Practical Management Science 5e Wayne L Winston S Christian Albright
Chapter 9 – Decision Making under Uncertainty
1. All problems related to decision making under uncertainty have three common elements:
a. 
the mean, median, and mode 
b. 
the set of decisions, the cost of each decision and the profit that can be made from each decision 
c. 
the set of possible outcomes, the set of decision variables and the constraints 
d. 
the set of decisions, the set of possible outcomes, and a value model that prescribes results 

2. Expected monetary value (EMV) is:
a. 
the average or expected value of the decision if you knew what would happen ahead of time 
b. 
the weighted average of possible monetary values, weighted by their probabilities 
c. 
the average or expected value of the information if it was completely accurate 
d. 
the amount that you would lose by not picking the best alternative 

3. Probabilities on the branches of a chance node may be ____ events that have occurred earlier in the decision tree.
a. 
marginal due to 
b. 
conditional on 
c. 
averaged with 
d. 
increased by 

4. Which of the following statements is true concerning decision tree conventions?
a. 
Time proceeds from right to left. 
b. 
The trees are composed of circles, triangles and ovals. 
c. 
The nodes represent points in time. 
d. 
Probabilities of outcomes are shown to the right of the end nodes. 

5. The solution procedure for solving decision trees is called:
a. 
sensitivity analysis 
b. 
policy iteration 
c. 
risk profiling 
d. 
folding back 

6. The strategy region graph is a type of sensitivity analysis chart that:
a. 
is useful in determining whether the optimal decision changes over the range of the input variable. 
b. 
ranks the sensitivity of the EMV to the input variables. 
c. 
reflects how the value of information changes over a range of probabilities. 
d. 
None of these 

7. Bayes’ rule is used to:
a. 
update the prior probabilities once new information is observed. 
b. 
turn the given conditional probabilities (i.e. likelihoods) around. 
c. 
update the posterior probabilities once new information is observed. 
d. 
All of the above are uses for Bayes’ rule. 

8. The denominator of Bayes’ rule:
a. 
is the same as the simple probability of an outcome O. 
b. 
decomposes the probability of the new information I into all possibilities. 
c. 
is sometimes called the law of complementary probabilities. 
d. 
is unique for each possible outcome. 

9. Which of the following are probabilities that are conditioned on information that is obtained?
a. 
Prior probabilities 
b. 
Posterior probabilities 
c. 
Marginal probabilities 
d. 
Objective probabilities 

10. A utility function for risk averse individuals is ____ and/or ____.
a. 
decreasing, linear 
b. 
decreasing, convex 
c. 
increasing, linear 
d. 
increasing, concave 

11. In general, the expected monetary value (EMV) of a decision will be equal to one of the possible payoffs.

12. For each possible decision and each possible outcome, the payoff table lists the associated monetary value.

13. The expected monetary value (EMV) criterion represents the longrun average of uncertain outcomes, so it should only be used for recurring decisions.

14. The risk profile shows the probability distribution of monetary outcomes in both graphical and tabular form.

15. The expected value of information (EVI) is the difference between the EMV obtained with free sample information and the EMV obtained without any information.

16. The expected value of perfect information (EVPI) is a largely irrelevant concept since perfect information is almost never available at any price.

17. Bayes’ rule is used for updating the probability of an uncertain outcome after observing the results of a test or study.

18. Prior probabilities are sometimes called likelihoods, the probabilities that are influenced by information about the outcome of an earlier uncertainty.

19. The certainty equivalent is the certain dollar amount a riskaverse decision maker would accept in order to avoid a gamble altogether.

