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COST- EFFECTIVE HEPATITIS B INTERVENTIONS Hepatitis B is a viral disease that can lead to death and liver cancer if not treated. It is especially prevalent in Asian populations. The disease chronically infects approximately 8% to 10% of people in China and a similar percentage of Americans of Asian descent. It often infects newborns and children, in which case it is likely to become a life- long infection. Chronic infection is often asymptomatic for decades, but if left untreated, about 25% of the chronically infected will die of liver diseases such as cirrhosis or liver cancer.

A hepatitis B vaccine became available in the 1980s, but it is costly (thousands of dollars per year) and it does not cure the disease. Vaccination of children in the United States is widespread, and only about 0.5% of the general popu- lation is infected. This percentage, however, jumps to about 10% for U.S. adult Asian and Pacific Islanders, where the rate of liver cancer is more than three times that of the general U.S. population. The situation is even worse in China, where it is estimated that approxi- mately 300,000 die each year from liver disease caused by hepatitis B. Although rates of newborn vaccination in China have increased in recent years, about 20% of 1- to 4-year olds and 40% of 5- to 19-year olds still remain unprotected. In a pilot program in the Qinghai province, the feasibility of a vaccination “catch-up” program was demonstrated, but China’s public health officials worried about the cost effectiveness of a country-wide catch-up program.

The article by Hutton et al. (2011) reports the results of the work his team carried out over several years with the Asian Liver Center at Stanford University. They used decision analysis and other quantitative methods to analyze the cost effectiveness of several inter- ventions to combat hepatitis B in the United States and China. They addressed two policy questions in the study: (1) What combination of screening, treatment, and vaccination is most cost effective for U.S. adult Asian and Pacific Islanders; and (2) Is it cost effective to provide hepatitis B catch-up vaccination for children and adolescents in China?

For the first question, the team first considered the approach usually favored by the medical community, clinical trials, but they decided it was infeasible because of expense and the time (probably decades) required. Instead, they used decision analysis with a deci- sion tree very much like those in this chapter. The initial decision is whether to screen people for the disease. If this initial decision is no, the next decision is whether to vacci- nate. If this decision is no, they wait to see whether infection occurs. On the other side, if the initial decision to screen is yes, there are three possible outcomes: infected, immune, or susceptible. If infected, the next decision is whether to treat. If immune, no action is necessary. If susceptible, the sequence is the same as after the “no screen” decision. At end nodes with a possibility (or certainty) of being infected, the team used a probabilistic “Markov” model to determine future health states.

The various decision strategies were compared in terms of incremental cost to gain an incremental unit of health, with units of health expressed in quality-adjusted life years (QALYs). Interventions are considered cost effective if they cost less per QALY gained than three times a country’s per capita GDP; they are considered very cost effective if they cost less than a country’s per capita GDP.

CHAPTER 6 Decision Making Under Uncertainty

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6-1 Introduction 2 4 3

The study found that it is cost effective to screen Asian and Pacific Islanders so that they can receive treatment, and it is also cost effective to vaccinate those in close contact with infected individuals so that they can be protected from infection. Specifically, they estimated that this policy costs from $36,000 to $40,000 per QALY gained, whereas an intervention that costs $50,000 per QALY gained is considered cost effective in the United States. However, they found that it is not cost effective to provide universal vaccination for all U.S. adult Asian and Pacific Islanders, primarily because the risk of being exposed to hepatitis B for U.S. adults is low.

For the second question, the team used a similar decision analysis to determine that providing catch-up vaccination for children up to age 19 not only improves health out- comes but saves costs. Using sensitivity analysis, they found that catch-up vaccination might not be cost saving if the probability of a child becoming infected is one-fifth as high as the base-case estimate of 100 out of 100,000 per year. This is due to the high level of newborn vaccination coverage already achieved in some urban areas of China. They also found if treatment becomes cheaper, the cost advantages of vaccination decrease. However, treatment costs would have to be halved and infection risk would have to be five times lower than in their base case before the cost of providing catch-up vaccination would exceed $2500 per QALY gained (roughly equal to per capita GDP in China).

In any case, their analysis influenced China’s 2009 decision to expand free catch-up vaccination to all children in China under the age of 15. This decision could result in about 170 million children being vaccinated, and it could prevent hundreds of thousands of chronic infections and close to 70,000 deaths from hepatitis B.

6-1 Introduction This chapter provides a formal framework for analyzing decision problems that involve uncertainty. Our discussion includes the following:

• criteria for choosing among alternative decisions • how probabilities are used in the decision-making process • how early decisions affect decisions made at a later stage • how a decision maker can quantify the value of information • how attitudes toward risk can affect the analysis

Throughout, we employ a powerful graphical tool—a decision tree—to guide the analysis. A decision tree enables a decision maker to view all important aspects of the problem at once: the decision alternatives, the uncertain outcomes and their probabilities, the eco- nomic consequences, and the chronological order of events. Although decision trees have been used for years, often created with paper and pencil, we show how they can be imple- mented in Excel with the PrecisionTree add-in from Palisade.

Many examples of decision making under uncertainty exist in the business world, including the following:

• Companies routinely place bids for contracts to complete a certain project within a fixed time frame. Often these are sealed bids, where each company presents a bid for complet- ing the project in a sealed envelope. Then the envelopes are opened, and the low bidder is awarded the bid amount to complete the project. Any particular company in the bidding competition must deal with the uncertainty of the other companies’ bids, as well as possible uncertainty regarding their cost to complete the project if they win the bid. The trade-off is between bidding low to win the bid and bidding high to make a larger profit.

• Whenever a company contemplates introducing a new product into the market, there are a number of uncertainties that affect the decision, probably the most important being the customers’ reaction to this product. If the product generates high customer demand, the company will make a large profit. But if demand is low—and the vast majority of new products do poorly—the company could fail to recoup its development costs. Because the level of customer demand is critical, the company might try to gauge this level by

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test marketing the product in one region of the country. If this test market is a success, the company can then be more optimistic that a full-scale national marketing of the product will also be successful. But if the test market is a failure, the company can cut its losses by abandoning the product.

• Whenever manufacturing companies make capacity expansion decisions, they face uncertain consequences. First, they must decide whether to build new plants. If they don’t expand and demand for their products is higher than expected, they will lose rev- enue because of insufficient capacity. If they do expand and demand for their prod- ucts is lower than expected, they will be stuck with expensive underutilized capacity. Companies also need to decide where to build new plants. This decision involves a whole new set of uncertainties, including exchange rates, labor availability, social stabil- ity, competition from local businesses, and others.

• Banks must continually make decisions on whether to grant loans to businesses or indi- viduals. Many banks made many very poor decisions, especially on mortgage loans, during the years leading up to the financial crisis in 2008. They fooled themselves into thinking that housing prices would only increase, never decrease. When the bottom fell out of the housing market, banks were stuck with loans that could never be repaid.

• Utility companies must make many decisions that have significant environmental and economic consequences. For these companies it is not necessarily enough to conform to federal or state environmental regulations. Recent court decisions have found com- panies liable—for huge settlements—when accidents occurred, even though the compa- nies followed all existing regulations. Therefore, when utility companies decide whether to replace equipment or mitigate the effects of environmental pollution, they must take into account the possible environmental consequences (such as injuries to people) as well as economic consequences (such as lawsuits). An aspect of these situations that makes decision analysis particularly difficult is that the potential “disasters” are often extremely unlikely; hence, their probabilities are difficult to assess accurately.

• Sports teams continually make decisions under uncertainty. Sometimes these decisions involve long-run consequences, such as whether to trade for a promising but as yet untested pitcher in baseball. Other times these decisions involve short-run consequences, such as whether to go for a fourth down or kick a field goal late in a close football game. You might be surprised at the level of quantitative sophistication in today’s professional sports. Management and coaches typically do not make important decisions by gut feel- ing. They employ many of the tools in this chapter and in other chapters of this book.

Although the focus of this chapter is on business decisions, the approach discussed in this chapter can also be used in important personal decisions you have to make. As an example, if you are just finishing an undergraduate degree, should you go immediately into a graduate program, or should you work for several years and then decide whether to pursue a graduate degree? As another example, if you currently have a decent job but you have the option to take another possibly more promising job that would require you and your family to move to another part of the country, should you stay or move?

You might not have to make too many life-changing decisions like these, but you will undoubtedly have to make a few. How will you make them? You will probably not use all the formal methods discussed in this chapter, but the discussion provided here should at least motivate you to think in a structured way before making your final decisions.

6-2 Elements of Decision Analysis Although decision making under uncertainty occurs in a wide variety of contexts, the problems we discuss in this chapter are alike in the following ways:

1. A problem has been identified that requires a solution. 2. A number of possible decisions have been identified. 3. Each decision leads to a number of possible outcomes.

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4. There is uncertainty about which outcome will occur, and probabilities of the possible outcomes are assessed.

5. For each decision and each possible outcome, a payoff is received or a cost is incurred. 6. A “best” decision must be chosen using an appropriate decision criterion.

We now discuss these elements in some generality.1

Identifying the Problem When something triggers the need to solve a problem, you should think carefully about the problem that needs to be solved before diving in. Perhaps you are just finishing your undergrad- uate degree (the trigger), and you want to choose the Business School where you should get your MBA degree. You could define the problem as which MBA program you should attend, but maybe you should define it more generally as what you should do next now that you have your undergraduate degree. You don’t necessarily have to enter an MBA program right away. You could get a job and then get an MBA degree later, or you could enter a graduate program in some area other than Business. Maybe you could even open your own business and forget about graduate school. The point is that by changing the problem from deciding which MBA program to attend to deciding what to do next, you change the decision problem in a fundamental way.

Possible Decisions The possible decisions depend on the previous step: how the problem is specified. But after you identify the problem, all possible decisions for this problem should be listed. Keep in mind that if a potential decision isn’t in this list, it won’t have a chance of being chosen as the best decision later, so this list should be as comprehensive as possible. Some problems are of a multistage nature, as discussed in Section 6.6. In such problems, a first-stage decision is made, then an uncertain outcome is observed, then a second-stage decision is made, then a second uncertain outcome is observed, and so on. (Often there are only two stages, but there could be more.) In this case, a “decision” is really a “strategy” or “contingency plan” that prescribes what to do at each stage, depending on prior deci- sions and observed outcomes. These ideas are clarified in Section 6.6.

Possible Outcomes One of the main reasons why decision making under uncertainty is difficult is that decisions have to be made before uncertain outcomes are revealed. For example, you must place your bet at a roulette wheel before the wheel is spun. Or you must decide what type of auto insurance to purchase before you find out whether you will be in an accident. However, before you make a decision, you must at least list the possible outcomes that might occur. In some cases, the outcomes will be a small set of discrete possibilities, such as the 11 possible sums (2 through 12) of the roll of two dice. In other cases, the outcomes will be a continuum of possibilities, such as the possible damage amounts to a car in an accident. In this chapter, we generally allow only a small discrete set of possible outcomes. If the actual set of outcomes is a continuum, we typically choose a small set of representative outcomes from this continuum.

Probabilities of Outcomes A list of all possible outcomes is not enough. As a decision maker, you must also assess the likelihoods of these outcomes with probabilities. These outcomes are generally not equally likely. For example, if there are only two possible outcomes, rain or no rain, when you are deciding whether to carry an umbrella to work, there is no generally no reason to assume that each of these outcomes has a 50-50 chance of occurring. Depending on the weather report, they might be 80-20, 30-70, or any of many other possibilities.

There is no easy way to assess the probabilities of the possible outcomes. Sometimes they will be determined at least partly by historical data. For example, if demand for your

1 For an interesting discussion of decision making at a very nontechnical level, we recommend the book by Hammond et al. (2015).

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product is uncertain, with possible outcomes “low,” “medium,” and “high,” you might assess their probabilities as 0.5, 0.3, and 0.2 because past demands have been low about 50% of the time, medium about 30% of the time, and high about 20% of the time.2 How- ever, this product might be a totally new product, unlike any of your previous products. Then data on past demands will probably not be relevant, and your probability assess- ments for demand of the new product will necessarily contain a heavy subjective compo- nent—your best guesses based on your experience and possibly the inputs of the marketing experts in your company. In fact, probabilities in most real business decision-making problems are of the subjective variety, so managers must make the probability assessments most in line with the data available and their gut feeling.

To complicate matters, probabilities sometimes change as more information becomes available. For example, suppose you assess the probability that the Golden State Warriors will win the NBA championship this year. Will this assessment change if you hear later that Steph Curry has suffered a season-ending injury? It almost surely will, probably quite a lot. Sometimes, as in this basketball example, you will change your probabilities in an informal way when you get new information. However, in Section 6.6, we show how probabilities can be updated in a formal way by using an important law of probabilities called Bayes’ rule.

Payoffs and Costs Decisions and outcomes have consequences, either good or bad. These must be assessed before intelligent decisions can be made. In our problems, these will be monetary payoffs or costs, but in many real-world decision problems, they can be nonmonetary, such as environmental damage or loss of life. Nonmonetary consequences can be very difficult to quantify, but an attempt must be made to do so. Otherwise, it is impossible to make mean- ingful trade-offs.

Decision Criterion Once all of these elements of a decision problem have been specified, you must make some difficult trade-offs. For example, would you rather take a chance at receiving $1 million, with the risk of losing $2 million, or would you rather play it safer? Of course, the answer depends on the probabilities of these two outcomes, but as you will see later in the chapter, if very large amounts of money are at stake (relative to your wealth), your attitude toward risk can also play a key role in the decision-making process.

In any case, for each possible decision, you face a number of uncertain outcomes with given probabilities, and each of these leads to a payoff or a cost. The result is a probability distribution of payoffs and costs. For example, one decision might lead to the following: a payoff of $50,000 with probability 0.1, a payoff of $10,000 with probability 0.2, and a cost of $5000 with probability 0.7. (The three outcomes are mutually exclusive; their probabil- ities sum to 1.) Another decision might lead to the following: a payoff of $5000 with prob- ability 0.6 and a cost of $1000 with probability 0.4. Which of these two decisions do you favor? The choice is not obvious. The first decision has more upside potential but more downside risk, whereas the second decision is safer.

In situations like this—the same situations faced throughout this chapter—you need a decision criterion for choosing between two or more probability distributions of payoff/ cost outcomes. Several methods have been proposed:

• Look at the worst possible outcome for each decision and choose the decision that has the least bad of these. This is relevant for an extreme pessimist.

• Look at the 5th percentile of the distribution of outcomes for each decision and choose the decision that has the best of these. This is also relevant for a pessimist—or a com- pany that wants to limit its losses. (Any percentile, not just the 5th, could be chosen.)

2 As discussed in the previous chapter, there are several equivalent ways to express probabilities. As an example, you can state that the probability of your team winning a basketball game is 0.6. Alternatively, you can say that the probability of them winning is 60%, or that the odds of them winning are 3 to 2. These are all equivalent. We will generally express probabilities as decimal numbers between 0 and 1, but we will some- times quote percentages.

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• Look at the best possible outcome for each decision and choose the decision that has the best of these. This is relevant for an extreme optimist.

• Look at the variance (or standard deviation) of the distribution of outcomes for each decision and choose the decision that has the smallest of these. This is relevant for mini- mizing risk but it treats upside risk and downside risk in the same way.

• Look at the downside risk (however you want to define it) of the distribution of outcomes for each decision and choose the decision with the smallest of these. Again, this is relevant for minimizing risk, but now it minimizes only the part of the risk you really want to avoid.

The point here is that a probability distribution of payoffs and costs has several summary measures that could be used a decision criterion, and you could make an argument for any of the measures just listed. However, the measure that has been used most often, and the one that will be used for most of this chapter, is the mean of the probability distribution, also called its expected value. Because we are dealing with monetary outcomes, this crite- rion is generally known as the expected monetary value, or EMV criterion. The EMV criterion has a long-standing tradition in decision-making analysis, both at a theoretical level (hundreds of scholarly journal articles) and at a practical level (used by many busi- nesses). It provides a rational way of making decisions, at least when the monetary pay- offs and costs are of “moderate” size relative to the decision maker’s wealth. (Section 6.7 presents another decision criterion when the monetary values are not “moderate.”)

The expected monetary value, or EMV, for any decision is a weighted average of the possible payoffs/costs for this decision, weighted by the probabilities of the outcomes. Using the EMV criterion, you choose the decision with the largest EMV. This is sometimes called “playing the averages.”

The EMV criterion is also easy to operationalize. For each decision, you take a weighted sum of the possible monetary outcomes, weighted by their probabilities, to find the EMV. Then you identify the largest of these EMVs. For the two decisions listed earlier, their EMVs are as follows:

• Decision 1: EMV 5 50000(0.1) 1 10000(0.3) 1 (25000)(0.6) 5 $3500

• Decision 2: EMV 5 5000(0.6) 1 (21000)(0.4) 5 $2600

Therefore, according to the EMV criterion, you should choose decision 1.

6-3 EMV and Decision Trees Because the EMV criterion plays such a crucial role in decision making under uncertainty, it is worth exploring in more detail.

First, if you are acting according to the EMV criterion, you value a decision with a given EMV the same as a sure monetary outcome with the same EMV. To see how this works, sup- pose there is a third decision in addition to the previous two. If you choose this decision, there is no risk at all; you receive a sure $3000. Should you make this decision, presumably to avoid risk? According to the EMV criterion, the answer is no. Decision 1, with an EMV of $3500, is equivalent (for an EMV maximizer) to a sure $3500 payoff. Hence, it is favored over the new riskless decision. (Read this paragraph several times and think about its consequences. It is sometimes difficult to accept this logic in real decision-making problems, which is why not everyone uses the EMV criterion in every situation.)

Second, the EMV criterion doesn’t guarantee good outcomes. Indeed, no criterion can guarantee good outcomes. If you make decision 1, for example, you might get lucky and make $50,000, but there is a 70% chance that you will lose $5000. This is the very nature of decision making under uncertainty: you make a decision and then you wait to see the consequences. They might be good and they might be bad, but at least by using the EMV criterion, you know that you have proceeded rationally.

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Third, the EMV criterion is easy to operationalize in a spreadsheet. This is shown in Figure 6.1. (See the file Simple Decision Problem Finished.xlsx.) For any decision, you list the possible payoff/cost values and their probabilities. Then you calculate the EMV with a SUMPRODUCT function. For example, the formula in cell B7 is

5SUMPRODUCT(A3:A5,B3:B5)

Figure 6.1 EMV Calculations in Excel

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A B C D E F G H Decision 1

Payoff/Cost $50,000 $10,000 –$5,000

EMV $3,500 EMV$2,600EMV $3,000

0.1 $5,000 −$1,000

0.6 $3,000 1 0.40.2

0.7

Probability Payoff/Cost Probability Probability Decision 2

Payoff/Cost Decision 3

The advantage to calculating EMVs in a spreadsheet is that you can easily perform sensitivity analysis on any of the inputs. For example, Figure 6.2 shows what happens when the good outcome for decision 2 becomes more probable (and the bad outcome becomes less probable). Now the EMV for decision 2 is the largest of the three EMVs, so it is the best decision.

