EXERCISES Extensions of chapter examples 4-1 Reconsider the drilling decision situation presented in Table 4-5. A reassessment of a complete success indicates that the payoffs have changed and are as follows: Percent of profit Fixed lease Combination Complete success 1700 600 1400 (a) Using the expected payoff criterion, what is the preferred alternative? (b) What does this suggest about the sensitivity of the decision to estimates of future revenues? 4-2 Arriving at the probability of each outcome in the drilling problem was in part a subjective process. Although we have numeric estimates, we should be attuned to their subjective origin. The reasoning that generated the probability values .25, .20, and .55 might just as validly have generated the values .21, .20, and .59, which are not all that different. Using this new set of probability estimates and the payoffs of Table 4-1, determine the alternative preferred by the expected value criterion. What does this suggest about the sensitivity of the decision to the estimated probability values? 4-3 Assume you are an expected value decision maker. Noting the results of exercises 4-1 and 4-2, what decision alternative would you choose? 4-4 In the southeast Texas hospital planning problem, the basis for the dollar value of excess beds is the $202,000 operating cost for each excess bed. Suppose the true cost is only $20,200 for each excess bed, or just IO percent of the original estimate. Reconstruct the decision table and the expected payoffs, using the probabilities of Table 4-9. How might these changes have affected the council's choice of decision alternative I? 82 QUANTITATIVE METHODS FOR PUBLIC DECISION MAKING 4-5 In problem 4-4, suppose the true cost is $181,800 for each excess bed, or just 10 percent less than the original estimate. How might these changes have affected the council's choice of decision alternative I? 4-6 Assume your answer to problem 4-4 suggests that the council would have chosen an alternative other than alternative I, and your answer to problem 4-5 suggests they would have stayed with alternative I. Determine the level of "operating cost for each excess bed" at which they would change their decision. Discuss. 4-7 Refer to the last section of Chapter 4 and the illustration on choosing a technical assistance firm. It is sometimes claimed that in-house costs to support contractual services are sunk costs. They are costs already incurred and will not change whether the result of the contract is satisfactory or unsatisfactory. What may change is the delay in delivering required studies and the level of complaint and harassment. Reconstruct Table 4-10 with the following changes: The consequences of satisfactory work are no delays for Flash, Inc. and a I-week delay for Penser, Ltd. The consequences for unsatisfactory work are a 4-week delay for Flash, Inc. and a 2-week delay for Penser, Ltd. (a) What is the numerical expected payoff or consequence? (b) What would the decision be according to the expected value, maximax, and minimax principles? (c) What would your decision be? Other applications 4-8 The following letter to the Editor appeared in The New York Times of February 28, 1971. The writer directed the theoretical division of the Los Alamos Scientific Laboratory from April 1943 to January 1946. Hans Bethe was the 1967 Nobel laureate in physics. Yalta: Lack of Communication on Bomb To the Editor: Under the title "The Truth about Yalta," C. L. Sulzberger [column of February 14] discussed the assessment by Ambassador Charles Bohlen of the chief problems that faced President Roosevelt and the U.S. delegation at the time of the Yalta Conference, February 1945. The third point of this assessment reads in part as follows: "While Roosevelt and a handful of advisers knew about the Manhattan Project, no one could be certain the atomic bomb would in fact explode or how effective a weapon it would be." This problem looked different as seen from the Los Alamos Scientific Laboratory charged with the development of the bomb. By February 1945, it appeared to me and to other fully informed scientists that there was a better than 90 percent probability that the atomic bomb would in fact explode, that it would be an extremely effective weapon, and that there would be enough material to build several bombs in the course of a few months. Thus even if the first bomb should have failed, the project was bound to succeed in a relatively short time. Few things in war and even fewer in politics have as good a chance as 90 percent. That the full flavor of this conviction of the scientists was not transmitted to the decision-makers was a failure of communication-excessive secrecy and the absence ofdirect channels between scientists and high Government officials were responsible. Because of this failure of direct communication, the U.S. at Yalta urged Russia to participate in the assault on Japan, with grave consequences for the future of the political situation in the Far East. Suppose there had been good communication. Should the U.S. Government have acted on a 90 percent probability of technical success? In my opinion, definitely yes. Again, in 1958 we had a chance to arrange a ban on the testing of nuclear weapons at a time when the U.S. had a clear advantage over the Soviet Union in weapon design. However, we were afraid of the possibility of clandestine underground Russian tests of small nuclear DECISION THEORY: A FRAMEWORK FOR DECISION MAKING 83 weapons and insisted therefore on ironclad safeguards. These were unacceptable to the U.S.S.R., and no agreement was reached by 1961. In 1961, the Russians conducted a series of nuclear weapon tests in which they managed to equal, in most of the important aspects, the perlormance of U.S. thermonuclear weapons. Thus here again, by insisting on certainty, the U.S. lost a clear advantage. This letter is not meant to imply that our foreign policy should center on advantage over the U.S.S.R. I merely wish to argue that if and when the seeking of such an advantage is part of our policy, we should act on high technical probability rather than requiring certainty and should have easy communication between the knowledgeable persons and the decisionmakers. Hans Bethe Ithaca, New York, February 16, 1971 Comment on the letter, 4 placing emphasis on the decision philosophies (a) Apparently used at top levels of government (b) Suggested by Bethe (c) Recommended by you. Support your recommendation. 