings are favorable with oil being fairly likely, or unfavorable with oil being quite unlikely.
12.6-3 Conditional probabilities need to be estimated.
12.6-4 The five kinds of probabilities considered are prior, conditional, joint, unconditional, and posterior.
12.6-5 P(state and finding) = P(state) * P(finding | state).
12.6-6 P(finding) = sum of P(state and finding) for each state.
12.6-7 P(state | finding) = P(state and finding) / P(finding).
12.6-8 Bayes’ theorem is used to calculate posterior probabilities.
12.7-1 A decision tree provides a graphical display of the progression of decisions and random events for a problem.
12.7-2 A decision needs to be made at a decision node.
12.7-3 A random event will occur at a event node.
12.7-4 The probabilities of random events and the payoffs need to be inserted before beginning analysis.
12.7-5 When performing the analysis, start at the right side of the decision tree and move left one column at a time.
12.7-6 For each event node, calculate its expected payoff by multiplying the payoff of each branch by the probability of that branch and then summing these products.
12.7-7 For each decision node, compare the expected payoffs of its branches and choose the alternative whose branch has the largest expected payoff.
12.8-1 Consolidate the data and results into one section of the spreadsheet.
12.8-2 Performing sensitivity analysis on a piece of data should require changing a value in only one place on the spreadsheet.
12.8-3 A data table can consider changes in only one or two data cells.
12.8-4 One.
12.8-5 Yes. The spider graph can consider changes in many data cells at a time.
12.8-6 SensIt’s spider graph assumes that each data value varies by the same amount. Sensit’s tornado diagram overcomes this limitation.
12.9-1 Utilities are intended to reflect the true value of an outcome to the decision-maker.
12.9-2
12.9-3 Under the assumptions of utility theory, the decision-maker’s utility function for money has the property that the decision-maker is indifferent between two alternative courses of action if the two alternatives have the same expected utility.
12.9-4 The decision-maker is offered two hypothetical alternatives and asked to identify the point of indifference between the two.
12.9-5 The point of indifference is the value of p where the decision-maker is indifferent between the two hypothetical alternatives.
12.9-6 The value obtained to evaluate each node of the tree is the expected utility.
12.9-7 Max decided to do the seismic survey and to sell if the result is unfavorable or drill if the result is favorable.
12.10-1 The Goferbroke problem contained the same elements as typical applications of decision analysis but is oversimplified.
12.10-2 An influence diagram complements the decision tree for representing and analyzing decision analysis problems.
12.10-3 Typical participants include management, an analyst, and a group facilitator.
12.10-4 A manager can go to a management consulting firm that specializes in decision analysis.
12.10-5 Decision analysis is widely used around the world.
Problems
12.1 a) Max(A1) = 6, Max(A2) = 4, Max(A3) = 8. Maximax = 8 with alternative A3.
b) Min(A1) = 2, Min(A2) = 3, Min(A3) = 1. Maximin = 3 with alternative A2.
12.2 a) Max(A1) = 30, Max(A2) = 31, Max(A3) = 22, Max(A4) = 29. Maximax = 31 with A2.
b) Min(A1) = 20, Min(A2) = 14, Min(A3) = 22, Min(A4) = 21. Maximin = 22 with A3.
12.3 a)
State of Nature
Alternative
Sell 10 cases
Sell 11 cases
Sell 12 cases
Sell 13 cases
Buy 10 cases
$50
$50
$50
$50
Buy 11 cases
$47
$55
$55
$55
Buy 12 cases
$44
$52
$60
$60
Buy 13 cases
$41
$49
$57
$65
Prior Probability
b) Max(Buy 10) = $50, Max(Buy 11) = $55, Max(Buy 12) = $60, Max(Buy 13) = $65.
Maximax = $65 with buying 13 cases.
c) Min(Buy 10) = $50, Min(Buy 11) = $47, Min(Buy 12) = $44, Min(Buy 13) = $41.
Maximin = $50 with buying 10 cases.
d) The most likely state of nature is to sell 11 cases. Under this state, she should buy 11 cases with a payoff of $55.
e)
Jean should buy 12 cases. The maximum expected payoff is $53.60.
f)
Jean should purchase 12 cases. The maximum expected payoff is $55.20.
Jean should purchase 12 cases. The maximum expected payoff is $54.40.
Jean should purchase 11 cases. The maximum expected payoff is $53.40.
12.4 a) Max(Conservative) = $30 million
Max(Speculative) = $40 million
Max(Countercyclical) = $15 million
Maximax = $40 million with the speculative investment
b) Min(Conservative) = –$10 million
Min(Speculative) = –$30 million
Min(Countercyclical) = –$10 million
Maximin = –$10 million with either the conservative of countercyclical investment.
c) The stable economy is the most likely state of nature.
The speculative investment has the maximum payoff for this state ($10 million).
d) The countercyclical investment has the maximum expected payoff of $5 million.
12.5 a) The countercyclical investment has the maximum expected payoff of $8 million.
b) The speculative investment has the maximum expected payoff of $5 million.
c&d)
e)
f) Part a)Part b)
g)
h)
Counter-cyclical and conservative cross at approximately p=0.62.