Usually, the most important information from a sensitivity analysis is whether the best decision continues to be best as one or more inputs change.

Figure 6.2 EMV Calculations with Different Inputs

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A B C D E F G H Decision 1

Payoff/Cost $50,000 $10,000 –$5,000

EMV $3,500 EMV$3,800EMV $3,000

0.1 $5,000 –$1,000

0.8 $3,000 1 0.20.2

0.7

Probability Payoff/Cost Probability Payoff/Cost Probability Decision 2 Decision 3

You might still be wondering why we choose the EMV criterion in the first place. One way of answering this is that EMV represents a long-run average. If—and this is a big if—the decision could be repeated many times, all with the same monetary values and probabilities, the EMV is the long-run average of the outcomes you would observe. For example, by making decision 1, you would gain $50,000 about 10% of the time, you would gain $10,000 about 20% of the time, and you would lose $5000 about 70% of the time. In the long run, your average net gain would be about $3500.

This argument might or might not be relevant. For a company that routinely makes many decisions of this type, even though they are not identical, long-term averages make sense. Sometimes they win, and sometimes they lose, but it makes sense for them to be concerned only with long-term averages. However, a particular decision problem is often a “one-shot deal.” It won’t be repeated many times in the future; in fact, it won’t be repeated at all. In this case, you might argue that a long-term average criterion makes no sense and that some other criterion should be used instead. This has been debated by decision analysts, including many academics, for years, and the arguments continue. Nevertheless, most analysts agree that when “moderate” amounts of money are at stake, the EMV crite- rion provides a rational way of making decisions, even for one-shot deals. Therefore, we use the EMV criterion in most of this chapter.

A decision problem evolves through time. A decision is made, then an uncertain out- come is observed, then another decision might need to be made, then another uncertain outcome might be observed, and so on. All the while, payoffs are being received or costs are being incurred. It is useful to show all these elements of the decision problem, includ- ing the timing, in a type of graph called a decision tree. A decision tree not only allows everyone involved to see the elements of the decision problem in an intuitive format, but it also provides a straightforward way of making the necessary EMV calculations.

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The decision tree for the simple decision problem discussed earlier appears in Figure 6.3.

What It Means to Be an eMV Maximizer

An EMV maximizer, by definition, is indifferent when faced with the choice between entering a gamble with a given EMV and receiving a sure dollar amount in the amount of the EMV. For example, consider a gamble where you flip a fair coin and win $0 or $1000 depending on whether you get a head or a tail. If you are an EMV maximizer, you are indifferent between entering this gamble, which has EMV $500, and receiving $500 for sure. Similarly, if the gamble is between losing $1000 and winning $500, based on the flip of the coin, and you are an EMV maximizer, you are indifferent between entering this gamble, which has EMV 2$250, and pay- ing a sure $250 to avoid the gamble. (This latter scenario is the basis of insurance.)

Fundamental Insight

Figure 6.3 Simple Decision Tree

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50000

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�5000

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�1000

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Decision 1

Decision 2

Decision 3

3000

This decision tree was actually created in Excel by using its built-in shape tools on a blank worksheet, but you could just as well draw it on a piece of paper. Alternatively, you could use the Palisade PrecisionTree add-in that we discuss later in the chapter. The important thing for now is how you interpret this decision tree. It is important to realize that decision trees such as this one have been used for over 50 years. They all use the fol- lowing basic conventions:

Decision Tree Conventions

1. Decision trees are composed of nodes (circles, squares, and triangles) and branches (lines).

2. The nodes represent points in time. A decision node (a square) represents a time when you make a decision. A probability node (a circle) represents a time when the result of an uncertain outcome becomes known. An end node (a triangle) indicates that the problem is completed—all decisions have been made, all uncertainty has been resolved, and all payoffs and costs have been incurred. (When people draw decision trees by hand, they often omit the actual triangles, as we have done in Figure 6.3. However, we still refer to the right-hand tips of the branches as the end nodes.)

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The decision tree in Figure 6.3 follows these conventions. The decision node comes first (to the left) because you must make a decision before observing any uncertain out- comes. The probability nodes then follow the decision branches, and the probabilities appear above their branches. (Actually, there is no need for a probability node after deci- sion 3 branch because its monetary value is a sure $3000.) The ultimate payoffs or costs appear next to the end nodes, to the right of the probability branches. The EMVs above the probability nodes are for the various decisions. For example, the EMV for the decision 1 branch is $3500. The maximum of the EMVs corresponds to the decision 1 branch, and this maximum is written above the decision node. Because it corresponds to decision 1, we put a notch on the decision 1 branch to indicate that this decision is best.

This decision tree is almost a direct translation of the spreadsheet model in Figure 6.1. Indeed, a decision tree is overkill for such a simple problem; the spreadsheet model provides all of the required information. However, as you will see later, especially in Section 6.6, decision trees provide a useful view of more complex problems. In addition, decision trees provides a framework for doing all of the EMV calculations. They allow you to use the following folding-back procedure to find the EMVs and the best decision.

3. Time proceeds from left to right. This means that any branches leading into a node (from the left) have already occurred. Any branches leading out of a node (to the right) have not yet occurred.

4. Branches leading out of a decision node represent the possible decisions; you get to choose the branch you prefer. Branches leading out of probability nodes represent the possible uncertain outcomes; you have no control over which of these will occur.

5. Probabilities are listed on probability branches. These probabilities are conditional on the events that have already been observed (those to the left). Also, the probabilities on branches leading out of any probability node must sum to 1.

6. Monetary values are shown to the right of the end nodes. (As we discuss shortly, some monetary values can also be placed under the branches where they occur in time.)

7. EMVs are calculated through a “folding-back” process, discussed next. They are shown above the various nodes. It is then customary to mark the optimal decision branch(es) in some way. We have marked ours with a small notch.

Folding-Back Procedure

Starting from the right of the decision tree and working back to the left:

1. At each probability node, calculate an EMV—a sum of products of monetary values and probabilities.

2. At each decision node, take a maximum of EMVs to identify the optimal decision.3

3 Some decision problems involve only costs. In that case it is more convenient to label the tree with positive costs and take minimums of expected costs at the decision nodes.

This is exactly what we did in Figure 6.3. At each probability node, we calculated EMVs in the usual way (sums of products) and wrote them above the nodes. Then at the decision node, we took the maximum of the three EMVs and wrote it above this node.

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6-4 One-Stage Decision problems 2 5 1

Although this procedure requires more work for more complex decision trees, the same two steps—taking EMVs at probability nodes and taking maximums at decision nodes— are the only arithmetic operations required. In addition, the PrecisionTree add-in discussed later in the chapter performs the folding-back calculations for you.

The folding-back process is a systematic way of calcu- lating EMVs in a decision tree and thereby identifying the best decision strategy.

b. Let the probability of the worst outcome for the first decision, the value in cell B5, vary from 0.7 to 0.9 in increments of 0.025, and use formulas in cells B3 and B4 to ensure that they remain in the ratio 1 to 2 and the three probabilities for decision 1 continue to sum to 1.

c. Use a two-way data table to let the inputs in parts a and b vary simultaneously over the indicated ranges.

Level B 3. Some decision makers prefer decisions with low risk, but

this depends on how risk is measured. As we mentioned in this section, variance (see the definition in problem 1) is one measure of risk, but it includes both upside and downside risk. That is, an outcome with a large positive payoff contributes to variance, but this type of “risk” is good. Consider a decision with some possible payoffs and some possible costs, with given probabilities. How might you develop a measure of downside risk for such a decision? With your downside measure of risk, which decision in Figure 6.1 do you prefer, decision 1 or deci- sion 2? (There is no single correct answer.)

Problems Solutions for problems whose numbers appear within a colored box can be found in the Student Solution Files.

Level A 1. Several decision criteria besides EMV are suggested in

the section. For each of the following criteria, rank all three decisions in Figure 6.1 from best to worst. a. Look only at the worst possible outcome for each

decision. b. Look only at the best possible outcome for each

decision. c. Look at the variance of the distribution of outcomes

for each decision, which you want to be small. (The variance of a probability distribution is the weighted sum of squared differences from the mean, weighted by the probabilities.)

2. For the decision problem in Figure 6.1, use data tables to perform the following sensitivity analyses. The goal in each is to see whether decision 1 continues to have the largest EMV. In each part, provide a brief explanation of the results. a. Let the payoff from the best outcome, the value in cell

A3, vary from $30,000 to $50,000 in increments of $2500.

6-4 One-Stage Decision Problems Many decision problems are similar to the simple decision problem discussed in the previous section. You make a decision, then you wait to see an uncertain outcome, and a payoff is received or a cost is incurred. We refer to these as single-stage decision problems because you make only one decision, the one right now. They all unfold in essentially the same way, as indicated by the spreadsheet model in Figure 6.1 or the deci- sion tree in Figure 6.3. The following example is typical of one-stage decision problems. This example is used as a starting point for more complex examples in later sections.

EXAMPLE

6.1 NEW PRODUCT DECISIONS AT ACME The Acme Company must decide whether to market a new product. As in many new-product situations, there is considerable uncertainty about the eventual success of the product. The product is currently part way through the development process, and some fixed development costs have already been incurred. If the company decides to continue development and then market the product, there will be additional fixed costs, and they are estimated to be $6 million. If the product is marketed, its unit margin (selling price minus variable cost) will be $18. Acme classifies the possible market results as “great,” “fair,” and “awful,” and it estimates the probabilities of these outcomes to be 0.45, 0.35, and 0.20, respectively. Finally, the company

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As before, a decision tree is probably overkill for this problem, but it is shown in Figure 6.5. (All monetary and sales volumes are shown in thousands.) This tree indicates one of at least two equivalent ways to show the EMV calculations. The values at the end nodes ignore the fixed cost, which is instead shown under the decision branch as a negative number. There- fore, the 7074 value above the probability node is the expected net revenue, not including the fixed cost. Then the fixed cost is subtracted from this to obtain the 1074 value above the decision node.

Figure 6.6 shows an equivalent tree, where the fixed cost is still shown under the decision branch but is subtracted from each end node. Now the EMV above the probability node is after subtraction of the fixed cost. The two trees are equivalent and either is perfectly acceptable. However, the second tree provides the insight that two of the three outcomes result in a net loss to Acme, even though the weighted average, the EMV, is well in the positive range. (Besides, as you will see in the next section, the second tree is the way the PrecisionTree add-in does it.)

estimates that the corresponding sales volumes (in thousands of units sold) from these three outcomes are 600, 300, and 90, respectively. Assuming that Acme is an EMV maximizer, should it finish development and then market the product, or should it stop development at this point and abandon the product?4

Objective To use the EMV criterion to help Acme decide whether to go ahead with the product.

Where Do the Numbers Come From? Acme’s cost accountants should be able to estimate the monetary inputs: the fixed costs and the unit margin. (Any fixed costs already incurred are sunk and therefore have no relevance to the current decision.) The uncertain sales volume is really a con- tinuous variable but, as in many decision problems, Acme has replaced the continuum by three representative possibilities. The assessment of the probabilities and the sales volumes for these three possibilities might be based partly on historical data and market research, but they almost surely have a subjective component.

Solution The elements of the decision problem appear in Figure 6.4. (See the files New Product Decisions - Single-Stage 1a Finished.xlsx and New Product Decisions - Single-Stage 1b Finished.xlsx.) If the company decides to stop development and abandon the product, there are no payoffs, costs, or uncertainties; the EMV is $0. (Actually, this isn’t really an expected value; it is a sure $0.) On the other hand, if the company proceeds with the product, it incurs the fixed cost and receives $18 for every unit it sells. The probability distribution of sales volume given in the problem statement appears in columns A to C, and each sales volume is multiplied by the unit margin to obtain the net revenues in column D. Finally, the formula for the EMV in cell B12 is

5SUMPRODUCT(D8:D10,B8:B10)-B4

Because this EMV is positive, slightly over $1 million, the company is better off marketing the product than abandoning it.

Figure 6.4 Spreadsheet Model for Single-Stage New Product Decision

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16

A B C D E F G

Decision 1: Continue development and market the new product

Acme single-stage new product decision

Fixed cost Unit margin

% decrease in all sales volumes EMV for decision 1

Sensitivity analysis to percentage decrease in all sales volumes

5% 10% 15% 20%

$1,074,000 $720,300 $366,600

$12,900 –$340,800

Market Great Fair Awful

EMV

$6,000,000 $18

0% $1,074,000

% decrease EMV for decision 1Probability 0.45 0.35 0.20

$1,074,000

Sales volume 600,000 300,000

90,000

Net revenue $10,800,000

$5,400,000 $1,620,000

Decision 2: Stop development and abandon product No payoffs, no costs, no uncertainty EMV

2 5 2 C h a p t e r 6 D e c i s i o n M a k i n g U n d e r U n c e r t a i n t y

4 To keep the model simple, we ignore taxes and the time value of money.

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Then a data table is used in the usual way, with cell G3 as the column input cell, to calculate the EMV for various percentage decreases. As you can see, the EMV stays positive, so that marketing remains best, for decreases up to 15%. But if the decrease is 20%, the EMV becomes negative, meaning that the best decision is to abandon the product. In this case, the possible gains from marketing are not large enough to offset the fixed cost.

Figure 6.5 Decision Tree for New Product Model

7074

600(18) = 10800

300(18) = 5400

90(18) = 1620

Awful 0.20

Fair 0.35

Great 0.45

1074

Market product –6000

Abandon product

Figure 6.6 Equivalent Decision Tree

1074

600(18) – 6000 = 4800

300(18) – 6000 = –600

90(18) – 6000 = –4380

Awful 0.20

Fair 0.35

Great 0.45

1074

Market product –6000

Abandon product

Figure 6.7 Sensitivity Analysis 3 4 5 6 7 8 9

10 11 12

F G % decrease in all sales volumes

Sensitivity analysis to percentage decrease in all sales volumes

EMV for decision 1

% decrease

0% $1,074,000

$1,074,000 $720,300 $366,600

$12,900 –$340,800

5% 10% 15% 20%

EMV for decision 1

6-4 One-Stage Decision problems 2 5 3

Using the spreadsheet model in Figure 6.4, it is easy to perform a sensitivity analysis. Usually, the main purpose of such an analysis is to see whether the best decision changes as one or more inputs change. As an example, we will see whether the best decision continues to be “proceed with marketing” if the total market decreases. Specifically, we let each of the potential sales volumes decrease by the same percentage and we keep track of the EMV from marketing the product. The results appear in Figure 6.7. For any percentage decrease in cell G3, the EMV from marketing is calculated in cell G4 with the formula

5(1-G3)*SUMPRODUCT(D8:D10,B8:B10)-B4

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2 5 4 C h a p t e r 6 D e c i s i o n M a k i n g U n d e r U n c e r t a i n t y

The Acme problem is a prototype for all single-stage decision problems. When only a single decision needs to be made, and all of the elements of the decision problem have been specified, it is easy to calculate the required EMVs for the possible decisions and hence determine the EMV-maximizing decision in a spreadsheet model. The problem and the calculations can also be shown in a decision tree, although this doesn’t really provide any new information except possibly to give everyone involved a better “picture” of the decision problem. In the next section, we examine a multistage version of the Acme problem, and then the real advantage of decision trees will become evident.

Level B 7. Sometimes a “single-stage” decision can be broken

down into a sequence of decisions, with no uncertainty resolved between these decisions. Similarly, uncer- tainty can sometimes be broken down into a sequence of uncertain outcomes. Here is a typical example. A com- pany has a chance to bid on a government project. The company first decides whether to place a bid, and then if it decides to place a bid, it decides how much to bid. Once these decisions have been made, the uncertainty is resolved. First, the company observes whether there are any competing bids. Second, if there is at least one com- peting bid, the company observes the lowest competing bid. The lowest of all bids wins the contract. Draw a decision tree that reflects this sequence. There should be two “stages” of decision nodes, followed by two “stages” of probability nodes. Then label the tree with some reasonable monetary values and probabilities, and perform the folding back process to find the company’s best strategy. Note that if the company wins the contract, its payoff is its bid amount minus its cost of complet- ing the project minus its cost of preparing the bid, where these costs are assumed to be known.

Problems Solutions for problems whose numbers appear within a colored box can be found in the Student Solution Files.

Level A 4. The fixed cost of $6 million in the Acme problem is

evidently not large enough to make Acme abandon the product at the current time. How large would the fixed cost need to be to make the abandon option the best option? Explain how the decision tree, especially the version in Figure 6.5, answers this question easily.

5. Perform a sensitivity analysis on the probability of a great market. To do this, enter formulas in cells B9 and B10 (see Figure 6.4) to ensure that the probabilities of “fair” and “awful” remain in the same ratio, 35 to 20, and that all three probabilities continue to sum to 1. Then let the prob- ability of “great” vary from 0.25 to 0.50 in increments of 0.05. Is it ever best to abandon the product in this range?

6. Sometimes it is possible for a company to influence the uncertain outcomes in a favorable direction. Suppose Acme could, by an early marketing blitz, change the prob- abilities of “great,” “fair,” and “awful” from their current values to 0.75, 0.15, and 0.10. In terms of EMV, how much would the company be willing to pay for such a blitz?

6-5 The PrecisionTree Add-In Decision trees present a challenge for Excel. The challenge is to take advantage of Excel’s calculation capabilities (to calculate EMVs, for example) and its graphical capabilities (to draw the decision tree). Using only Excel’s built-in tools, this is virtually impossible (or at least very painful) to do. Fortunately, Palisade has developed an Excel add-in called PrecisionTree that makes the process relatively straightforward. This add-in not only enables you to draw and label a decision tree, but it also performs the folding-back procedure automatically and then allows you to perform sensitivity analysis on key input parameters.

The first thing you must do to use PrecisionTree is to “add it in.” We assume you have already installed the Palisade DecisionTools Suite. Then to run PrecisionTree, you have two options:

• If Excel is not currently running, you can open Excel and PrecisionTree by selecting PrecisionTree from the Palisade group in the list of programs on your computer.

• If Excel is currently running, the first option will open PrecisionTree on top of Excel.

In either case, you will see the Welcome screen in Figure 6.8. Note the Quick Start link. We will come back to this shortly.

Once you click OK to dismiss the Welcome screen, you will know that PrecisionTree is loaded because of the new PrecisionTree tab and associated ribbon shown in Figure 6.9.