20. For a risk averse decision maker, the certainty equivalent is less than the expected monetary value (EMV).

Exhibit 91
A farmer must decide whether to take protective action to limit damage to his grapefruit crop in the event that the overnight temperature falls to a level well below freezing. If the temperature drops too low he runs the risk of losing his entire crop, valued at $75,000. Based on the National Weather Service, the probability of such a temperature drop is 60%. He can insulate his crop by spraying water on all the trees, which will cost $20,000. This action might succeed in protecting the crop, with the following possible outcomes:
Probability 
Damage 
0.30 
$0 
0.15 
$5,000 
0.10 
$10,000 
0.15 
$15,000 
0.30 
$20,000 

21. Refer to Exhibit 91. Construct a decision tree to help the farmer make his decision. What should he do? Explain your answer.
ANSWER: 
The solved decision tree below shows that it is optimal for the farmer to insulate, since the expected cost is $26,000 if he insulates, versus $45,000 if he doesn’t insulate. 
POINTS: 
1 

22. Refer to Exhibit 91. Find the highest cost of insulating the grapefruits for which the farmer prefers to insulate his crop.
ANSWER: 
By trial and error (or by looking at the difference in expected values) the decision would switch to “No” (don’t insulate) if the cost exceeds $39,000. 
POINTS: 
1 

23. Refer to Exhibit 91. Suppose the farmer is uncertain about the reliability of the National Weather Service forecast. If he thinks the probability of a freeze occurring could be anywhere between 40% and 80%, would that change his decision?
ANSWER: 
No, the strategy region chart below shows the best option is to insulate over this entire range of probability. 
POINTS: 
1 

24. Refer to Exhibit 91. Construct a risk profile and from that determine the probability that no additional cost is incurred if the decision to insulate at a cost of $20,000 is made.
ANSWER: 
The risk profile is shown below. The probability of incurring no additional cost once the decision to insulate is made is 58%.

POINTS: 
1 

25. Refer to Exhibit 91. Suppose the farmer is not riskneutral, but instead his behavior can be modeled using an exponential utility function with a risk tolerance parameter of 100,000. What is the most he would be willing to pay for insulation in that case?
ANSWER: 
The risk averse decision tree below shows he would be willing to pay more (˜$45,000). 
POINTS: 
1 

Exhibit 92
A customer has approached a local credit union for a $20,000 1year loan at a 10% interest rate. If the credit union does not approve the loan application, the $20,000 will be invested in bonds that earn a 6% annual return. Without additional information, the credit union believes that there is a 5% chance that this customer will default on the loan, assuming that the loan is approved. If the customer defaults on the loan, the credit union will lose the $20,000. 
26. Refer to Exhibit 92. Construct a decision tree to help the credit union decide whether or not to make the loan. Make sure to label all decision and chance nodes and include appropriate costs, payoffs and probabilities.

27. Refer to Exhibit 92. What should the credit union do? What is their expected profit?
ANSWER: 
The tree shows that the best alternative is not to make the loan, and to invest the funds in bonds earning 6% interest instead. The EMV of this option is $1,200. 
POINTS: 
1 

28. Refer to Exhibit 92. The bank can thoroughly investigate the customer’s credit record and obtain a favorable or unfavorable recommendation. If the credit report is perfectly reliable, what is the most the credit union should be willing to pay for the report?
ANSWER: 
The tree above shows that if the credit report is favorable, the credit union should make the loan, and if the report is unfavorable, it should not make the loan. This increases the expected value to $1,960 with the credit information. Thus, the EVPI is $760, which is the most the credit union should be willing to pay. 
POINTS: 
1 

29. Refer to Exhibit 92. Should the credit union purchase the report if it costs $150?
ANSWER: 
The tree above shows that the credit report is still valuable, even in the imperfect case. The information increases the expected value to $1,543, thus, the EVSI is $343 ($1,543 − $1,200). Therefore, the credit union would be justified in purchasing the report for $150. 
POINTS: 
1 

30. Refer to Exhibit 92. Suppose that an actual (not perfectly reliable) credit report has the following characteristics based on historical data; in cases where the customer did not default on the approved loan, the probability of receiving a favorable recommendation on the basis of the credit investigation was 80%, while in cases where the customer defaulted on the approved loan, the probability of receiving a favorable recommendation on the basis of the credit investigation was 25%. Given this information, what are the posterior probabilities that default will and will not occur, given the credit report?
ANSWER: 
The worksheet above shows that if the report predicts default, the probabilities of actual default and no default are 0.165 and 0.835, respectively, while if the report predicts no default, the probabilities of default and no default are 0.016 and 0.984, respectively. 
POINTS: 
1 

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