Conservative and speculative cross at approximately p = 0.68.
i) Let p = prior probability of stable economy
For the conservative option:
EP = (0.1)(30) + p(5) + (1–0.1–p)(–10)
= 3 + 5p – 9 +10p
= 15p – 6
For the speculative option:
EP = (0.1)(40) + p(10) + (1–0.1–p)(–30)
= 4 + 10p – 27 +30p
= 40p – 23
For the counter-cyclical option:
EP = (0.1)(–10) + p(0) + (1–0.1–p)(15)
= –1 + 0 + 13.5 – 15p
= –15p + 12.5
Counter-cyclical and conservative cross when
–15p + 12.5 = 15p – 6 or 30p = 18.5 or p = 0.617
Conservative and speculative cross when
15p – 6 = 40p – 23 or 25p = 17 or p = 0.68
They should choose the counter-cyclical option when p < 0.617, the conservative option when 0.617 ≤ p < 0.68, and the speculative option when p ≥ 0.68.
12.6 a) Max(A1) = 80, Max(A2) = 50, Max(A3) = 60.
Maximax = $80 thousand when choosing alternative A1.
b) Min(A1) = 25, Min(A2) = 30, Min(A3) = 40.
Maximin = $40 thousand when choosing alternative A3.
c) S2 is the most likely outcome. For this state, the maximum payoff of $50 thousand occurs with alternative A2.
d) Alternative A3 has the highest expected payoff of $48 thousand.
e)
f) When the prior probability of S1 is 0.2, alternative A2 should be chosen, with an expected payoff of $46 thousand.
When the prior probability of S1 is 0.6, alternative A1 should be chosen, with an expected payoff of $58 thousand.
g)
12.7 a) Max(A1) = $220 thousand, Max(A2) = $200 thousand.
Maximax = $220 thousand when choosing alternative A1.
b) Min(A1) = $110 thousand, Min(A2) = $150 thousand.
Maximin = $150 thousand when choosing alternative A2.
c) S1 is the most likely outcome. For this state, the maximum payoff of $220 thousand occurs with alternative A1.
d) Alternative A1 has the highest expected payoff of $194 thousand.
e & f)
g)
Let p = prior probability of S1.
For A1:
EP = p(220) + (1 – 0.1 – p)(170) + (0.1)(110)
= 220p + 153 – 170p + 11
= 50p + 164
For A2:
EP = p(200) + (1 – 0.1 – p)(180) + (0.1)(150)
= 200p + 162 – 180p + 15 = 20p + 177
A1 and A2 cross when 50p + 164 = 20p + 177 or 30p = 13 or p=0.433.
They should choose A2 when p ≤ 0.433, A1 when p > 0.433.
h)
Let p = prior probability of S1.
For A1:
EP = p(220) + (0.3)(170) + (1 – 0.3 – p)(110)
= 220p + 51 + 77 – 110p
= 110p + 128
For A2:
EP = p(200) + (0.3)(180) + (1 – 0.3 – p)(150)
= 200p + 54 + 105 – 150p
= 50p + 159
A1 and A2 cross when 110p + 128 = 50p + 159 or 60p = 31 or p = 0.517.
They should choose A2 when p ≤ 0.517, A1 when p > 0.517.
i)
Let p = prior probability of S2.
For A1:
EP = (0.6)(220) + p(170) + (1 – 0.6 – p)(110)
= 132 + 170p + 44 – 110p
= 60p + 176
For A2:
EP = (0.6)(200) + p(180) + (1 – 0.6 – p)(150)
= 120 + 180p + 60 – 150p
= 30p + 180
A1 and A2 cross when 60p + 176 = 30p + 180 or 30p = 4 or p = 0.133.
They should choose A2 when p ≤ 0.133, A1 when p > 0.133.
j) Alternative A1 should be chosen.
12.8 a)
State of Nature (Weather)
Alternative
Dry
Moderate
Damp
Crop 1
20
35
40
Crop 2
30
45
Crop 3
30
25
25
Crop 4
20
20
20
Prior Probability
b)
c) Crop 1 has the highest expected payoff of $31,500.
d) When the prior probability of moderate weather is 0.2, Crop 2 has the highest expected payoff of $35,250.
When the prior probability of moderate weather is 0.3, Crop 2 has the highest expected payoff of $33,750.
When the prior probability of moderate weather is 0.4, Crop 2 has the highest expected payoff of $32,250.
When the prior probability of moderate weather is 0.6, Crop 1 has the highest expected payoff of $31,000.
12.9 When x = 50, alternative A3 has the highest expected payoff of $5,600.
When x = 75, alternative A1 has the highest expected payoff of $7,400.
Barbara Miller should pay a maximum of $1,800 to increase x to 75.
12.10 a) Alternative A2 has the highest expected payoff of $1,000.
b) With perfect information, choose A1 for when the state is S1, A2 when the state is S2, and A3 when the state is S3.
EP(with perfect information) = (0.2)(4) + (0.5)(2) + (0.3)(1) = $2,100
EVPI = EP(with perfect information) – EP (without more information)
= $2,100 – $1,000 = $1,100.
c)
EVPI = EP(with perfect information) – EP (without more information)
= $2,100 – $1,000 = $1,100.
d) Since the information will cost $1,000 and the value is no more than $1,100, it might be worthwhile to spend the money.
12.11 a) Alternative A1 has the highest expected payoff of $35.
b) With perfect information, choose A1 for when the state is S1, A1 when the stat
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