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6-5 the precisiontree add-In 2 5 5

Although PrecisionTree is quite easy to use once you are familiar with it, you have to learn the basics. The easiest way to do this is to run a series of Quick Start videos. To do this, you can bring up the Welcome screen in Figure 6.8 at any time through the Precision- Tree Help dropdown list. Then you can click the Quick Start link on the Welcome screen. This opens an example file shown in Figure 6.10. The five buttons on the left each launch a video that explains the basic features of PrecisionTree. Rather than repeat this information here, we urge you to watch the videos and practice the steps—as often as you like. From here on, we assume that you have done so.

Figure 6.8 PrecisionTree Welcome Screen

Figure 6.9 PrecisionTree Ribbon

Figure 6.10 PrecisionTree Quick Start Buttons

1 2 3 4

$7,500 $150,000

75%

20% 40% 30% 10%

5 6 7 8 9

10 11 12 13 14 15

18 19

17 16

A B C D E F G H I J Bidding for a government contract

Known inputs Cost of placing a bid Cost of completing project Probability of any competing bid(s)

Probability distribution of low competitor bid (if any) Assuming at least one competitor bid... Value Probability Our bid

$160,000 $170,000 $180,000

80% 40% 10%

Probability of winning Less than $160,000 Between $160,000 and $170,000 Between $170,000 and $180,000 Greater than $180,000

click the following buttons in the order shown to see videos of the steps in the analysis:

Step 2 Build Skeleton

of Tree

Step 3 Enter Values and

Probabilities

Step 4 Examine Optimal

Strategy

Step 5 Perform Sensitivity

Analysis

More PrecisionTree Examples

Read PDF Instructions

Step 1 Plan Decision Tree

Model

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2 5 6 C h a p t e r 6 D e c i s i o n M a k i n g U n d e r U n c e r t a i n t y

It is instructive to examine PrecisionTree’s decision tree for Acme’s single stage prob- lem. The completed tree appears in Figure 6.11. (See the file New Product Decisions - Single-Stage - 1c Finished.xlsx.) It is essentially a mixture of the trees in Figures 6.4 and 6.5, and it is equivalent to each of them. As in Figure 6.4, the fixed cost is entered as a negative number below the decision branch, and the net revenues are entered below the probability branches. Then PrecisionTree calculates the net value—the sum of the monetary values on any path through the tree—to the right of the correspond- ing triangle end nodes. For the folding back process, it uses these net values. Specifi- cally, the 1074 value to the right of the probability node is calculated (automatically) as (4800)(0.45) 1 (2600)(0.35) 1 (24380)(0.20).5 Then the 1074 value to the right of the decision node is calculated as the maximum of 1074 and 0.

In other words, PrecisionTree draws essentially the same tree and makes the same calculations that you could do by hand. Its advantages are that (1) it generates a nice-looking tree with all of the relevant inputs displayed, (2) it performs the folding-back calculations automatically, and (3) it permits quick sensitivity analyses on any of the model inputs. Also, you can easily identify the best decisions by following the TRUE branches. We will continue to use PrecisionTree in the rest of the chapter for trees that are considerably more complex than the one in Figure 6.11.

single-stage_decision_ tree video.

Formatting Numbers

If you are careful about formatting numbers in Excel, you might spend a lot of time formatting all of the numbers in a decision tree just the way you like them. However, there is a much quicker way in PrecisionTree. From the Settings dropdown on the PrecisionTree ribbon, select Model Settings and then the Format tab. By entering the formats you prefer here, the entire tree is formatted appropriately.

PrecisionTree Tip

Figure 6.11 Decision Tree from PrecisionTree

1 Acme single-stage new product decision

Inputs

Fixed cost �$6,000,000

$18

A B C D

Unit margin

Market Probability

0.45

0.35

0.20

Sales volume

600,000

300,000

90,000

Net revenue

$10,800,000

$5,400,000

$1,620,000

45.0%

$4,800,000

35.0%

�$600,000

�$4,380,000

Great

Fair

Awful

21 New Product Decision Continue with product?

$1,074,000

�$6,000,000 TRUE Sales volume

$1,074,000

45.0%

$10,800,000

35.0% $5,400,000

20.0% $1,620,000

FALSE

0.0%

The best decision is to continue with the product. Its EMV is $1,074,000.

Great

Fair

Awful

Yes

5 PrecisionTree refers to the “probability” nodes and branches as “chance” nodes and branches. The terms are equivalent.

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6-6 Multistage Decision problems 2 5 7

We finish this section with one important reminder discussed in the Quick Start vid- eos. PrecisionTree reserves the cells with colored font (green, red, and blue) for its special formulas, so you should not change these cells. Your entries—probabilities and monetary values—should all be in the cells with black font, and it is a good practice to cell reference these inputs whenever possible. For example, we didn’t enter 45% in cell C12; we entered a link to cell B8.

9. Use PrecisionTree’s Sensitivity Analysis tools to per- form the sensitivity analysis requested in problem 5 of the previous section. (Watch the Step 5 video in Figure 6.10 if necessary.)

Level B 10. Use PrecisionTree to solve problem 7 of the previous

section.

Problems Solutions for problems whose numbers appear within a colored box can be found in the Student Solution Files.

Level A 8. Explain in some detail how the PrecisionTree calculations

in Figure 6.11 for the Acme problem are exactly the same as those for the hand-drawn decision tree in Figure 6.6. In other words, explain exactly how PrecisionTree gets the monetary values in the colored cells in Figure 6.11.

6-6 Multistage Decision Problems Many real-world decision problems evolve through time in stages. A company first makes a decision. Then it observes an uncertain outcome that provides some infor- mation. Based on this information, the company then makes another decision. Then it observes another uncertain outcome. This process could continue for more stages, but we will limit the number of stages to two: a first decision, a first uncertain outcome, a second decision, and a second uncertain outcome. As time unfolds, payoffs are received and costs are incurred, depending on the decisions made and the uncertain outcomes observed. The objective is again to maximize EMV, but now we are searching for an EMV-maximizing strategy, often called a contingency plan, that specifies which deci- sion to make at each stage.

As you will see shortly, a contingency plan tells the company which decision to make at the first stage, but the company won’t know which decision to make at the second stage until the information from the first uncertain outcome is known. For example, if the infor- mation is bad news about a product, then the company might decide at the second stage to abandon the product, but if the news is good, the company might decide to continue with the product. This is the essence of a contingency plan: it specifies what do for each possi- ble uncertain outcome.

An important aspect of multistage decision problems is that probabilities can change through time. After you receive the information from the first-stage uncertain outcome, you might need to reassess the probabilities of future uncertain outcomes. As an example, if a new product is observed to do very poorly in a regional test market, your assessment of the probability that it will do well in a national market will almost surely decrease. Some- times this reassessment of probabilities can be done in an informal subjective manner. But whenever possible, it should be done with a probability law called Bayes’ rule. This rule provides a mathematical way of updating probabilities as new information becomes avail- able. We explain how it works in this section.

Another important aspect of multistage decision problems is the value of informa- tion. Sometimes the first-stage decision is to buy information that will help in making the second-stage decision. The question then is how much this information is worth. If you knew what the information would be, there would be no point in buying it. However, you

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2 5 8 C h a p t e r 6 D e c i s i o n M a k i n g U n d e r U n c e r t a i n t y

virtually never know what the information will be; you can only assess the probabilities of various information outcomes. In such cases, the goal is to calculate the expected value of the information—how much better you would be with the information than without it—and then compare this to the actual cost of buying the information to see whether it is worth buying. Again, we explain how it works in this section.

We now show one way the Acme decision problem can be extended to two stages. Later in this section, we examine another multistage version of Acme’s problem.

EXAMPLE

6.2 NEW PRODUCT DECISIONS AT ACME WITH TECHNOLOGICAL UNCERTAINTY

In this version of the example, we assume as before that the new product is still in the development stage. However, we now assume that there is a chance that the product will be a failure for technological reasons, such as a new drug that fails to meet FDA approval. At this point in the development process, Acme assesses the probability of technological failure to be 0.2. The $6 million fixed cost from before is now broken down into two components: $4 million for addition development costs and $2 million for fixed costs of marketing, the latter to be incurred only if the product is a technological success and the company decides to market it. The unit margin and the probability distribution of the product’s sales volume if it is marketed are the same as before. How should Acme proceed?

Objective To use a decision tree to find Acme’s EMV-maximizing strategy for this two-stage decision problem.

Where Do the Numbers Come From? The probability of technological failure might be based partly on historical data—the technological failure rate of similar prod- ucts in the past—but it is probably partly subjective, based on how the product’s development has proceeded so far. The prob- ability distribution of sales volume is a more difficult issue. When Acme makes its first decision, right now, it must look ahead to see how the market might look in the future, after the development stage, which could be quite a while from now. (The same issue is relevant in Example 6.1, although we didn’t discuss it there.) This a difficult assessment, and it is an obvious candidate for an eventual sensitivity analysis.6

Solution The reason this is a two-stage decision problem is that Acme can decide right away to stop development and abandon the prod- uct, thus saving further fixed costs of development. However, if Acme decides to continue development and the product turns out to be a technological success, a second decision on whether to market the product must still be made.

A spreadsheet model such as in Figure 6.1 for the single-stage problem could be developed to calculate the relevant EMVs, but this isn’t as easy as it sounds. A much better way is to use a decision tree, using the PrecisionTree add-in. The fin- ished tree appears in Figure 6.12. (See the file New Product Decisions - Technological Uncertainty Finished.xlsx.) The first decision is whether to continue development. If “Yes,” the fixed development cost is incurred, so it is entered on this branch. Then there is a probability node for the technological success or failure. If it’s a failure, there are no further costs, but the fixed development cost is lost. If it’s a success, Acme must decide whether to market the product. From this point, the tree is exactly like the single-stage tree, except that the fixed development cost has been incurred.

By following the TRUE branches, you can see Acme’s best strategy. The company should continue development, and if the product is a technological success, it should be marketed. The EMV, again the weighted average of all possible monetary outcomes with this strategy, is $59,200. However, this is only the expected value, or mean, of the probability distribution of monetary outcomes. You can see the full probability distribution by requesting a risk profile from PrecisionTree (through the Decision Analysis dropdown). This appears, both in graphical and tabular form, in Figure 6.13. Note that Acme has a 64% chance of incurring a net loss with this strategy, including a possible loss of $4.38 million. This doesn’t sound good. However, the company has a 36% chance of a net gain of $4.8 million and, in an expected value sense, this more than offsets the possi- ble losses.

6 We have purposely avoided the use of Bayes’ rule for now. It will be used in the next version of the Acme problem.

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Figure 6.13 Risk Profile from Best Strategy

Figure 6.12 Decision Tree with Possible Technological Failure

A 1 2 3 Inputs

Acme multistage new product decisions with technological uncertainty

Probability of technological failure Fixed development cost Fixed marketing cost Unit margin

Market Probability Sales volume Net revenue $10,800,000

$5,400,000 $1,620,000

Probability of technological success

Great Fair Awful

4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

FALSE

TRUE

�$4,000,000�$4,000,000

0.0%

20.0%

80.0%80.0%

20.0%

–4,000,000

Market product?

45.0%

$10,800,000

35.0%

$5,400,000

$1,620,000

20.0%

4,800,000

�600,000

16.0%

�4,380,000

0.0%

–4,000,000

36.0%

28.0%

FALSE

TRUE

�$2,000,000�$2,000,000

31 32 33 34

B C D E F

Continue development?

Sales volume

59,200

Yes

1,074,000

New Product Decisions

Fair

Great

Awful

1,074,000

Technological success?

59,200

The best strategy is to continue development and, if there is technological success, market the product. The EMV from this strategy is $59,200.

600,000 300,000

90,000

0.45

0.2 0.8

–$4,000,000 –$2,000,000

$18

0.35 0.20

Chart Data Optimal Path

Value �4,380,000#1

#2 #3 #4

16.0000% 20.0000% 28.0000% 36.0000%

�4,000,000 �600,000

4,800,000

Probability

Probabilities for Decision Tree ‘New Product Decisions’ Optimal Path of Entire Decision Tree

40%

35%

30%

25%

20%

15% +

10%

�5 ,0

00 ,0

Pr ob

ab ili

�4 ,0

00 ,0

�3 ,0

00 ,0

�2 ,0

00 ,0

�1 ,0

00 ,0

1, 00

0, 00

2, 00

0, 00

4, 00

0, 00

5, 00

0, 00

3, 00

0, 00

6-6 Multistage Decision problems 2 5 9

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We won’t perform any systematic sensitivity analyses on this model (we ask you to do some in the problems), but it is easy to show that the best strategy is quite sensitive to the probability of technological success. If you change this probability from 0.8 to 0.75 in cell B4, the tree automatically recalculates, with the results in Figure 6.14. With just this small change, the best decision changes completely. Now the company should discontinue development and abandon the product. There is evidently not a large enough chance of recovering the fixed development cost.

Placement of Results

When you request a risk profile or other PrecisionTree reports, they are placed in a new workbook by default. If you would rather have them placed in the same workbook as your decision tree, select Application Settings from the Utilities dropdown list on the PrecisionTree ribbon, and change the “Place Reports In” setting to Active Workbook. You only have to do this once.

PrecisionTree Tip

A B C D E F 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

New Product Decisions

FALSE

�$4,000,000

Technological success?

�194,500

25.0%

75.0% Market product?

1,074,000

FALSE

TRUE Sales volume

1,074,000

45.0%

$10,800,000

35.0%

$5,400,000

20.0%

$1,620,000

�$2,000,000

0.0%

4,800,000

0.0%

�600,000

0.0%

�4,380,000

0.0%

�4,000,000

0.0%

�4,000,000

100.0%

TRUE

Continue development?

Yes

Great

Fair

Awful

Figure 6.14 Decision Tree with Larger Probability of Failure

Modeling Issues We return to the probability distribution of eventual sales volume. The interpretation here is that at the time of the first decision, Acme has assessed what the market might look like after the development stage, which could be quite a while from now. Again, this is a difficult assessment. Acme could instead break this assessment into parts. It could first assess a probability distribution for how the general market for such products might change—up, down, or no change, for example—by the time development is completed. Then for each of these general markets, it could assess a probability distribution for the sales volume of its new product. By breaking it up in this way, Acme might be able to make a more accurate assessment, but the decision tree would be somewhat more complex. We ask you to explore this in one of the problems.

2 6 0 C h a p t e r 6 D e c i s i o n M a k i n g U n d e r U n c e r t a i n t y

The next example illustrates another possible multistage extension of the Acme deci- sion problem. This example provides an opportunity to introduce two important topics discussed earlier: Bayes’ rule for updating probabilities and the value of information.

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6-6 Multistage Decision problems 2 6 1

EXAMPLE

6.3 NEW PRODUCT DECISIONS AT ACME WITH AN OPTION TO BUY INFORMATION

Suppose now that Acme has just about finished the development process on the new product, so that fixed development costs are no longer an issue, and technological failure is no longer a possibility. The only question is whether Acme should mar- ket the product, given the uncertainty about the eventual sales volume. If the company decides to market the product, it will incur fixed marketing costs of $4 million. To keep the model simple, we now assume that there are only two possible market outcomes, good or bad. The sales volumes for these two possible outcomes are 600,000 units and 100,000 units, and Acme assesses that their probabilities are 0.4 and 0.6. However, before making the ultimate decision, Acme has the option to hire a well-respected marketing research firm for $150,000. If Acme decides to use this option, the result will be a prediction of good or bad. That is, the marketing research firm will predict that either “We think the market for this product will be good” or “We think the market for this product will be bad.” Acme has used this firm before, so it has a sense of the prediction accuracy, as indicated in Table 6.1. Each row in this table indicates the actual market outcome, and each column indicates the prediction. If the actual market is good, the prediction will be good with probability 0.8 and bad with probability 0.2. If the actual market is bad, the prediction will be bad with probability 0.7 and good with probability 0.3. What should Acme do to maximize its EMV?

actual/predicted Good Bad

Good 0.8 0.2

Bad 0.3 0.7

Table 6.1 Prediction Accuracy of Marketing Research Firm

Objective To use a decision tree to see whether the marketing research firm is worth its cost and whether the product should be marketed.

Where Do the Numbers Come From? The main question here concerns the probabilities. Acme’s assessment of the probabilities of good or bad markets, 0.4 and 0.6, would be assessed as in earlier examples, probably subjectively. The probabilities in Table 6.1 are probably based partly on historical dealings with the marketing research firm—perhaps they have been right in about 75% of their predictions, give or take a bit—and some subjectivity. In any case, all of these probabilities are prime candidates for sensitivity analysis.

Solution Acme must first decide whether to hire the marketing research firm. If it decides not to, it can then immediately decide whether to market the product. On the other hand, if it decides to hire the firm, it must then wait for the firm’s prediction. After the pre- diction is received, Acme can then make the ultimate decision on whether to market the product. However, when making this ultimate decision, Acme should definitely take the firm’s prediction into account.

A “skeleton” for the appropriate decision tree, without any of the correct probabilities or monetary val- ues, appears in Figure 6.15. For now, just focus on the structure of the tree, and how time flows from left to right. The tree is not symmetric. If Acme doesn’t hire the firm, this “No” decision branch is followed imme- diately by another decision node for whether to market the product. However, if Acme does hire the firm, this “Yes” decision branch is followed by a probability node for the firm’s prediction. Then after a good or a bad prediction, there is a decision node for whether to market the product.

Now it is time to label the tree with probabilities and monetary values. The monetary values present no problems, as will be seen shortly. However, the probabilities require a digression so that we can discuss Bayes’ rule. The problem is that the given probabilities in Table 6.1 are not the probabilities required in the tree. After we discuss Bayes’ rule, we will return to Acme’s decision problem.

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Figure 6.15 Skeleton of Decision Tree with Option to Buy Information

17 A B C D E F

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

New Product Decisions

50.0%

Bad

50.0%

Yes

50.0%

25.0%

0.0%

25.0%

0.0%

Market product?

Prediction

Sales volume

Bad

Good

Yes

Good

Yes

Good

Bad

Good

Bad

TRUE

FALSE

Hire firm?

FALSE

Market product?

TRUE

Market product?

TRUE

FALSE

TRUE

Figure 6.16 Bayesian Updating Process

Prior probabilities Information

observed Bayes’ rule Bayes’ rule

Posterior probabilities (which become priors for later information)

Later information observed

New posterior probabilities

6.6a Bayes’ Rule Bayes’ rule, named after the Reverend Thomas Bayes from the 1700s, is a formal mathe- matical mechanism for updating probabilities as new information becomes available. The general idea is simple, as illustrated in Figure 6.16. The original probabilities are called prior probabilities. Then information is observed and Bayes’ rule is used to update the prior prob- abilities to posterior probabilities. As the diagram indicates, the terms prior and posterior are relative. If later information is observed, the posterior probabilities in the middle play the role of priors. They are used, along with Bayes’ rule, to calculate new posterior probabilities.