4-9 The highway police patrol the interstate highways in order to prevent accidents and to service them when they do occur. Stopping speeders may be seen as a means to the end. Two types of patrol are available: (1) standard patrol, in which cars combine roadside standing to identify speeders and highway driving to catch them, and (2) cruising patrol, in which cars drive almost constantly at the posted speed limit to set a traffic pace. The operating cost per shift is $600 for the standard patrol and $800 for the cruising patrol. It is estimated that the mean cost to the patrol of servicing an accident is $100. In response to the right leading questions, the troop commander estimates that with the standard patrol, the chances of averaging Oor 2 accidents per shift are about the same, and either is twice as likely as averaging 6 accidents. With the cruising patrol, averaging Oaccidents is four times as likely as 2 accidents, and averaging 6 accidents just will not happen. Assume no other average is possible. (a) Develop a dollar-cost table for the two decision alternatives and the three states of nature. (b) Determine the probabilities of the states of nature based on the given descriptive likelihoods. (Note that they are different for each decision alternative. If a state of nature for the first alternative has Oprobability for the second alternative, it is not considered a state of nature for the second alternative.) For each of the following criteria determine the better alternative; I. Minimax. 2. Maximax. 3. Expected value. 4. Minimax regret. Comment on the appropriateness of this criterion. 4-10 It is suggested that the revenue that patrols generate in issuing citations to speeders should be taken into consideration. Reassess problem 4-9 if standard patrols generate 8 citations per shift (averaging $25 each) and cruising patrols generate only 2 per shift, because their cruising keeps traffic at a proper speed. 4-11 No matter how many relevant facts are considered, there are usually some that have been neglected. It is further noted that the cost of issuing and adjudicating a citation is about $50. Now reconsider problem 4-10. What other information should be considered in choosing the type of patrol? Should the dollar costs be considered at all? 4-12 Cy Linder is responsible for the medical supply room of a large medical center. Overall he is responsible for having supplies on hand as they are needed; however, he is also accountable for controlling expenses. Medicines acquired by him and subsequently requested for a patient are charged to that patient, and hence do not affect his budget. Supplies lost, stolen, or discarded do 4 1971 by The New York Times Company. Reprinted by permission. 84 QUANTITATIVE METHODS FOR PUBLIC DECISION MAKING affect it. Cy has observed that a widely used but perishable medication has been overstocked for the past few weeks. Records reveal that the demand for the medication over the past 4 years (200 weeks) is as follows: Vials of me 130 140 150 160 170 dicine used per week Number of weeks 20 80 70 20 10 Total number of weeks 200 The medication costs $5 a vial and is ordered and delivered weekly. The shelf life of the medicine is I week. (Supplies on hand at the end of the week must be discarded.) (a) Set up a loss table for the possible states of nature and decision alternatives. (b) Keeping in mind Cy's overall responsibilities, what alternative would you recommend? (c) What is his expected dollar loss for that alternative? 4-13 In problem 4-12, if Cy were able to arrange two deliveries per week, would his expected dollar loss be approximately halved? Discuss the overall effects of such an arrangement. 4-14 The U.S. Army Corps of Engineers has proposed to the Congress that the cost of maintaining the inland waterway system should be in part financed by the users of the system. They have also presented four different alternatives for assessing this charge: a tax on the fuel used by the tugs, a lockage fee for each time a barge passes through a lock, a toll on the barges for each ton per mile, and a license fee for each barge. While none of these alternatives will return the complete cost of maintaining the system, they were set at a level that would not significantly reduce the traffic. The corps also predicts that there is a 30 percent chance of a major increase of traffic, a 30 percent chance of a moderate increase, a 30 percent chance of a slight decrease, and a 10 percent chance of a major decrease. The consequences are displayed in the following table. Decision table for inland waterway revenue* Decision alternatives State of nature p Fuel tax Lockage fee License fee Toll Major increase .3 $3.7 B $3.0 B $2.7 B $4.1 B Slight increase .3 3.0 2.5 2.3 3.3 Slight decrease .3 2.4 2.1 2.1 2.4 Major decrease .1 1.9 1.9 2.0 1.7 * B = billion. (a) Which alternative is preferred according to each of the applicable criteria? (b) Which alternative do you recommend? Why? (c) Inasmuch as the probability for increases and decreases are estimates, what kind of sensitivity analysis would you recommend? Project problems 4-15 Recent years have seen some wide-scale public efforts to prevent the occurrence of some