The actual updating mechanism can be done in two ways: with frequencies (counts) or with probabilities. We (and our students) believe that the frequency approach is much easier to understand, so we will present it first. But because the probability approach is useful for spreadsheet calculations, we will present it as well.

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Bayes’ Rule: Frequency Approach Consider the following situation. During a routine physical exam, a middle-aged man named Joe tests positive for a certain disease. There were previously no indications that Joe had this disease, but the positive test sends him into a panic. He “knows” now that he has the disease. Or does he? Suppose that only 1% of all undiagnosed middle-aged men

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Figure 6.17 Frequency Approach for Updating Probabilities

10,000 middle-aged men

9,900 without disease

8910 test negative

95 test positive

100 with disease

5 test negative

95 %

10 % 90%

990 test positive

1% 99%

Joe’s chance of having the disease, given a positive test = 95/(95+990) = 8.75% Joe’s chance of having the disease, given a negative test = 5/(5+8910) = 0.056%

Joe’s chance of testing positive = (95+990)/10000 = 10.9%

have this disease. Also, suppose the test Joe took is not entirely accurate. For middle-aged men who don’t have the disease, there is a 10% chance they will test positive (the false positive rate). For middle-aged men who have the disease, there is a 5% chance they will test negative (the false negative rate). So with a positive test result, what is the probability that Joe has the disease?

A frequency approach starts with a large number, say 10,000, of middle-aged men and follows them according to the stated percentages. A diagram of this appears in Figure 6.17. The percentages on the links are the given percentages, and the number in each box is the percentage times the number in the box above it. We know that Joe’s status is in one of the two gray boxes because he tested positive. Therefore, the probability that he has the dis- ease, that is, the probability that his status is in the leftmost gray box, is 95>(95 1 990), or about 8.75%.

6-6 Multistage Decision problems 2 6 3

We suspect that you are surprised by this low posterior probability. If so, you’re not alone. Highly trained physicians, when posed with this same problem, have given widely ranging answers—almost the entire range from 0% to 100%. The problem is that we humans don’t have very good intuition about probabilities. However, there are two sim- ple reasons why Joe’s posterior probability of having the disease is as low as it is. First, not many men in his age group, only 1%, have the disease. Therefore, we tend to believe that Joe doesn’t have the disease unless we see convincing evidence to the contrary. The second reason is that this test has fairly large error rates, 10% false positives and 5% false negatives. Therefore, the evidence from the test is not entirely convincing.

What if Joe tested negative? Then using the same numbers in the diagram, his poste- rior probability of having the disease would drop quite a lot from the prior of 1%; it would be 5>(5 1 8910), or about 0.06%. We were fairly sure he didn’t have the disease before the test (because only 1% of his age group has it), and a negative test result convinces us even more that he doesn’t have the disease.

One final calculation that will become useful in Acme’s problem appears at the bot- tom of Figure 6.17. This is the probability that Joe tests positive in the first place. (In general, it is the probability of any potential piece of information.) Of the 10,000 men, 95 1 990 test positive, so this probability is (95 1 990)>10000, or about 10.9%. Most of these are false positives.

This frequency approach is quite straightforward, even though it often yields sur- prising and unintuitive results. You are asked to apply it to several other scenarios in the problems. For now, we apply it to Acme’s problem in Example 6.3. The prior probabilities

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You should be aware that three complementary probabilities are implied by the prob- abilities shown. After a good prediction, the market will be either good or bad, so the posterior probability of a bad market is 36%. Similarly, the posterior probability of a bad market after a bad prediction is 84%, and the probability of a bad prediction is 50%.

If you like this frequency approach for updating probabilities, you can keep using it. It is a perfectly acceptable way to update probabilities as information is received, and it is equivalent to a more formal use of Bayes’ rule, which we now present.

Bayes’ Rule: Probability Approach For any possible outcome O, let P(O) be the probability of O. This implicitly indicates the probability of O occurring, given all the information currently available. If we want to indicate that new information, I, is available, we write the probability as P(OuI). This is called a conditional probability. The vertical bar is read “given that,” so this is the proba- bility of O, given that we have information I.

The typical situation is that there are several outcomes such as “good market” and “bad market.” In general, denote these outcomes as O1 to On, assuming there are n pos- sibilities. Then we start with n prior probabilities, P(O1) to P(On), that sum to 1. Next, we observe new information, I, such as a market prediction, and we want the n updated posterior probabilities, P(O1uI) to P(OnuI), that sum to 1. We assume the “opposite” con- ditional probabilities, P(IuO1) to P(IuOn), are given. In Bayesian terminology, these are called likelihoods. These likelihoods are the accuracy probabilities in Table 6.1. For example, one of these is the probability of seeing a good prediction, given that the mar- ket will be good.

However, these likelihoods are not what we need in the decision tree. Because time goes from left to right, we first need the (unconditional) probabilities of possible predictions and then we need the posterior probabilities of market outcomes, given the

of good or bad markets are 40% and 60%. The probabilities for the accuracy of the pre- dictions are given in Table 6.1. From these, we can follow 1000 similar products and predictions, exactly as we did with Joe’s disease. (We use 1000 rather than 10,000 for variety. The number chosen doesn’t influence the results at all.) The diagram appears in Figure 6.18. The probability of a good market increases from 40% to 64% with a good prediction, and it decreases to 16% with a bad prediction. Also, the probability that the prediction will be good is 50%.

Figure 6.18 Frequencies for Updating Probabilities in Acme Problem

1000 new products

600 with bad market

420 with bad prediction

320 with good prediction

400 with good market

80 with bad prediction

80 %

30 % 70%

180 with good prediction

20%

40% 60%

Chance of a good market, given a good prediction = 320/(320+180) = 64% Chance of a good market, given a bad prediction = 80/(80+420) = 16%

Chance of a good prediction = (320+180)/1000 = 50%

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In words, Bayes’ rule says that the posterior is the likelihood times the prior, divided by a sum of likelihoods times priors. As a side benefit, the denominator in Bayes’ rule is also useful in multistage decision trees. It is the probability P(I) of the information outcome.

predictions (see Figure 6.15). So Bayes’ rule is a formal rule for turning these conditional probabilities around. It is given in Equation (6.1) for any i from 1 to n.

Bayes’ Rule for Two Outcomes

P(OuI) 5 P(IuO)P(O)

P(IuO)P(O) 1 P(IuNot O)P(Not O) (6.3)

This formula is important in its own right. For I to occur, it must occur along with one of the O’s. Equation (6.2) decomposes the probability of I into all of these possibilities. It is sometimes called the law of total probability.

In the special case where there are only two O’s, labeled as O and Not O, Bayes’ rule takes the following form:

6-6 Multistage Decision problems 2 6 5

Denominator of Bayes’ Rule (Law of Total Probability)

P(I) 5 P(IuO1)P(O1) 1 g 1 P(I|On)P(On) (6.2)

Bayes’ Rule

P(OiuI) 5 P(IuOi)P(Oi)

P(IuO1)P(O1) 1 g 1 P(IuOn)P(On) (6.1)

Implementing Bayes’ Rule for Acme These formulas are actually fairly easy to implement in Excel, as shown in Figure 6.19 for the Acme problem. (See the file New Product Decisions – Information Option Finished. xlsx.) It is important to be consistent in the use of rows and columns for the outcomes (the O’s) and the predictions (the I’s). If you examine Figure 6.19 closely, we have always put the Good/Bad labels for market outcomes down columns, and we have always put the Good/Bad labels for predictions across rows. You could do it in the opposite way, but you should be consistent. This allows you to copy formulas. The given probabilities, the priors and the likelihoods, are in the left section, columns B and C. The Bayes’ rule calculations are in the right section, columns G and H.

We first implement Equation (6.2) in cells G7 and H7. The formula in cell G7, a sum of likelihoods times priors, is

5SUMPRODUCT(B14:B15,$B$9:$B$10)

and this can be copied to cell H7. Next, we implement Equation (6.3) in the range G11:H12 to calculate the posteriors. Each is a likelihood times a prior, divided by one of the sums in row 7. The formula in cell G11 is

5B14*$B9/G$7 bayesian_revision video.

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and because of the careful use of relative/absolute addresses, this can be copied to the range G11:H12. (The “Sum checks” indicate which probabilities should sum to 1.) Fortunately, this procedure is perfectly general. If you always enter outcomes down columns and infor- mation across rows, the same essential formulas given here will always work. Of course, you can check that these Bayes’ rule calculations give the same results as the frequency approach in Figure 6.18.

Completing the Acme Decision Tree Now that we have the required probabilities, we can label Acme’s decision tree from Figure 6.15 with monetary values and probabilities. The completed tree appears in Figure 6.20. You should examine this tree carefully. The cost of hiring the marketing research firm is entered when the hiring takes place, in cell B28, the fixed cost of marketing the prod- uct is entered when the decision to market occurs, in cells D20, D32, and C44, and the net revenues are enter in the branches to the right as before. As for probabilities, the priors are entered in the bottom section of the tree, the part where the marketing research firm is not hired. Then the probabilities of the predictions are entered in column C (they just happen to be 50-50), and the posterior probabilities are entered in column E. Of course, all of these entries are cell references to the inputs and calculations in Figure 6.19.

As before, once all of the monetary values and probabilities are entered in the tree, PrecisionTree automatically performs all the folding-back calculations, as you can see in Figure 6.20. Then you can follow the TRUEs. In this case, the marketing research firm should be hired. If its prediction is “good,” Acme should market the product. If its prediction is “bad,” Acme should abandon the product. The EMV from this strategy is $1.63 million.

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Figure 6.19 Bayes’ Rule Calculations for Acme Problem

Acme multistage new product decisions with an option to buy information

Inputs Cost of market research

Probabilities that indicate the accuracy of the predictions

Fixed marketing cost Unit margin

Market Good

–$150,000 –$4,000,000

$18

Prior probability 0.40 0.60

0.8 0.3

Sales volume Net revenue $10,800,000

$1,800,000 600,000 100,000

5 6 7 8

10 9

11 12 13 14 15

A B C D E F G H I

Bad

Bayes’ rule calculations Probabilities of predictions

Posterior probabilities, given predictions Actual\Predicted Good

GoodActual\Predicted Good Bad

0.2 0.7

Bad 1 1

Sum check

Bad

Good

Good Bad Sum check

0.64 0.16 0.36 0.84

0.5 Bad Sum check

10.5

Making Sequential Decisions

Whenever you have a chance to make several sequential decisions and you will learn useful information between decision points, the decision you make initially depends on the decisions you plan to make in the future, and these depend on the information you will learn in the meantime. In other words, when you decide what to do initially, you should look ahead to see what your future options will be, and what your decision will be under each option. Such a contingency plan is typically superior to a myopic (short-sighted) plan that doesn’t take into account future options in the initial decision making.

Fundamental Insight

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6-6b The Value of Information In a decision-making context, information is usually bought to reduce the uncertainty about some outcome. The information always comes with a price tag, in this case $150,000, and the question is whether the information is worth this much. Alternatively, the ques- tion is how much you would be willing to pay for the information. In some situations the answer is clear. To see such a situation, change the fixed marketing cost from $4 million to $2 million to get the decision tree in Figure 6.21. Then the marketing research is worth nothing to Acme. It is not even worth $1, let alone $150,000. Do you see why? The key is that Acme should now make the same decision, market the product, regardless of the pre- diction the marketing research firm makes. Essentially, Acme will now ignore the firm’s prediction, so it makes no sense for Acme to pay for a prediction it intends to ignore.

The reason the marketing research firm’s prediction is worthless is partly because it’s inaccurate and partly because marketing continues to be better than not marketing even when the probability of a bad market is high. In other words, it would take a really bad market prediction to persuade to Acme not to market.

However, the behavior is different in Figure 6.20, where the fixed marketing cost is $4 million. Now the fixed cost more than offsets the possible gain from a good market unless the probability of a good market is fairly high—64% and 40% are high enough, but 16% isn’t. In this case, the marketing research firm should be hired for $150,000, and then Acme should market the product only if it hears a good prediction.

In this case, we know that the marketing research firm is worth its $150,000 price, but how much would Acme be willing to pay to hire the firm. This amount is called the expected value of information, or EVI, and it is given by Equation (6.4).7

7 The traditional term is EVSI, where S stands for “sample,” meaning “imperfect.” We think “sample” has too much of a statistical connotation, so we omit it.

6-6 Multistage Decision problems 2 6 7

A B C D E F 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

Good

Bad

Good

Bad

Good

Bad

Good

Bad

Yes

New Product Decisions

0.0%

32.0%

6650000

18.0%

�2350000

36.0%

1800000

64.0%

10800000

0.0%

�2350000

84.0%

1800000

0.0%

6650000

16.0%

10800000

�$150,000

–$4,000,000

50.0%

40.0%

$10,800,000 6800000

0.0%

�150000

50.0%

�150000

0.0%

�2200000

0.0%60.0%

$1,800,000

Sales volume

1630000

Prediction

3410000

Sales volume

�910000

Sales volume

1400000

–$4,000,000

Hire firm?

1400000

Market product?

3410000

Market product?

�150000

Market product?

FALSE

1630000

FALSE

TRUE

FALSE

TRUE

FALSE

–$4,000,000

The best strategy is to hire the market research firm and then market the product only if its prediction is “good.” The EMV from this strategy is $1,630,000.

Figure 6.20 Completed Decision Tree for Acme Problem

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2 6 8 C h a p t e r 6 D e c i s i o n M a k i n g U n d e r U n c e r t a i n t y

The calculation of EVI is quite easy, given the completed decision tree in Figure 6.20. The $1.63 million value in cell C28 is Acme’s net EMV after paying $150,000 to the firm. If Acme could get this information for free, its EMV would be 1.630 1 0.150, or $1.78 million. On the other hand, the bottom section of the tree shows that Acme’s EMV with no information is $1.4 million. Therefore, according to Equation (6.4), EVI is

EVI 5 1.78 2 1.4 5 $380,000

In other words, the marketing research firm could charge up to $380,000, and Acme would still be willing to hire them. You can prove this to yourself. With the fixed market- ing cost at $4 million, change the cost of the hiring the firm to $379,000. The decision tree should still have a TRUE for the hiring decision. Then change the cost to $381,000. Now there should be a FALSE for the hiring decision.

Although the calculation of EVI is straightforward once the decision tree has been created, the decision tree itself requires a lot of probability assessments and Bayes’ rule

The EVI is the most you would be willing to pay for the sample information.

Equation for EVI

EVI 5 EMV with (free) information 2 EMV without information (6.4)

Figure 6.21 Completed Tree with a Smaller Fixed Marketing Cost

17 A B C D E F

18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

New Product Decisions

50.0%

16.0% 0.0%

8650000

0.0%

�350000

40.0%

8800000

60.0%

�200000

10800000

84.0%

1800000

0.0%

0 �150000

40.0%

$10,800,000

60.0%

$1,800,000

Bad

�$2,000,000

50.0%

0.0%

�$2,000,000

Yes

�$2,000,000

64.0% 0.0%

8650000

0.0%

�350000

10800000

0.0%

36.0%

1800000

�150000

�$150,000

Prediction

Sales volume

1090000

Sales volume

3400000

Sales volume

5410000

3250000

Bad

Good

Yes

Good

Yes

Good

Bad

Good

Bad

FALSE

Hire firm?

3400000

TRUE

FALSE

Market product?

3400000

TRUE

Market product?

1090000

TRUE

Market product?

FALSE

TRUE

5410000

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6-6 Multistage Decision problems 2 6 9

To calculate EVPI for the Acme problem, forget about the marketing research firm and its possible prediction. All you have is the bottom of the tree in Figure 6.20, which uses Acme’s prior probabilities of market outcomes. So Acme’s EMV with no infor- mation is $1.4 million. To find the EMV with free perfect information, imagine that Acme will be told, truthfully, whether the market will be good or bad before the mar- keting decision has to be made. With probability 0.4, the prior probability, it will be told “good,” and with probability 0.6, it will be told “bad.” If it knows the market will be good, it should market the product because the net payoff is $10.8 million from sales minus $4 million for the fixed cost, or $6.8 million, a positive value. On the other hand, if it knows the market will be bad, it should not market the product because this would lead to a net payoff of $1.8 million from sales minus $4 million for the fixed cost, which is less than the 0 value it could get from not marketing the product. Therefore, according to Equation (6.5), EVPI is

EVPI 5 (0.4)(6.8) 1 (0.6)(0) 2 1.4 5 $1.32 million

In words, no information, regardless of its form or accuracy, could be worth more than $1.32 million to Acme. This calculation is often performed because it is easy and it provides an upper limit on the value of any information. As we saw, however, some infor- mation, such as the marketing research firm’s prediction, can be worth considerably less than $1.32 million. This is because the firm’s predictions are not perfect.

Let’s make the EVPI calculation once more, for the problem where the fixed market- ing cost is reduced to $2 million. In this case, the value of the market research firm was 0, but will EVPI also be 0? The answer is no. Referring to the bottom section of Figure 6.21, the calculation is

EVPI 5 (0.4)*Max(10.8@2,0) 1 (0.6)*Max(1.8@2,0) 2 3.4

5 (0.4)(8.8) 1 (0.6)(0)23.4 5 $0.12 million

Each “Max” in this equation indicates the best of marketing and not marketing, and the second max is 0 because the fixed marketing cost is greater than the sales revenue if the market is bad.

In this case, Acme shouldn’t pay anything for the marketing research firm’s infor- mation, but other types of information could be worth up to $0.12 million. The intuition here is the following. The marketing research firm’s information is worthless because Acme will ignore it, marketing the product regardless of the information. But perfect information is worth something because Acme will act one way, market the product, if the information is good, and it will act another way, don’t market the product, if the infor- mation is bad. Presumably, other types of imperfect information could have the same effect and hence be worth something, but they can’t be worth more than $0.12 million.

calculations. These can be difficult, depending on the type of information available. There- fore, it is sometimes useful to ask how much any information could be worth, regardless of its form or accuracy. The result is called the expected value of perfect information, or EVPI, and it is given by Equation (6.5).

The EVPI is the most you would be willing to pay for perfect information.

Equation for EVPI

EVPI 5 EMV with (free) perfect information 2 EMV without information (6.5)

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6-6c Sensitivity Analysis We have already performed one sensitivity analysis, simply by changing the fixed market- ing cost in cell B5 from $4 million to $2 million and seeing how the tree changes. You can perform any number of similar “ad hoc” sensitivity analyses in this way, provided that the entries in the tree are linked to the input section of your worksheet. But you might also like to perform a more formal sensitivity analysis with PrecisionTree’s powerful tools. We will show the results of several such sensitivity analyses, each with the fixed marketing cost set to $4 million. (We won’t provide the step-by-step details of how to perform these in Preci- sionTree, but if you need help, you can watch the Quick Start step 5 video we referenced earlier.)

The first sensitivity analysis is on the prior probability, currently 0.4, of a good mar- ket. To do this, you should make sure cell B10 contains a formula, 512B9. The reason is that as the probability in cell B9 varies, we want probability in cell B10 to vary accord- ingly. (In general, if you want to perform a sensitivity analysis on a probability, you should use formulas to guarantee that the relevant probabilities continue to sum to 1. In the file for this example, you will see that there are also formulas in cells C14 and C15 for this same purpose.)

The strategy region graph in Figure 6.22 shows how Acme’s EMVs from hiring and not hiring the firm vary as the prior probability of a good market varies from 0.2 to 0.6. Acme wants the largest EMV, which corresponds to the Yes line for small probabilities (up to about 0.48) and the No line for large probabilities. Why would Acme hire the firm only when the prior probability of a good market is small? The reason is that this is when

the Value of Information

The amount you should be willing to spend for information is the expected increase in EMV you can obtain from having the information. If the actual price of the information is less than or equal to this amount, you should purchase it; otherwise, the information is not worth its price. In addition, information that never affects your decision is worthless, and it should not be purchased at any price. Finally, the value of any information can never be greater than the value of perfect information that would eliminate all uncertainty.

Fundamental Insight

Figure 6.22 Sensitivity Analysis on Prior Probability of Good Market

2000

1500

500

1000

2500

3500

3000

Strategy Region of Decision Tree ‘New Product Decisions’ Expected Value of Node ‘Hire firm?’ (B40)

With Variation of Good (B9)

0. 15

0. 65

0. 60

0. 55

0. 50

0. 40

0. 45

0. 35

0. 30

0. 25

0. 20

Ex pe

ct ed

V al

Good (B9)

–500

Yes No

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6-6 Multistage Decision problems 2 7 1

the firm’s predictions are most helpful. On the right side of the graph, Acme is already fairly sure that the market is good, so the marketing research firm’s predictions in this case are less useful. Of course, the value of a sensitivity analysis such as this is that your intuition might not be so good. It tells you what you might not have been able to figure out on your own.

If you don’t want to use PrecisionTree’s Sensitivity Analysis, which is admittedly a bit confusing, you can use data tables that capture the TRUE/FALSE values on decision branches. One of these is illustrated in Figure 6.23. It tells exactly the same story as Figure 6.22.

Figure 6.23 Sensitivity Analysis with a Data Table 16

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

H Sensitivity of hiring decision to prior probability of Good P(Good) Hire?

0.38 0.39 0.40 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.50 0.51 0.52 0.53

TRUE TRUE Link to cell B27

TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE FALSE FALSE FALSE FALSE FALSE FALSE

I J K L M

Figure 6.24 Sensitivity Analysis for Marketing Decision after a Bad Prediction

–1000

–500

500

Strategy Region of Decision Tree ‘New Product Decisions’ Expected Value of Node ‘Market product?’ (D36)

With Variation of Good (B9)

–1500

0. 15

0. 65

0. 60

0. 55

0. 50

0. 40

0. 45

0. 35

0. 30

0. 25

0. 20

Ex pe

ct ed

V al

Good (B9)

–2000

Yes No

When you fill out PrecisionTree’s Sensitivity Analysis dialog box, you can choose a “starting node” other than the “Entire Model.” We repeated the analysis by choosing cell D36 as the starting node. (See Figure 6.20.) The idea here is that Acme has already decided to hire the firm and has then seen a bad prediction. The strategy region graph in Figure 6.24 shows the EMVs from marketing and not marketing the product, from this point on, as the prior probability of a good market varies as before.

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As you can see, Acme should not market the product after a bad prediction unless the prior probability of a good market is approximately 0.55 or higher. This makes intuitive sense.

In one final sensitivity analysis, a two-way analysis, we let the prior probability of a good market vary as before, and we let the fixed marketing cost vary from 25% below to 25% above its current value of $4 million. Also, we choose “Entire Model” as the starting node. The results are shown in Figure 6.25. The diamonds correspond to input values where Acme should hire the firm, and the triangles correspond to input values where Acme shouldn’t hire the firm. The pattern indicates that hiring is best only when the fixed marketing cost is high and/or the prior probability of a good market is low. For example, if the prior probability of a good market is 0.5, the Acme should hire the firm only if its fixed marketing cost is $4.2 mil- lion or above. As another example, if Acme’s fixed marketing cost is $3.8 million, it should hire the firm only if the prior probability of a good market is 0.4 or below.

Yes

Strategy Region for Node ‘Hire firm?’

Good (B9)

$3,000

$5,000

$4,800

$4,600

$4,400

$4,200

$4,000

$3,800

$3,600

$3,400

$3,200

Fi xe

d m

ar ke

tin g

co st

(B 5)

0. 20

0. 25

0. 30

0. 35

0. 40

0. 45

0. 50

0. 55

0. 60

Figure 6.25 Two-Way Sensitivity Analysis

One of the most important benefits of using PrecisionTree (or Excel data tables) is that once you have built the decision tree, you can quickly run any number of sensitivity analyses such as the ones shown. They often provide important insights that help you bet- ter understand the decision problem. You are asked to perform other sensitivity analyses on this example (and Example 6.2) in the problems.

12. In Example 6.2, the fixed costs are split $4 million for development and $2 million for marketing. Per- form a sensitivity analysis where the sum of these two fixed costs remains at $6 million but the split changes. Specifically, let the fixed cost of development vary from $1 million to $5 million in increments of $0.5 million. Does Acme’s best strategy change in this range? Use either a data table or PrecisionTree’s Sensitivity Analysis tools to answer this question.

Problems Solutions for problems whose numbers appear within a colored box can be found in the Student Solution Files.

Level A 11. In Example 6.2, Acme’s probability of technological suc-

cess, 0.8, is evidently large enough to make “continue development” the best decision. How low would this probability have to be to make the opposite decision best?

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6-6 Multistage Decision problems 2 7 3

13. In Example 6.2, use a two-way PrecisionTree sensitiv- ity analysis to examine the changes in both of the two previous problems simultaneously. Let the probability of technological success vary from 0.6 to 0.9 in incre- ments of 0.05, and let the fixed cost of development vary as indicated in the previous problem. Explain in a short memo exactly what the results of the sensitivity analysis imply about Acme’s best strategy.

14. In the file Bayes Rule for Disease.xlsx, explain why the probabilities in cells B9 and B10 (or those in cells C9 and C10) do not necessarily sum to 1, but why the prob- abilities in cells B9 and C9 (or those in cells B10 and C10) do necessarily sum to 1.

15. In using Bayes’ rule for the presence of a disease (see Figure 6.17 and the file Bayes Rule for Disease.xlsx), we assumed that there are only two test results, positive or negative. Suppose there is another possible test result, “maybe.” The 2 3 2 range B9:C10 in the file should now be replaced by a 2 3 3 range, B9:D10, for positive, maybe, and negative (in that order). Let the correspond- ing probabilities in row 9 be 0.85, 0.10, and 0.05, and let those in row 10 be 0.05, 0.15, and 0.80. Redo the Bayes’ rule calculations with both the frequency approach and the probability approach.

16. The finished version of the file for Example 6.3 contains two “Strategy B9” sheets. Explain what each of them indicates and how they differ.

17. Starting with the finished version of the file for Example 6.3, get back into PrecisionTree’s One-Way Sensitivity Analysis dialog box and add three more inputs. (These will be in addition to the two inputs already there, cells B9 and B5.) The first should be the unit margin in cell B6, varied from $15 to $21 in incre- ments of $1, the second should be the sales volume from a good market in cell C9, varied from 400 to 700 in increments of 50, and the third should be the probability of a good prediction, given a good market, in cell B14, varied from 0.7 to 0.9 in increments of 0.05. Make sure the analysis type is one-way, all five of the inputs are checked, and the starting node is “Entire Model.” In the “Include Results” section, check the Strategy Region, Tornado Graph, and Spider Graph options, and then run the analysis. Interpret the resulting outputs. Specifically, what do the tornado and spider graphs indicate?

18. Starting with the finished version of the file for Example 6.3, change the probabilities in cells B9 (make it smaller), B14 (make it larger), and B15 (make it smaller) in some systematic way (you can choose the details) and, for each combination, calculate the EVI. Does EVI change in the way you’d expect? Why?

19. Suppose you are a heterosexual white male and are going to be tested to see if you are HIV positive. Assume that if you are HIV positive, your test will always come back positive. Assume that if you are not HIV positive, there is still a 0.001 chance that your test will indicate that you are HIV positive. In reality, 1 of 10,000 heterosexual

white males is HIV positive. Your doctor calls and says that you have tested HIV positive. He is sorry but there is a 99.9% (1 2 0.001) chance that you have HIV. Is he correct? What is the actual probability that you are HIV positive?

Level B 20. If you examine the decision tree in Figure 6.12 (or any

other decision trees from PrecisionTree), you will see two numbers (in blue font) to the right of each end node. The bottom number is the combined monetary value from following the corresponding path through the tree. The top number is the probability that this path will be followed, given that the best strategy is used. With this in mind, explain (1) how the positive probabilities following the end nodes are calculated, (2) why some of the probabilities following the end nodes are 0, and (3) why the sum of the probabilities following the end nodes is necessarily 1.

21. In Example 6.2, a technological failure implies that the game is over—the product must be abandoned. Change the problem so that there are two levels of technological failure, each with probability 0.1. In the first level, Acme can pay a further development cost D to fix the product and make it a technological success. Then it can decide whether to market the product. In the second level, the product must be abandoned. Modify the decision tree as necessary. Then answer the following questions. a. For which values of D should Acme fix the product

and then market it, given that the first level of techno- logical failure occurs?

b. For which values of D is Acme’s best first decision still to “continue development”?

c. Explain why these two questions are asking different things and can have different answers.

22. The model in Example 6.3 has only two market out- comes, good and bad, and two corresponding pre- dictions, good and bad. Modify the decision tree by allowing three outcomes and three predictions, good, fair, and bad. You can change the inputs to the model (monetary values and probabilities) in any reasonable way you like. Then you will also have to modify the Bayes’ rule calculations. You can decide whether it is easier to modify the existing tree or start from scratch with a new tree.

23. The terms prior and posterior are relative. Assume that the test in Figure 6.17 (and in the file Bayes Rule for Disease.xlsx) has been performed, and the outcome is positive, which leads to the posterior probabilities shown. Now assume there is a second test, independent of the first, that can be used as a follow-up. Assume that its false-positive and false-negative rates are 0.02 and 0.06. a. Use the posterior probabilities as prior probabilities in

a second Bayes’ rule calculation. (Now prior means

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prior to the second test.) If Joe also tests positive in this second test, what is the posterior probability that he has the disease?

b. We assumed that the two tests are independent. Why might this not be realistic? If they are not indepen- dent, what kind of additional information would you need about the likelihoods of the test results?

24. In the OJ Simpson trial it was accepted that OJ had bat- tered his wife. OJ’s lawyer tried to negate the impact of

this information by stating that in a one-year period, only 1 out of 2500 battered women are murdered, so the fact that OJ battered his wife does not give much evidence that he was the murderer. The prosecution (foolishly!) let this go unchallenged. Here are the relevant statistics: In a typical year 6.25 million women are battered, 2500 are battered and murdered, and 2250 of the women who were battered and murdered were killed by the batterer. How should the prosecution have refuted the defense’s argument?

6-7 The Role of Risk Aversion Rational decision makers are sometimes willing to violate the EMV maximization cri- terion when large amounts of money are at stake. These decision makers are willing to sacrifice some EMV to reduce risk. Are you ever willing to do so personally? Consider the following scenarios.

• You have a chance to enter a lottery where you will win $100,000 with probability 0.1 or win nothing with probability 0.9. Alternatively, you can receive $5000 for certain. Would you take the certain $5000, even though the EMV of the lottery is $10,000? Or change the $100,000 to $1,000,000 and the $5000 to $50,000 and ask yourself whether you’d prefer the sure $50,000.

• You can buy collision insurance on your expensive new car or not buy it. The insurance costs a certain premium and carries some deductible provision. If you decide to pay the premium, then you are essentially paying a certain amount to avoid a gamble: the possi- bility of wrecking your car and not having it insured. You can be sure that the premium is greater than the expected cost of damage; otherwise, the insurance company would not stay in business. Therefore, from an EMV standpoint you should not purchase the insurance. But would you drive without this type of insurance?

These examples, the second of which is certainly realistic, illustrate situations where rational people do not behave as EMV maximizers. Then how do they act? This ques- tion has been studied extensively by many researchers, both mathematically and behavior- ally. Although there is still not perfect agreement, most researchers believe that if certain basic behavioral assumptions hold, people are expected utility maximizers—that is, they choose the alternative with the largest expected utility. Although we will not go deeply into the subject of expected utility maximization, the discussion in this section presents the main ideas.

risk aversion

When large amounts of money are at stake, most of us are risk averse, at least to some extent. We are willing to sacrifice some EMV to avoid risk. The exact way this is done, using utility functions and expected utility, can be difficult to imple- ment in real situations, but the idea is simple. If you are an EMV maximizer, you are indifferent between a gamble with a given EMV and a sure dollar amount equal to the EMV of the gamble. However, if you are risk averse, you prefer the sure dollar amount to the gamble. That is, you are willing to accept a sure dollar amount that is somewhat less than the EMV of the gamble, just to avoid risk. The more EMV you are willing to give up, the more risk averse you are.

Fundamental Insight

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6-7 the role of risk aversion 2 7 5

There are two aspects of implementing expected utility maximization in a real deci- sion analysis. First, an individual’s (or company’s) utility function must be assessed. This is a time-consuming task that typically involves many trade-offs. It is usually carried out by experts in the field, and we do not discuss the details of the process here. Second, the resulting utility function is used to find the best decision. This second step is relatively straightforward. You substitute utility values for monetary values in the decision tree and then fold back as usual. That is, you calculate expected utilities at probability branches and take maximums of expected utilities at decision branches. We will look at a numerical example later in this section.

6-7b Exponential Utility Utility assessment is not easy. Even in the best of circumstances, when a trained consultant attempts to assess the utility function of a single person, the process requires the person to make a series of choices between hypothetical alternatives involving uncertain outcomes. Unless the person has some training in probability, these choices will probably be difficult to understand, let alone make, and it is unlikely that the person will answer consistently as the questioning proceeds. The process is even more difficult when a company’s utility function is being assessed. Because different company executives typically have different attitudes toward risk, it can be difficult for these people to reach a consensus on a common utility function.

For these reasons, classes of ready-made utility functions are available. One important class is called exponential utility and has been used in many financial investment deci- sions. An exponential utility function has only one adjustable numerical parameter, called the risk tolerance, and there are straightforward ways to discover an appropriate value of this parameter for a particular individual or company. So the advantage of using an expo- nential utility function is that it is relatively easy to assess. The drawback is that exponen- tial utility functions do not capture all types of attitudes toward risk. Nevertheless, their ease of use has made them popular.

6-7a Utility Functions We begin by discussing an individual’s utility function. This is a mathematical function that transforms monetary values—payoffs and costs—into utility values. Essentially, an individual’s utility function specifies the individual’s preferences for various monetary payoffs and costs and, in doing so, it automatically encodes the individual’s attitudes toward risk. Most individuals are risk averse. Intuitively, this means they are willing to sacrifice some EMV to avoid risky gambles. In terms of the utility function, this means that every extra dollar of payoff is worth slightly less than the previous dollar, and every extra dollar of cost is considered slightly more costly (in terms of utility) than the previous dollar. The resulting utility functions are shaped as in Figure 6.26. Mathematically, these functions are said to be increasing and concave. The increasing part means that they go uphill—everyone prefers more money to less money. The concave part means that they increase at a decreasing rate. This is the risk-averse behavior.

Figure 6.26 Risk-Averse Utility Function

–7 –6 –5 –4 –3 –2 –1 $5,000

Utility values (vertical axis) for $ amounts

(horizontal axis)

1 2

–$5,000 $10,000–$10,000 0

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To assess a person’s (or company’s) exponential utility function, only one number, the value of R, needs to be assessed. There are two tips for doing this. First, it has been shown that the risk tolerance is approximately equal to the dollar amount R such that the decision maker is indifferent between the following two options:

• Option 1: Receive no payoff at all. • Option 2: Receive a payoff of R dollars or incur a cost of R/2 dollars, depending on the

flip of a fair coin.

For example, if you are indifferent between a bet where you win $1000 or lose $500, with probability 0.5 each, and not betting at all, your R is approximately $1000. From this crite- rion it certainly makes intuitive sense that a wealthier person (or company) ought to have a larger value of R. This has been found in practice.

A second tip for finding R is based on empirical evidence found by Ronald Howard, a prominent decision analyst. Through his consulting experience with large companies, he discovered tentative relationships between risk tolerance and several financial variables: net sales, net income, and equity. [See Howard (1988).] Specifically, he found that R was approximately 6.4% of net sales, 124% of net income, and 15.7% of equity for the compa- nies he studied. For example, according to this prescription, a company with net sales of $30 million should have a risk tolerance of approximately $1.92 million. Howard admits that these percentages are only guidelines. However, they do indicate that larger and more profitable companies tend to have larger values of R, which means that they are more will- ing to take risks involving large dollar amounts.

We illustrate the use of the expected utility criterion, and exponential utility in partic- ular, in the following version of the Acme decision problem.

Finding the appropriate risk tolerance value for any company or individual is not necessarily easy, but it is easier than assessing an entire utility function.

An exponential utility function has the following form:

Here x is a monetary value (a payoff if positive, a cost if negative), U(x) is the utility of this value, and R 7 0 is the risk tolerance. As the name suggests, the risk tolerance measures how much risk the decision maker will accept. The larger the value of R, the less risk averse the decision maker is. That is, a person with a large value of R is more willing to take risks than a person with a small value of R. In the limit, a person with an extremely large value of R is an EMV maximizer.

Exponential utility

U(x) 5 1 2 e2x>R (6.6)

The risk tolerance for an exponential utility function is a single number that specifies an individual’s aversion to risk. The higher the risk tolerance, the less risk averse the individual is.