epidemic that was considered a real possibility. An epidemic, obviously, is a widespread disaster. A public program taken to forestall or prevent it is necessarily a widespread program. A vaccination DECISION THEORY: A FRAMEWORK FOR DECISION MAKING 85 program carries with it not only the purported benefit of preventing the illness but also the recognized risk of a severe reaction in certain types of individuals. Suppose the National Center for Disease Control indicates that there is a possibility of a widespread influenza epidemic within the next few months. Specifically, suppose there is a .02 probability that the epidemic will reach 20 million persons, a .08 probability that it will reach 2 million persons, and a 90 percent chance that there will be only a trace, that is, that the disease will reach 1,000 or fewer persons. Subjective estimates indicate that the influenza will be fatal to I in 1,000 persons afflicted. Hence, the death toll due to the influenza could go as high as 20,000 people without any prevention program. In the face of such a dismal forecast, the federal government is faced with a decision concerning whether to fund a public vaccination program. It has three alternatives: to have no program; to have a select program, under which 20 million high-risk persons could be vaccinated; and to have a widespread vaccination program making the vaccine available to all who desire it. Estimates in this case are that 100 million persons would avail themselves of the vaccination. The cost is estimated to be $5 per vaccination. It is expected that there will be rather widespread reaction to the vaccine. It is believed that this reaction to the vaccine will occur only in those who have not been exposed to the influenza, so that the more widespread the influenza is, the fewer reactions there will be. The reaction is considered to be fatal to one of every 100,000 vaccinated people who are not also exposed. If a selective vaccination program is undertaken, those considered most likely to get the influenza illness will be among those vaccinated. Since the influenza is believed to affect heart muscles, persons with chronic heart disease will be given the opportunity to be vaccinated. All of the various estimates, although in part subjective, can be summarized in the following table. The entries for "epidemic= 1,000" and "no program" are derived from the above information. The remaining entries are estimated but are not directly calculable from the given information alone. Decision table for influenza vaccination program; Number of fatalities due to influenza and vaccination reaction Type of program, Number vaccinated, Cost Extent of epidemic Probability Full program, 100 million, $500 million Select program, 20 million, $100 million No program, 0. $0 20 million 2 million 1,000 .02 .08 .90 1,000 + 200 100 + 600 0 + 1,000 10,000 + 50 1,000 + 150 0 + 200 20,000 + 0 2,000 + 0 1 + 0 (a) For purposes of comparison, find the alternative that would be chosen according to each of the decision criteria applied to estimated fatalities: maximax, minimax, minimax regret, expected value decision rule, maximum likelihood. (b) What would your decision be and why? Did you consider the dollar cost in your decision? 4-16 Refer to problem 4-15. There are many subjective estimates in this decision situation and, hence, sensitivity analysis of the estimates is called for. (a) If the probability estimates were corrected to be .01, .09, and .90 respectively, how would the expected value change for each alternative? (b) Which alternative do you recommend? (c) Did you make use of the cost in choosing your alternative? 86 QUANTITATIVE METHODS FOR PUBLIC DECISION MAKING (d) Would a decrease in cost to $1 per vaccination or an increase to $100 per vaccination have any effect on your decision choice? Is there any unit cost per vaccination at which you would include costs as a consideration? (e) If you were a decision maker, or someone charged with making recommendations to the decision makers, what other information would you like to have before making a decision or recommendation? Would it take a long time to get this information? What effect might waiting have on decision making? 4-17 Refer to problem 4-15. Suppose that a consequence of the influenza and the vaccination reaction is not death, but rather critical illness. Since it is not completely known how the influenza affects the heart muscles, the severity of the illness is uncertain. The uncertainty includes whether the effect is temporary or permanent, whether the effect is on the heart muscles in general or on a particular muscle, and how the influenza in combination with some other preexisting condition will affect the individual. Presumably, some persons with particular respiratory or circulatory difficulties would succumb. Under these new conditions, which type of program would you recommend? (Assume no numeric changes from problem 4-15.) 4-18 Think of a decision situation in which you have been involved. (a) Was there an identifiable objective? (b) Was there a workable criterion? (c) Were alternatives explicitly sought and found? List them. (d) Were uncontrollable but related events (states of nature) uncovered? What were they? (e) Were the consequences of the alternatives investigated? Summarize them. (f) Can you summarize the decision situation in a decision table? (g) Is there an appropriate probability distribution for the states of nature (the uncontrollables)? (h) Which alternative would you pick based on the summary? (i) Which alternative was chosen? (j) If there is a discrepancy between the answers to parts h and i, is it due to a problem feature that cannot be adequately represented in the decision table? (k) Discuss the role of such a summary in decision making.

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