EXAMPLE

6.4 NEW PRODUCT DECISIONS WITH RISK AVERSION This example is the same as Acme’s single-stage decision problem in Example 6.1, but we now assume that Acme is risk averse and that it assesses its utility function as exponential with risk tolerance $5 million. How does this risk aversion affect its decision-making progress?

Objective To see how risk aversion affects Acme’s decision-making process.

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Where Do the Numbers Come From? The only new number is the risk tolerance. According to the earlier discussion, this value implies that Acme is indifferent between entering a gamble where it could win $5 million or lose $2.5 million on the flip of a fair coin, or not enter the gamble at all. Because the risk tolerance is a difficult number to assess, it is a good candidate for a sensitivity analysis.

Solution Starting with the decision tree from the previous example (see Figure 6.11), there are only two changes to make. First, it is use- ful to add the risk tolerance as a new input, as shown in cell B12 of Figure 6.27. Second, you must tell PrecisionTree that you want to use expected utility. To do so, select Model Settings from the PrecisionTree Settings dropdown list, select the Utility Function tab, and fill it in as shown in Figure 6.28. The most important part is to check the “Use Utility Function” option. Then you can choose any of three display options. If you choose Expected Value, you will see EMVs on the tree (in the colored cells), but expected utility will actually be maximized in the background. This is useful if you want to see how much EMV you are sacrificing by being risk-averse. Alternatively, if you choose Expected Utility, as is done here, you will see utility values rather than monetary values. The Certainty Equivalent option will be discussed shortly.

Figure 6.27 Risk Tolerance as an Extra Input 1 Acme single-stage new product decision with risk aversion

Inputs –$6,000,000

$18

0.45 600000 $10,800,000 $5,400,000 $1,620,000

300000 90000

0.35 0.20

$5,000,000

Probability Sales volume Net revenue

Unit margin Fixed cost

Market Great Fair Awful

Risk tolerance

2 3 4 5 6 7 8 9

10 11 12

A B C D

Figure 6.28 PrecisionTree Utility Function Dialog Box

The resulting tree that displays expected utilities appears in Figure 6.29. If you compare this tree to the EMV tree in Figure 6.11, you will see that Acme’s best strategy has been reversed. As an EMV maximizer, Acme should market the product. As a risk-averse expected utility maximizer, it should not market the product. The expected utility from marketing the product is negative, 20.047, and the utility from not marketing the product is 0. (We formatted these as decimal numbers in the tree.

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They are not expressed in dollars.) Neither of these expected utilities is meaningful in an absolute sense; only their relative values are important. The larger corresponds to the best decision.

Figure 6.29 Decision Tree Displaying Expected Utilities

FALSE –$6,000,000

45.0% 0.0%

0.0%35.0%

20.0%

$5,400,000

$1,620,000

$10,800,000 0.617

–0.127

–1.401 $0

13 14

17 18

A B C

Sales volume –0.047

100.0% 0.000

New Product Decision

Yes

0.000

TRUE 0

Continue with product?

Great

Awful

Fair

Calculations with utilities

The PrecisionTree utility calculations are not as mysterious as they might seem. PrecisionTree takes the (blue) monetary values that originally follow the end nodes and transforms them to utility values by using Equation (6.6). Then it folds back in the usual way, except that it uses utility values, not monetary values, in the folding-back calculations. This isn’t just PrecisionTree’s way of doing it; it is the way any decision tree software should handle utilities.

PrecisionTree Tip

6-7c Certainty Equivalents The reversal in Acme’s decision can be understood better by looking at certainty equiv- alents. For a risk-averse person, the certainty equivalent of a gamble is the sure dol- lar amount the person would accept to avoid the gamble. In other words, the person is indifferent between taking this sure amount and taking the gamble. By selecting Certainty Equivalent in Figure 6.28, you can see the certainty equivalents in the tree, as shown in Figure 6.30. (We formatted the certainty equivalents as currencies because that’s what they really are.)

Recall from Figure 6.11 that the EMV for marketing the product is $1.074 million. However, this is not how the risk-averse Acme sees the decision to market the product. Acme now sees this gamble as equivalent to a loss of approximately $230,000. In other words, Acme would pay up to $230,000 to avoid this gamble. Of course, it doesn’t have to pay anything. The company can simply decide to abandon the product and receive a sure payoff of $0; hence, TRUE on the bottom branch indicates the best decision.

Again, risk aversion is all about giving up some EMV to avoid a gamble. In this case, Acme is able to obtain an EMV of $1.074 million by marketing the product, but the com- pany is willing to trade this for a sure $0 payoff because of its aversion to risk.

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6-7d Is Expected Utility Maximization Used? The previous discussion indicates that expected utility maximization is a somewhat involved task. The question, then, is whether the effort is justified. Theoretically, expected utility maximization might be interesting to researchers, but is it really used in the business world? The answer appears to be: not very often. For example, one article on the practice of decision making [see Kirkwood (1992)] quotes Ronald Howard—the same person we quoted previously—as having found risk aversion to be of practical concern in only 5% to 10% of business decision analyses. This same article quotes the president of a Fortune 500 company as saying, “Most of the decisions we analyze are for a few million dollars. It is adequate to use expected value (EMV) for these.”

What happens if Acme is less risk-averse, with a larger risk tolerance? You can check, for example, that if the company’s risk tolerance doubles from $5 million to $10 million, the best decision reverts back to the original “market the product” decision, and the cer- tainty equivalent of this decision increases to about $408,000. That is, Acme would be willing to accept a sure $408,000 to avoid a gamble with EMV $1.074 million. Only when the company’s risk tolerance becomes huge will the certainty equivalent be near the origi- nal EMV of $1.074 million.

A B C

FALSE –$6,000,000

45.0% 0.0%

0.0%35.0%

20.0%

$5,400,000

$1,620,000

$10,800,000 $4,800,000

–$600,000

–$4,380,000 $0

Sales volume –$230,508

100.0% $0

New Product Decision

Yes

TRUE 0

Continue with product?

Great

Awful

Fair

Figure 6.30 Decision Tree Displaying Certainty Equivalents

continuing with the product is well above 0. Using this same risk tolerance, experiment with the sales volume from a great market in cell C8 to see approximately how large it has to be for Acme to prefer the “continue with product” decision. (One way to do this is with a data table, using the TRUE/FALSE value in cell B14 as the single output.) With this sales volume, what is the certainty equivalent of the “continue with product” decision? How does it compare to the decision’s EMV? Explain the difference between the two.

Level B 28. Starting with the finished version of Example 6.2,

change the decision criterion to “maximize expected utility,” using an exponential utility function with risk tolerance $5 million. Display certainty equivalents on the tree.

Problems Level A 25. Explain what it means in general when we say a

risk-averse decision maker is willing to give up some EMV to avoid risk? How is this apparent in certainty equivalents of gambles?

26. Using the finished version of the file for Example 6.4, use a data table to perform a sensitivity analysis on the risk tolerance. Specifically, let the risk tolerance in cell B12 vary from $5 million to $20 million and keep track of two outputs in the data table, the TRUE/FALSE value in cell B14 and the certainty equivalent in cell C15. Interpret the results.

27. You saw in Example 6.4 how Acme prefers to aban- don the product when the risk tolerance in cell B12 is $5 million. This is despite the fact that the EMV from

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a. Keep doubling the risk tolerance until the company’s best strategy is the same as with the EMV criterion— continue with development and then market if success- ful. Comment on the implications of this.

b. With this final risk tolerance, explain exactly what the certainty equivalents in cells B31, C27, D23, and E17 (that is, those to the right of the various nodes) really mean. You might phrase these explanations something like, “If Acme were at the point where …, they would be willing to trade … for ….”

29. Starting with the finished version of Example 6.3, change the decision criterion to “maximize expected utility,” using an exponential utility function with risk tolerance $5 million. Display certainty equivalents on the tree. Is the company’s best strategy the same as with the EMV criterion? What is the EVI? (Hint: EVI is still the most Acme would be willing to pay in dollars for the marketing research firm’s predictions, and it can be calculated from certainty equivalents.)

6-8 Conclusion In this chapter we have discussed methods that can be used in decision-making problems where uncertainty is a key element. Perhaps the most important skill you can gain from this chapter is the ability to approach decision problems with uncertainty in a systematic manner. This systematic approach requires you to list all possible decisions or strategies, list all possible uncertain outcomes, assess the probabilities of these outcomes (possibly with the aid of Bayes’ rule), calculate all necessary monetary values, and finally perform the necessary calculations to obtain the best decision. If large dollar amounts are at stake, you might also need to perform a utility analysis, where the decision maker’s attitudes toward risk are taken into account. Once the basic analysis has been completed, using best guesses for the various parameters of the problem, you should perform a sensi- tivity analysis to see whether the best decision continues to be best within a range of input parameters.

Summary of Key Terms TERM EXPLANATION EXCEL PAGES EQUATION

expected monetary value (eMV)

The weighted average of the possible payoffs from a decision, weighted by their probabilities

228

eMV criterion Choose the decision with the maximum EMV 228

Decision tree A graphical device for illustrating all of the aspects of the decision problem and for finding the optimal decision (or decision strategy)

230

Folding-back procedure Calculation method for decision tree; starting at the right, take EMVs at probability nodes, maxi- mums of EMVs at decision nodes

231

precisiontree Excel add-in developed by Palisade for building and analyzing decision trees

Has its own ribbon

236

Contingency plan A decision strategy where later decisions depend on earlier decisions and outcomes observed in the meantime

240

risk profile Chart that represents the probability distribution of monetary outcomes for any decision

242

Bayes’ rule Formula for updating probabilities as new infor- mation becomes available; prior probabilities are transformed into posterior probabilities

248 6.1

Law of total probability The denominator in Bayes’ rule, for calculating the (unconditional) probability of an information outcome

248 6.2

expected value of information (eVI)

The most the (imperfect) information (such as the results of a test market) would be worth

252 6.4

expected value of perfect information (eVpI)

The most perfect information on some uncertain outcome would be worth; represents an upper bound on any EVI

252 6.5

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TERM EXPLANATION EXCEL PAGES EQUATION

Strategy region graph Useful for seeing how the optimal decision changes as selected inputs vary

PrecisionTree 254

tornado and spider graphs Useful for seeing which inputs affect a selected EMV the most

PrecisionTree 256

expected utility maximization Choosing the decision that maximizes the expected utility; typically sacrifices EMV to avoid risk when large monetary amounts are at stake

258

Utility function A mathematical function that encodes an individu- al’s (or company’s) attitudes toward risk

258

exponential utility function, risk tolerance

A popular class of utility functions, where only a single parameter, the risk tolerance, has to be specified

259 6.6

Certainty equivalent The sure dollar value equivalent to the expected utility of a gamble

262

Why will these two probabilities not appear on the decision tree? Which probabilities will be on the deci- sion tree?

C.5. Your company has signed a contract with a good customer to ship the customer an order no later than 20 days from now. The contract indicates that the customer will accept the order even if it is late, but instead of paying the full price of $10,000, it will be allowed to pay 10% less, $9000, due to lateness. You estimate that it will take anywhere from 17 to 22 days to ship the order, and each of these is equally likely. You believe you are in good shape, reasoning that the expected days to ship is the average of 17 through 22, or 19.5 days. Because this is less than 20, you will get your full $10,000. What is wrong with your reasoning?

C.6. You must make one of two decisions, each with pos- sible gains and possible losses. One of these decisions is much riskier than the other, having much larger pos- sible gains but also much larger possible losses, and it has a larger EMV than the safer decision. Because you are risk averse and the monetary values are large rela- tive to your wealth, you base your decision on expected utility, and it indicates that you should make the safer decision. It also indicates that the certainty equivalent for the risky decision is $210,000, whereas its EMV is $540,000. What do these two numbers mean? What do you know about the certainty equivalent of the safer decision?

C.7. A potentially huge hurricane is forming in the Caribbean, and there is some chance that it might make a direct hit on Hilton Head Island, South Carolina, where you are in charge of emergency preparedness. You have made plans for evacuating everyone from the island, but such an evacuation is obviously costly and upsetting for all involved, so the decision to evac- uate shouldn’t be made lightly. Discuss how you would

Problems Solutions for problems whose numbers appear within a colored box can be found in the Student Solution Files.

Conceptual Questions C.1. Your company needs to make an important decision that

involves large monetary consequences. You have listed all of the possible outcomes and the monetary payoffs and costs from all outcomes and all potential decisions. You want to use the EMV criterion, but you realize that this requires probabilities and you see no way to find the required probabilities. What can you do?

C.2. If your company makes a particular decision in the face of uncertainty, you estimate that it will either gain $10,000, gain $1000, or lose $5000, with probabilities 0.40, 0.30, and 0.30, respectively. You (correctly) calcu- late the EMV as $2800. However, you distrust the use of this EMV for decision-making purposes. After all, you reason that you will never receive $2800; you will receive $10,000, $1000, or lose $5000. Discuss this reasoning.

C.3. In the previous question, suppose you have the option of receiving a check for $2700 instead of making the risky decision described. Would you make the risky decision, where you could lose $5000, or would you take the sure $2700? What would influence your decision?

C.4. In a classic oil-drilling example, you are trying to decide whether to drill for oil on a field that might or might not contain any oil. Before making this deci- sion, you have the option of hiring a geologist to per- form some seismic tests and then predict whether there is any oil or not. You assess that if there is actually oil, the geologist will predict there is oil with probability 0.85. You also assess that if there is no oil, the geolo- gist will predict there is no oil with probability 0.90.

Key Terms (continued)

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make such a decision. Is EMV a relevant concept in this situation? How would you evaluate the consequences of uncertain outcomes?

C.8. It seems obvious that if you can purchase information before making an ultimate decision, this information should generally be worth something, but explain exactly why (and when) it is sometimes worth nothing.

C.9. Insurance companies wouldn’t exist unless customers were willing to pay the price of the insurance and the insurance companies were making a profit. So explain how insurance is a win-win proposition for customers and the company.

C.10. You often hear about the trade-off between risk and reward. Is this trade-off part of decision making under uncertainty when the decision maker uses the EMV criterion? For example, how does this work in invest- ment decisions?

C.11. Can you ever use the material in this chapter to help you make your own real-life decisions? Consider the following. You are about to take an important and diffi- cult exam in one of your MBA courses, and you see an opportunity to cheat. Obviously, from an ethical point of view, you shouldn’t cheat, but from a purely mone- tary point of view, could it also be the wrong decision? To model this, consider the long-term monetary conse- quences of all possible outcomes.

Level A 30. The SweetTooth Candy Company knows it will need

10 tons of sugar six months from now to implement its production plans. The company has essentially two options for acquiring the needed sugar. It can either buy the sugar at the going market price when it is needed, six months from now, or it can buy a futures contract now. The contract guarantees delivery of the sugar in six months but the cost of purchasing it will be based on today’s market price. Assume that pos- sible sugar futures contracts available for purchase are for five tons or ten tons only. No futures con- tracts can be purchased or sold in the intervening months. Thus, SweetTooth’s possible decisions are to (1) purchase a futures contract for ten tons of sugar now, (2) purchase a futures contract for five tons of sugar now and purchase five tons of sugar in six months, or (3) purchase all ten tons of needed sugar in six months. The price of sugar bought now for delivery in six months is $0.0851 per pound. The transaction costs for five-ton and ten-ton futures contracts are $65 and $110, respec- tively. Finally, the company has assessed the probability distribution for the possible prices of sugar six months from now (in dollars per pound). The file P06_30.xlsx contains these possible prices and their corresponding probabilities. a. Identify the decision that minimizes SweetTooth’s

expected cost of meeting its sugar demand.

b. Perform a sensitivity analysis on the optimal decision, letting each of the three currency inputs vary one at a time plus or minus 25% from its base value, and sum- marize your findings. Which of the inputs appears to have the largest effect on the best decision?

31. Carlisle Tire and Rubber, Inc., is considering expanding production to meet potential increases in the demand for one of its tire products. Carlisle’s alternatives are to construct a new plant, expand the existing plant, or do nothing in the short run. The market for this particu- lar tire product may expand, remain stable, or contract. Carlisle’s marketing department estimates the probabili- ties of these market outcomes to be 0.25, 0.35, and 0.40, respectively. The file P06_31.xlsx contains Carlisle’s payoffs and costs for the various combinations of deci- sions and outcomes. a. Identify the strategy that maximizes this tire manufac-

turer’s expected profit. b. Perform a sensitivity analysis on the optimal decision,

letting each of the monetary inputs vary one at a time plus or minus 10% from its base value, and summa- rize your findings. Which of the inputs appears to have the largest effect on the best solution?

32. A local energy provider offers a landowner $180,000 for the exploration rights to natural gas on a certain site and the option for future development. This option, if exercised, is worth an additional $1,800,000 to the landowner, but this will occur only if natural gas is dis- covered during the exploration phase. The landowner, believing that the energy company’s interest in the site is a good indication that gas is present, is tempted to develop the field herself. To do so, she must contract with local experts in natural gas exploration and devel- opment. The initial cost for such a contract is $300,000, which is lost forever if no gas is found on the site. If gas is discovered, however, the landowner expects to earn a net profit of $6,000,000. The landowner estimates the probability of finding gas on this site to be 60%. a. Identify the strategy that maximizes the landowner’s

expected net earnings from this opportunity. b. Perform a sensitivity analysis on the optimal decision,

letting each of the inputs vary one at a time plus or minus 25% from its base value, and summarize your findings. Which of the inputs appears to have the larg- est effect on the best solution?

33. Techware Incorporated is considering the introduction of two new software products to the market. The company has four options regarding these products: introduce nei- ther product, introduce product 1 only, introduce product 2 only, or introduce both products. Research and devel- opment costs for products 1 and 2 are $180,000 and $150,000, respectively. Note that the first option entails no costs because research and development efforts have not yet begun. The success of these software products depends on the national economy in the coming year. The company’s revenues, depending on its decision and the

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state of the economy, are given in the file P06_33.xlsx. The probabilities of a strong, fair, or weak economy in the coming year are assessed to be 0.30, 0.50, and 0.20, respectively. a. Identify the strategy that maximizes Techware’s

expected net revenue. b. Perform a sensitivity analysis on the optimal decision,

letting each of the inputs vary one at a time plus or minus 25% from its base value, and summarize your findings. Which of the inputs appears to have the larg- est effect on the best solution?

34. An investor with $10,000 available to invest has the fol- lowing options: (1) he can invest in a risk-free savings account with a guaranteed 3% annual rate of return; (2) he can invest in a fairly safe stock, where the pos- sible annual rates of return are 6%, 8%, or 10%; or (3) he can invest in a more risky stock, where the pos- sible annual rates of return are 1%, 9%, or 17%. The investor can place all of his available funds in any one of these options, or he can split his $10,000 into two $5000 investments in any two of these options. The joint prob- ability distribution of the possible return rates for the two stocks is given in the file P06_34.xlsx. a. Identify the strategy that maximizes the investor’s

expected one-year earnings. b. Perform a sensitivity analysis on the optimal decision,

letting the amount available to invest and the risk-free return both vary, one at a time, plus or minus 100% from their base values, and summarize your findings.

35. A buyer for a large department store chain must place orders with an athletic shoe manufacturer six months prior to the time the shoes will be sold in the department stores. The buyer must decide on November 1 how many pairs of the manufacturer’s new- est model of tennis shoes to order for sale during the coming summer season. Assume that each pair of this new brand of tennis shoes costs the department store chain $45 per pair. Furthermore, assume that each pair of these shoes can then be sold to the chain’s custom- ers for $70 per pair. Any pairs of these shoes remaining unsold at the end of the summer season will be sold in a closeout sale next fall for $35 each. The probability distribution of consumer demand for these tennis shoes during the coming summer season has been assessed by market research specialists and is provided in the file P06_35.xlsx. Finally, assume that the department store chain must purchase these tennis shoes from the manu- facturer in lots of 100 pairs. a. Identify the strategy that maximizes the department

store chain’s expected profit earned by purchasing and subsequently selling pairs of the new tennis shoes. Is a decision tree really necessary? If so, what does it add to the analysis? If not, why not?

b. Perform a sensitivity analysis on the optimal decision, letting the three monetary inputs vary one at a time over reasonable ranges, and summarize your findings.

Which of the inputs appears to have the largest effect on the best solution?

36. Two construction companies are bidding against one another for the right to construct a new community cen- ter building. The first construction company, Fine Line Homes, believes that its competitor, Buffalo Valley Construction, will place a bid for this project according to the distribution shown in the file P06_36.xlsx. Fur- thermore, Fine Line Homes estimates that it will cost $160,000 for its own company to construct this build- ing. Given its fine reputation and long-standing service within the local community, Fine Line Homes believes that it will likely be awarded the project in the event that it and Buffalo Valley Construction submit exactly the same bids. Find the bid that maximizes Fine Line’s expected profit. Is a decision tree really necessary? If so, what does it add to the analysis? If not, why not?

37. You have sued your employer for damages suffered when you recently slipped and fell on an icy surface that should have been treated by your company’s physical plant department. Your injury was sufficiently serious that you, in consultation with your attorney, decided to sue your company for $500,000. Your company’s insur- ance provider has offered to settle this suit with you out of court. If you decide to reject the settlement and go to court, your attorney is confident that you will win the case but is uncertain about the amount the court will award you in damages. He has provided his assessment of the probability distribution of the court’s award to you in the file P06_37.xlsx. In addition, there are extra legal fees of $10,000 you will have to pay if you go to court. Let S be the insurance provider’s proposed out-of-court settlement (in dollars). For which values of S will you decide to accept the settlement? For which values of S will you choose to take your chances in court? Assume that your goal is to maximize the expected net payoff from this litigation.

38. Consider a population of 2000 people, 800 of whom are women. Assume that 300 of the women in this popula- tion earn at least $60,000 per year, and 200 of the men earn at least $60,000 per year. a. What is the probability that a randomly selected per-

son from this population earns less than $60,000 per year?

b. If a randomly selected person is observed to earn less than $60,000 per year, what is the probability that this person is a man?

c. If a randomly selected person is observed to earn at least $60,000 per year, what is the probability that this person is a woman?

39. Yearly automobile inspections are required for res- idents of the state of Pennsylvania. Suppose that 18% of all inspected cars in Pennsylvania have problems that need to be corrected. Unfortunately, Pennsylvania state inspections fail to detect these problems 12% of the time. On the other hand, assume that an inspection never

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detects a problem when there is no problem. Consider a car that is inspected and is found to be free of problems. What is the probability that there is indeed something wrong that the inspection has failed to uncover?

40. Referring to the landowner’s decision problem in Problem 32, suppose now that, at a cost of $90,000, the landowner can request a soundings test on the site where natural gas is believed to be present. The company that conducts the soundings concedes that the test has a 30% false-negative rate (it indicates no gas when gas is pres- ent) and a 10% false-positive rate (it indicates gas when no gas is present). a. If the landowner pays for the soundings test and the

test indicates that gas is present, what is the landown- er’s revised probability that the site contains gas?

b. If the landowner pays for the soundings test and the test indicates that gas is not present, what is the land- owner’s revised probability that there is no gas on the site?

c. Should the landowner request the soundings test at a cost of $90,000? Why or why not? If not, at what price (if any) would the landowner be willing to pay for the soundings test?

41. A customer has approached a bank for a $100,000 one-year loan at an 8% interest rate. If the bank does not approve this loan application, the $100,000 will be invested in bonds that earn a 6% annual return. Without additional information, the bank believes that there is a 4% chance that this customer will default on the loan, assuming that the loan is approved. If the customer defaults on the loan, the bank will lose $100,000. At a cost of $1000, the bank can thoroughly investigate the customer’s credit record and supply a favorable or unfavorable recommendation. Past experience indicates that the probability of a favorable recommendation for a customer who will eventually not default is 0.80, and the chance of a favorable recommendation for a customer who will eventually default is 0.15. a. Use a decision tree to find the strategy the bank should

follow to maximize its expected profit. b. Calculate and interpret the expected value of informa-

tion (EVI) for this decision problem. c. Calculate and interpret the expected value of perfect

information (EVPI) for this decision problem. d. How sensitive are the results to the accuracy of the

credit record recommendations? Are there any “rea- sonable” values of the error probabilities that change the optimal strategy?

42. A company is considering whether to market a new product. Assume, for simplicity, that if this product is marketed, there are only two possible outcomes: success or failure. The company assesses that the probabilities of these two outcomes are p and 12p, respectively. If the product is marketed and it proves to be a failure, the company will have a net loss of $450,000. If the product is marketed and it proves to be a success, the

company will have a net gain of $750,000. If the com- pany decides not to market the product, there is no gain or loss. The company can first survey prospective buyers of this new product. The results of the consumer survey can be classified as favorable, neutral, or unfavorable. Based on similar surveys for previous products, the company assesses the probabilities of favorable, neu- tral, and unfavorable survey results to be 0.6, 0.3, and 0.1 for a product that will eventually be a success, and it assesses these probabilities to be 0.1, 0.2, and 0.7 for a product that will eventually be a failure. The total cost of administering this survey is C dollars. a. Let p 5 0.4. For which values of C, if any, would this

company choose to conduct the survey? b. Let p 5 0.4. What is the largest amount this company

would be willing to pay for perfect information about the potential success or failure of the new product?

c. Let p 5 0.5 and C 5 $15,000. Find the strategy that maximizes the company’s expected net earnings. Does the optimal strategy involve conducting the sur- vey? Explain why or why not.

43. The U.S. government wants to determine whether immi- grants should be tested for a contagious disease, and it is planning to base this decision on financial consider- ations. Assume that each immigrant who is allowed to enter the United States and has the disease costs the country $100,000. Also, assume that each immigrant who is allowed to enter the United States and does not have the disease will contribute $10,000 to the national economy. Finally, assume that x% of all potential immi- grants have the disease. The U.S. government can choose to admit all immigrants, admit no immigrants, or test immigrants for the disease before determining whether they should be admitted. It costs T dollars to test a person for the disease, and the test result is either positive or negative. A person who does not have the dis- ease always tests negative. However, 10% of all people who do have the disease test negative. The government’s goal is to maximize the expected net financial benefits per potential immigrant. a. If x 5 5, what is the largest value of T at which the

U.S. government will choose to test potential immi- grants for the disease?

b. How does your answer to the question in part a change if x increases to 10?

c. If x 5 5 and T 5 $500, what is the government’s opti- mal strategy?

44. The senior executives of an oil company are trying to decide whether to drill for oil in a particular field in the Gulf of Mexico. It costs the company $600,000 to drill in the selected field. Company executives believe that if oil is found in this field its estimated value will be $3,400,000. At present, this oil company believes that there is a 45% chance that the selected field actu- ally contains oil. Before drilling, the company can hire a geologist at a cost of $55,000 to perform seismographic

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tests. Based on similar tests in other fields, the tests have a 25% false negative rate (no oil predicted when oil is present) and a 15% false positive rate (oil predicted when no oil is present). a. Assuming that this oil company wants to maximize its

expected net earnings, use a decision tree to determine its optimal strategy.

b. Calculate and interpret EVI for this decision prob- lem. Experiment with the accuracy probabilities of the geologist to see how EVI changes as they change.

c. Calculate and interpret EVPI for this decision problem.

45. A product manager at Clean & Brite (C&B) wants to determine whether her company should market a new brand of toothpaste. If this new product succeeds in the marketplace, C&B estimates that it could earn $1,800,000 in future profits from the sale of the new toothpaste. If this new product fails, however, the com- pany expects that it could lose approximately $750,000. If C&B chooses not to market this new brand, the prod- uct manager believes that there would be little, if any, impact on the profits earned through sales of C&B’s other products. The manager has estimated that the new toothpaste brand will succeed with probability 0.50. Before making her decision regarding this toothpaste product, the manager can spend $75,000 on a market research study. Based on similar studies with past prod- ucts, C&B believes that the study will predict a suc- cessful product, given that product would actually be a success, with probability 0.75. It also believes that the study will predict a failure, given that the product would actually be a failure, with probability 0.65. a. To maximize expected profit, what strategy should the

C&B product manager follow? b. Calculate and interpret EVI for this decision problem. c. Calculate and interpret EVPI for this decision

problem. 46. Ford is going to produce a new vehicle, the Pioneer,

and wants to determine the amount of annual capacity it should build. Ford’s goal is to maximize the profit from this vehicle over the next five years. Each vehicle will sell for $19,000 and incur a variable production cost of $16,000. Building one unit of annual capacity will cost $2000. Each unit of capacity will also cost $1000 per year to maintain, even if the capacity is unused. Demand for the Pioneer is unknown but marketing estimates the distribution of annual demand to be as shown in the file P06_46.xlsx. Assume that the number of units sold during a year is the minimum of capacity and annual demand. Which capacity level should Ford choose? Do you think EMV is the appropriate criterion?

47. Many decision problems have the following simple structure. A decision maker has two possible decisions, 1 and 2. If decision 1 is made, a sure cost of c is incurred. If decision 2 is made, there are two possible outcomes, with costs c1 and c2 and probabilities p and 12 p. We

assume that c1 , c , c2. The idea is that decision 1, the riskless decision, has a moderate cost, whereas decision 2, the risky decision, has a low cost c1 or a high cost c2. a. Calculate the expected cost from the risky decision. b. List as many scenarios as you can think of that have

this structure. (Here’s an example to get you started. Think of insurance, where you pay a sure premium to avoid a large possible loss.) For each of these scenar- ios, indicate whether you would base your decision on EMV or on expected utility.

48. A nuclear power company is deciding whether to build a nuclear power plant at Diablo Canyon or at Roy Rogers City. The cost of building the power plant is $10 mil- lion at Diablo and $20 million at Roy Rogers City. If the company builds at Diablo, however, and an earthquake occurs at Diablo during the next five years, construction will be terminated and the company will lose $10 mil- lion (and will still have to build a power plant at Roy Rogers City). Without further expert information the company believes there is a 20% chance that an earth- quake will occur, at Diablo during the next five years. For $1 million, a geologist can be hired to analyze the fault structure at Diablo Canyon. She will predict either that an earthquake will occur or that an earthquake will not occur. The geologist’s past record indicates that she will predict an earthquake with probability 0.95 if an earthquake will occur, and she will predict no earth- quake with probability 0.90 if an earthquake will not occur. Should the power company hire the geologist? Also, calculate and interpret EVI and EVPI.

49. Referring to Techware’s decision problem in Problem 33, suppose now that Techware’s utility function of net revenue x (measured in dollars) is U(x) 5 1 2 e2x/350000. a. Find the decision that maximizes Techware’s expected

utility. How does this optimal decision compare to the optimal decision with an EMV criterion? Explain any difference between the two optimal decisions.

b. Repeat part a when Techware’s utility function is U(x) 5 1 2 e2x/50000.

50. Referring to the bank’s customer loan decision problem in Problem 41, suppose now that the bank’s utility func- tion of profit x (in dollars) is U(x) 5 1 2 e2x/150000. Find the strategy that maximizes the bank’s expected utility. How does this optimal strategy compare to the optimal decision with an EMV criterion? Explain any difference between the two optimal strategies.

51. A television network earns an average of $25 million each season from a hit program and loses an average of $8 million each season on a program that turns out to be a flop. Of all programs picked up by this network in recent years, 25% turn out to be hits and 75% turn out to be flops. At a cost of C dollars, a market research firm will analyze a pilot episode of a prospective program and issue a report predicting whether the given pro- gram will end up being a hit. If the program is actually going to be a hit, there is a 75% chance that the market

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researchers will predict the program to be a hit. If the program is actually going to be a flop, there is only a 30% chance that the market researchers will predict the program to be a hit. a. What is the maximum value of C that the network

should be willing to pay the market research firm? b. Calculate and interpret EVPI for this decision

problem. 52. [Based on Balson et al. (1992).] An electric utility com-

pany is trying to decide whether to replace its PCB transformer in a generating station with a new and safer transformer. To evaluate this decision, the utility needs information about the likelihood of an incident, such as a fire, the cost of such an incident, and the cost of replacing the unit. Suppose that the total cost of replace- ment as a present value is $75,000. If the transformer is replaced, there is virtually no chance of a fire. However, if the current transformer is retained, the probability of a fire is assessed to be 0.0025. If a fire occurs, the cleanup cost could be high ($80 million) or low ($20 million). The probability of a high cleanup cost, given that a fire occurs, is assessed at 0.2. a. If the company uses EMV as its decision criterion,

should it replace the transformer? b. Perform a sensitivity analysis on the key parameters

of the problem that are difficult to assess, namely, the probability of a fire, the probability of a high cleanup cost, and the high and low cleanup costs. Does the optimal decision from part a remain optimal for a wide range of these parameters?

c. Do you believe EMV is the correct criterion to use in this type of problem involving environmental accidents?

53. The Indiana University basketball team trails by two points with eight seconds to go and has the ball. Should it attempt a two-point shot or a three-point shot? Assume that the Indiana shot will end the game and that no foul will occur on the shot. Assume that a three-point shot has a 30% chance of success, and a two-point shot has a 45% chance of success. Finally, assume that Indiana has a 50% chance of winning in overtime.

Level B 54. Mr. Maloy has just bought a new $30,000 sport utility

vehicle. As a reasonably safe driver, he believes there is only about a 5% chance of being in an accident in the coming year. If he is involved in an accident, the damage to his new vehicle depends on the severity of the acci- dent. The probability distribution of damage amounts (in dollars) is given in the file P06_54.xlsx. Mr. Maloy is trying to decide whether to pay $170 each year for col- lision insurance with a $300 deductible. Note that with this type of insurance, he pays the first $300 in damages if he causes an accident and the insurance company pays the remainder.

a. Identify the decision that minimizes Mr. Maloy’s annual expected cost.

b. Perform a sensitivity analysis on the best decision with respect to the probability of an accident, the premium, and the deductible amount, and sum- marize your findings. (You can choose the ranges to test.)

55. The purchasing agent for a PC manufacturer is cur- rently negotiating a purchase agreement for a particular electronic component with a given supplier. This com- ponent is produced in lots of 1000, and the cost of pur- chasing a lot is $30,000. Unfortunately, past experience indicates that this supplier has occasionally shipped defective components to its customers. Specifically, the proportion of defective components supplied by this supplier has the probability distribution given in the file P06_55.xlsx. Although the PC manufacturer can repair a defective component at a cost of $20 each, the pur- chasing agent learns that this supplier will now assume the cost of replacing defective components in excess of the first 100 faulty items found in a given lot. This guar- antee may be purchased by the PC manufacturer prior to the receipt of a given lot at a cost of $1000 per lot. The purchasing agent wants to determine whether it is worthwhile to purchase the supplier’s guarantee policy. a. Identify the strategy that minimizes the expected total

cost of achieving a complete lot of satisfactory micro- computer components.

b. Perform a sensitivity analysis on the optimal deci- sion with respect to the number of components per lot and the three monetary inputs, and summarize your findings. (You can choose the ranges to test.)

56. A home appliance company is interested in market- ing an innovative new product. The company must decide whether to manufacture this product in house or employ a subcontractor to manufacture it. The file P06_56.xlsx contains the estimated probability dis- tribution of the cost of manufacturing one unit of this new product (in dollars) if the home appliance com- pany produces the product in house. This file also contains the estimated probability distribution of the cost of purchasing one unit of the product if from the subcontractor. There is also uncertainty about demand for the product in the coming year, as shown in the same file. The company plans to meet all demand, but there is a capacity issue. The subcontractor has unlimited capacity, but the home appliance company has capacity for only 5000 units per year. If it decides to make the product in house and demand is greater than capacity, it will have to purchase the excess demand from an external source at a premium: $225 per unit. Assuming that the company wants to minimize the expected cost of meeting demand in the coming year, should it make the new product in house or buy it from the subcontractor? Do you need a decision tree, or can you perform the required EMV calculations without

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b. For which values of X (where 10% , X , 20%) and Y (where 12.5% , Y , 17.5%), if any, will this inves- tor prefer to place all of her available funds in stocks? Use the same method as in part a for each combina- tion of X and Y.

59. A city in Ohio is considering replacing its fleet of gaso- line-powered automobiles with electric cars. The manu- facturer of the electric cars claims that this municipality will experience significant cost savings over the life of the fleet if it chooses to pursue the conversion. If the manufacturer is correct, the city will save about $1.5 million dollars. If the new technology employed within the electric cars is faulty, as some critics suggest, the conversion to electric cars will cost the city $675,000. A third possibility is that less serious problems will arise and the city will break even with the conversion. A con- sultant hired by the city estimates that the probabilities of these three outcomes are 0.30, 0.30, and 0.40, respec- tively. The city has an opportunity to implement a pilot program that would indicate the potential cost or sav- ings resulting from a switch to electric cars. The pilot program involves renting a small number of electric cars for three months and running them under typical con- ditions. This program would cost the city $75,000. The city’s consultant believes that the results of the pilot pro- gram would be significant but not conclusive; she sub- mits the values in the file P06_59.xlsx, a compilation of probabilities based on the experience of other cities, to support her contention. For example, the first row of her table indicates that if a conversion to electric cars will actually result in a savings of $1.5 million, the pilot program will indicate that the city saves money, loses money, and breaks even with probabilities 0.6, 0.1, and 0.3, respectively. What actions should the city take to maximize its expected savings? When should it run the pilot program, if ever?

60. Sharp Outfits is trying to decide whether to ship some customer orders now via UPS or wait until after the threat of another UPS strike is over. If Sharp Outfits decides to ship the requested merchandise now and the UPS strike takes place, the company will incur $60,000 in delay and shipping costs. If Sharp Outfits decides to ship the customer orders via UPS and no strike occurs, the company will incur $4000 in shipping costs. If Sharp Outfits decides to postpone shipping its customer orders via UPS, the company will incur $10,000 in delay costs regardless of whether UPS goes on strike. Let p represent the probability that UPS will go on strike and impact Sharp Outfits’s shipments. a. For which values of p, if any, does Sharp Outfits min-

imize its expected total cost by choosing to postpone shipping its customer orders via UPS?

b. Suppose now that, at a cost of $1000, Sharp Outfits can purchase information regarding the likelihood of a UPS strike in the near future. Based on similar strike threats in the past, the company assesses that if there will be a

one? (You can assume that neither the company nor the subcontractor will ever produce more than demand.)

57. A grapefruit farmer in central Florida is trying to decide whether to take protective action to limit damage to his crop in the event that the overnight temperature falls to a level well below freezing. He is concerned that if the temperature falls sufficiently low and he fails to make an effort to protect his grapefruit trees, he runs the risk of losing his entire crop, which is worth approxi- mately $75,000. Based on the latest forecast issued by the National Weather Service, the farmer estimates that there is a 60% chance that he will lose his entire crop if it is left unprotected. Alternatively, the farmer can insulate his fruit by spraying water on all of the trees in his orchards. This action, which would likely cost the farmer C dollars, would prevent total devastation but might not completely protect the grapefruit trees from incurring some damage as a result of the unusually cold overnight temperatures. The file P06_57.xlsx contains the assessed distribution of possible damages (in dol- lars) to the insulated fruit in light of the cold weather forecast. The farmer wants to minimize the expected total cost of coping with the threatening weather. a. Find the maximum value of C below which the farmer

should insulate his crop to limit the damage from the unusually cold weather.

b. Set C equal to the value identified in part a. Perform sensitivity analysis to determine under what condi- tions, if any, the farmer would be better off not spray- ing his grapefruit trees and taking his chances in spite of the threat to his crop.

c. Suppose that C equals $25,000, and in addition to this protection, the farmer can purchase insurance on the crop. Discuss possibilities for reasonable insurance policies and how much they would be worth to the farmer. You can assume that the insurance is relevant only if the farmer purchases the protection, and you can decide on the terms of the insurance policy.

58. A retired partner from a large brokerage firm has one million dollars available to invest in particular stocks or bonds. Each investment’s annual rate of return depends on the state of the economy in the coming year. The file P06_58.xlsx contains the distribution of returns for these stocks and bonds as a function of the econo- my’s state in the coming year. As this file indicates, the returns from stocks and bonds in a fair economy are listed as X and Y. This investor wants to allocate her one million dollars to maximize her expected value of the portfolio one year from now. a. If X 5 Y 5 15% find the optimal investment strat-

egy for this investor. (Hint: You could try a decision tree approach, but it would involve a massive tree. It is much easier to find an algebraic expression for the expected final value of the investment when a per- centage p is put in stocks and the remaining percent- age is put in bonds.)

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strike, the information will predict a strike with prob- ability 0.75, and if there will not be a strike, the infor- mation will predict no strike with probability 0.85. Provided that p 5 0.15, what strategy should Sharp Outfits pursue to minimize its expected total cost?

c. Using the analysis from part b, find the EVI when p 5 0.15. Then use a data table to find EVI for p from 0.05 to 0.30 in increments of 0.05, and chart EVI versus p.

d. Continuing part b, calculate and interpret the EVPI when p 5 0.15.

61. A homeowner wants to decide whether he should install an electronic heat pump in his home. Given that the cost of installing a new heat pump is fairly large, the home- owner wants to do so only if he can count on being able to recover the initial expense over five consecutive years of cold winter weather. After reviewing historical data on the operation of heat pumps in various kinds of win- ter weather, he computes the expected annual costs of heating his home during the winter months with and without a heat pump in operation. These cost figures are shown in the file P06_61.xlsx. The probabilities of expe- riencing a mild, normal, colder than normal, and severe winter are 0.2(1 2 x), 0.5(1 2 x), 0.3(1 2 x), and x, respectively. In words, as the last probability varies, the first three probabilities remain in the ratios 2 to 5 to 3, and all probabilities continue to sum to 1. a. Given that x 5 0.1, what is the most that the home-

owner is willing to pay for the heat pump? b. If the heat pump costs $500, how large must x be

before the homeowner decides it is economically worthwhile to install the heat pump?

c. Given that c 5 0.1, calculate and interpret EVPI when the heat pump costs $500.

62. Suppose an investor has the opportunity to buy the fol- lowing contract, a stock call option, on March 1. The contract allows him to buy 100 shares of ABC stock at the end of March, April, or May at a guaranteed price of $50 per share. He can exercise this option at most once. For example, if he purchases the stock at the end of March, he can’t purchase more in April or May at the guaranteed price. The current price of the stock is $50. Each month, assume that the stock price either goes up by a dollar (with probability 0.55) or goes down by a dollar (with probability 0.45). If the investor buys the contract, he is hoping that the stock price will go up. The reasoning is that if he buys the contract, the price goes up to $51, and he buys the stock (that is, he exercises his option) for $50, he can then sell the stock for $51 and make a profit of $1 per share. On the other hand, if the stock price goes down, he doesn’t have to exercise his option; he can just throw the contract away. a. Use a decision tree to find the investor’s optimal strat-

egy—that is, when he should exercise the option— assuming that he purchases the contract.

b. How much should he be willing to pay for such a contract?

63. (This problem assumes knowledge of the basic rules of football.) The ending of the game between the Indianapolis Colts and the New England Patriots (NFL teams) in Fall 2009 was quite controversial. With about two minutes left in the game, the Patri- ots were ahead 34 to 28 and had the ball on their own 28-yard line with fourth down and two yards to go. In other words, they were 72 yards from a touchdown. Their coach, Bill Belichick, decided to go for the first down rather than punt, contrary to conventional wis- dom. They didn’t make the first down, so possession went to the Colts, who then scored a touchdown to win by a point. Belichick was harshly criticized by most of the media, but was his unorthodox decision really a bad one? a. Use a decision tree to analyze the problem. You can

make some simplifying decisions: (1) the game would essentially be over if the Patriots made a first down, and (2) at most one score would occur after a punt or a failed first down attempt. (There are no monetary val- ues. However, you can assume the Patriots receive $1 for a win and $0 for a loss, so that maximizing EMV is equivalent to maximizing the probability that the Patriots win.)

b. Show that the Patriots should go for the first down if p . 1 2 q/r. Here, p is the probability the Patriots make the first down, q is the probability the Colts score a touchdown after a punt, and r is the proba- bility the Colts score a touchdown after the Patriots fail to make a first down. What are your best guesses for these three probabilities? Based on them, was Belichick’s decision justified?

64. (This problem assumes knowledge of the basic rules of baseball.) George Lindsey (1959) looked at box scores of more than 1000 baseball games and found the expected number of runs scored in an inning for each on-base and out situation to be as listed in the file P06_64.xlsx. For example, if a team has a man on first base with one out, it scores 0.5 run on average until the end of the inning. You can assume throughout this problem that the team batting wants to maximize the expected number of runs scored in the inning. a. Use this data to explain why, in most cases, bunting

with a man on first base and no outs is a bad decision. In what situation might bunting with a man on first base and no outs be a good decision?

b. Assume there is a man on first base with one out. What probability of stealing second makes an attempted steal a good idea?

65. (This problem assumes knowledge of the basic rules of basketball.) One controversial topic in basketball (college or any other level) is whether to foul a player deliberately with only a few seconds left in the game. Consider the following scenario. With about 10 sec- onds left in the game, team A is ahead of team B by three points, and team B is just about to inbound the

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ball. Assume team A has committed enough fouls so that future fouls result in team B going to the free-throw line. If team A purposely commits a foul as soon as possible, team B will shoot two foul shots (a point apiece). The thinking is that this is better than let- ting team B shoot a three-point shot, which would be their best way to tie the game and send it into overtime. However, there is a downside to fouling. Team B could make the first free throw, purposely miss the second, get the rebound, and score a two-point shot to tie the game, or it could even score a three-point shot to win the game. Examine this decision, using reasonable input parame- ters. It doesn’t appear that this deliberate fouling strategy is used very often, but do you think it should be used?

66. (This problem assumes knowledge of the basic rules of football.) The following situation actually occurred

in a 2009 college football game between Washington and Notre Dame. With about 3.5 minutes left in the game, Washington had fourth down and one yard to go for a touchdown, already leading by two points. Notre Dame had just had two successful goal-line stands from in close, so Washington’s coach decided not to go for the touchdown and the virtually sure win. Instead, Washing- ton kicked a field goal, and Notre Dame eventually won in overtime. Use a decision tree, with some reasonable inputs, to see whether Washington made a wise decision or should have gone for the touchdown. Note that the only “monetary” values here are 1 and 0. You can think of Washington getting $1 if they win and $0 if they lose. Then the EMV is 1*P(Win) 1 0*P(lose) 5 P(Win), so maximizing EMV is equivalent to maximizing the prob- ability of winning.

CASE 6.1 Jogger Shoe Company The Jogger Shoe Company is trying to decide whether to make a change in its most popular brand of running shoes. The new style would cost the same to produce and be priced the same, but it would incorporate a new kind of lac- ing system that (according to its marketing research people) would make it more popular.

There is a fixed cost of $300,000 for changing over to the new style. The unit contribution to before-tax profit for either style is $8. The tax rate is 35%. Also, because the fixed cost can be depreciated and will therefore affect the after-tax cash flow, a depreciation method is needed. You can assume it is straight-line depreciation.

The current demand for these shoes is 190,000 pairs annually. The company assumes this demand will continue for the next three years if the current style is retained. How- ever, there is uncertainty about demand for the new style, if it is introduced. The company models this uncertainty by assuming a normal distribution in year 1, with mean 220,000

and standard deviation 20,000. The company also assumes that this demand, whatever it is, will remain constant for the next three years. However, if demand in year 1 for the new style is sufficiently low, the company can always switch back to the current style and realize an annual demand of 190,000. The company wants a strategy that will maximize the expected net present value (NPV) of total cash flow for the next three years, where a 10% interest rate is used for the purpose of calculating NPV.

Realizing that the continuous normal demand distribu- tion doesn’t lend itself well to decision trees that require a discrete set of outcomes, the company decides to replace the normal demand distribution with a discrete distribution with five “typical” values. Specifically, it decides to use the 10th, 30th, 50th, 70th, and 90th percentiles of the given normal distribution. Why is it reasonable to assume that these five possibilities are equally likely? With this discrete approxi- mation, how should the company proceed?

CASE 6.2 Westhouser Paper Company The Westhouser Paper Company in the state of Washington currently has an option to purchase a piece of land with good timber forest on it. It is now May 1, and the current price of the land is $2.2 million. Westhouser does not actually need the timber from this land until the beginning of July, but its top executives fear that another company might buy the land between now and the beginning of July. They assess that there is a 5% chance that a competitor will buy the land during May. If this does not occur, they assess that there is a 10% chance that the competitor will buy the land during June. If Westhouser does not take advantage of its current option, it can attempt to buy the land at the beginning of June or the beginning of July, provided that it is still available.

Westhouser’s incentive for delaying the purchase is that its financial experts believe there is a good chance that the price of the land will fall significantly in one or both of the next two months. They assess the possible price decreases and their probabilities in Tables 6.2 and 6.3. Table 6.2 shows the probabilities of the possible price decreases during May. Table 6.3 lists the conditional probabilities of the possible price decreases in June, given the price decrease in May. For example, it indicates that if the price decrease in May is $60,000, then the possible price decreases in June are $0, $30,000, and $60,000 with respective probabilities 0.6, 0.2, and 0.2.

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If Westhouser purchases the land, it believes that it can gross $3 million. (This does not count the cost of purchasing the land.) But if it does not purchase the land, Westhouser believes that it can make $650,000 from alternative invest- ments. What should the company do?

price Decrease probability

$0 0.5

$60,000 0.3

$120,000 0.2

Table 6.2 Distribution of Price Decrease in May

Price Decrease in May

$0 $60,000 $120,000

June Decrease probability June Decrease probability June Decrease probability

$0 0.3 $0 0.6 $0 0.7

$60,000 0.6 $30,000 0.2 $20,000 0.2

$120,000 0.1 $60,000 0.2 $40,000 0.1

Table 6.3 Distribution of Price Decrease in June

CASE 6.3 Electronic Timing System for Olympics Sarah Chang is the owner of a small electronics company. In six months, a proposal is due for an electronic timing sys- tem for the next Olympic Games. For several years, Chang’s company has been developing a new microprocessor, a crit- ical component in a timing system that would be superior to any product currently on the market. However, progress in research and development has been slow, and Chang is unsure whether her staff can produce the microprocessor in time. If they succeed in developing the microprocessor (probability p1), there is an excellent chance (probability p2) that Chang’s company will win the $1 million Olympic con- tract. If they do not, there is a small chance (probability p3) that she will still be able to win the same contract with an alternative but inferior timing system that has already been developed.

If she continues the project, Chang must invest $200,000 in research and development. In addition, making a proposal (which she will decide whether to do after seeing whether the R&D is successful) requires developing a prototype timing system at an additional cost. This additional cost is $50,000 if R&D is successful (so that she can develop the new timing system), and it is $40,000 if R&D is unsuccess- ful (so that she needs to go with the older timing system).

Finally, if Chang wins the contract, the finished product will cost an additional $150,000 to produce.

a. Develop a decision tree that can be used to solve Chang’s problem. You can assume in this part of the problem that she is using EMV (of her net profit) as a decision crite- rion. Build the tree so that she can enter any values for p1, p2, and p3 (in input cells) and automatically see her optimal EMV and optimal strategy from the tree.

b. If p2 5 0.8 and p3 5 0.1, what value of p1 makes Chang indifferent between abandoning the project and going ahead with it?

c. How much would Chang benefit if she knew for certain that the Olympic organization would guarantee her the contract? (This guarantee would be in force only if she were successful in developing the product.) Assume p1 5 0.4, p2 5 0.8, and p3 5 0.1.

d. Suppose now that this is a relatively big project for Chang. Therefore, she decides to use expected utility as her criterion, with an exponential utility function. Using some trial and error, see which risk tolerance changes her initial decision from “go ahead” to “abandon” when p1 5 0.4, p2 5 0.8, and p3 5 0.1.

CASE 6.4 Developing a Helicopter Component for the Army The Ventron Engineering Company has just been awarded a $2 million development contract by the U.S. Army Aviation Systems Command to develop a blade spar for its Heavy Lift Helicopter program. The blade spar is a metal tube that runs the length of and provides strength to the helicopter blade. Due to the unusual length and size of the Heavy Lift Helicop- ter blade, Ventron is unable to produce a single-piece blade spar of the required dimensions using existing extrusion

equipment and material. The engineering department has prepared two alternatives for developing the blade spar: (1) sectioning or (2) an improved extrusion process. Ventron must decide which process to use. (Backing out of the con- tract at any point is not an option.) The risk report has been prepared by the engineering department. The information from this report is explained next.

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The sectioning option involves joining several shorter lengths of extruded metal into a blade spar of sufficient length. This work will require extensive testing and rework over a 12-month period at a total cost of $1.8 million. Although this process will definitely produce an adequate blade spar, it merely represents an extension of existing technology.

To improve the extrusion process, on the other hand, it will be necessary to perform two steps: (1) improve the mate- rial used, at a cost of $300,000, and (2) modify the extrusion press, at a cost of $960,000. The first step will require six months of work, and if this first step is successful, the second step will require another six months of work. If both steps are successful, the blade spar will be available at that time, that is, a year from now. The engineers estimate that the probabilities of succeeding in steps 1 and 2 are 0.9 and 0.75, respectively. However, if either step is unsuccessful (which will be known only in six months for step 1 and in a year for step 2), Ventron will have no alternative but to switch to the sectioning process—and incur the sectioning cost on top of any costs already incurred.

Development of the blade spar must be completed within 18 months to avoid holding up the rest of the con- tract. If necessary, the sectioning work can be done on an

accelerated basis in a six-month period, but the cost of sec- tioning will then increase from $1.8 million to $2.4 million. The director of engineering, Dr. Smith, wants to try devel- oping the improved extrusion process. He reasons that this is not only cheaper (if successful) for the current project, but its expected side benefits for future projects could be sizable. Although these side benefits are difficult to gauge, Dr. Smith’s best guess is an additional $2 million. (These side benefits are obtained only if both steps of the modified extrusion process are completed successfully.)

a. Develop a decision tree to maximize Ventron’s EMV. This includes the revenue from this project, the side benefits (if applicable) from an improved extrusion pro- cess, and relevant costs. You don’t need to worry about the time value of money; that is, no discounting or net present values are required. Summarize your findings in words in the spreadsheet.

b. What value of side benefits would make Ventron indif- ferent between the two alternatives?

c. How much would Ventron be willing to pay, right now, for perfect information about both steps of the improved extrusion process? (This information would tell Ventron, right now, the ultimate success or failure outcomes of both steps.)

APPENDIX Decision Trees with DADM_Tools When this edition of the book was being developed, we weren’t sure whether we would be able to offer Palisade’s DecisionTools Suite as in previous editions. If Palisade’s PrecisionTree add-in were not available, there wouldn’t be any feasible way to create complex decision trees—there are no built-in Excel tools for doing so. Therefore, Albright developed his own decision tree program, the first in a series of programs that eventually became the DADM_Tools add-in. This add-in, along with a help file, is freely avail- able at his website https://kelley.iu.edu/albrightbooks/Free_ downloads.htm.

There are several differences between the DADM_Tools decision tree program and Palisade’s PrecisionTree add-in:

• DADM_Tools is free. • The DADM_Tools decision tree program works on a

Mac; PrecisionTree doesn’t.

• The DADM_Tools decision tree program doesn’t have the sensitivity analysis tools available in PrecisionTree. However, the same sensitivity analyses can be per- formed with Excel data tables.

• PrecisionTree uses its own functions in the colored cells for the folding back process. The DADM_Tools deci- sion tree program uses regular Excel formulas to accom- plish the same thing. The difference is that the latter are more transparent; you can understand the folding back process better by studying these formulas.

• The DADM_Tools decision tree program uses diamonds instead of triangles for end nodes—a purely cosmetic difference.

Many of the solutions to the problems in this chapter are available in two versions: one with DADM_Tools and one with PrecisionTree.

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