DECISION MODELING AND ANALYSIS
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CHAPTER6.docxCHAPTER
6
Decision Making Under Uncertainty
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DECIDING WHETHER TO DEVELOP NEW DRUGS AT BAYER
The formal decision-making process discussed in this chapter is often used to make difficult decisions in the face of much uncertainty, large monetary values, and long-term consequences. Stonebraker (2002) chronicles one such decision-making process he performed for Bayer Pharmaceuticals in 1999. The development of a new drug is a time-consuming and expensive process that is filled with risks along the way. A pharmaceutical company must first get the proposed drug through preclinical trials, where the drug is tested on animals. Assuming this stage is successful (and only about half are), the company can then file an application with the Food and Drug Administration (FDA) to conduct clinical trials on humans. These clinical trials have three phases. Phase 1 is designed to test the safety of the drug on a small sample of healthy patients. Phase 2 is designed to identify the optimal dose of the new drug on patients with the disease. Phase 3 is a statistically designed study to prove the efficacy and safety of the new drug on a larger sample of patients with the disease. Failure at any one of these phases means that further testing stops and the drug is never brought to market. Of course, this means that all costs up to the failure point are lost. If the drug makes it through the clinical tests (and only about 25% of all drugs do so), the company can then apply to the FDA for permission to manufacture and market its drug in the United States. Assuming that the FDA approves, the company is then free to launch the drug in the marketplace.
The study involved the evaluation of a new drug for busting blood clots called BAY 57-9602, and it commenced at a time just prior to the first decision point: whether to con-duct preclinical tests. This was the company's first formal use of decision making for evaluating a new drug, so to convince the company of the worth of such a study, Stonebraker did exactly what a successful management scientist should do. He formulated the problem and its objectives; he identified risks, costs, and benefits; he involved key people in the organization to help provide the data needed for the decision analysis; and, because much of the resulting data consisted of educated guesses at best, he performed a thorough sensitivity analysis on the inputs. Although we are not told in the article how everything turned out, the analysis did persuade Bayer management to proceed in January 2000 with preclinical testing of the drug.
The article provides a fascinating look at how such a study should proceed. Because there is so much uncertainty, the key is determining probabilities and probability distributions for the various inputs. First, there are uncertainties in the various phases of testing. Each of these can be modeled with a probability of success. For example, the chance of making it through preclinical testing was assessed to be about 65% for BAY 57-9602, although management preferred to use the more conservative benchmark of 50% (based on historical data on other drugs) for the decision analysis. Many of the other uncertain quantities, such as the eventual market share, are continuous random variables. Because the decision tree approach discussed in this chapter requires discrete random variables, usually with only a few possible values, Stonebraker used a popular three-point approximation for all continuous quantities. He asked experts to assess the 10th percentile, the 50th percentile, and the 90th percentile, and he assigned probabilities 0.3, 0.4, and 0.3 to these three values. [The validity of such an approximation is discussed in Keefer and Bodily (1983).]
After getting all such estimates of uncertain quantities from the company experts, the author examined the expected net present value (NPV) of all costs and benefits from developing the new drug. To see which of the various uncertain quantities affected the expected NPV most, he varied each such quantity, one at a time, from its 10th percentile to its 90th percentile, leaving the other inputs at their base 50th percentile values. This identified several quantities that the expected NPV was most sensitive to, including the peak product share, the price per treatment in the United States, and the annual growth rate. The expected NPV was not nearly as sensitive to other uncertain inputs, including the product launch date and the production process yield. Therefore, in the final decision analysis, Stonebraker treated the sensitive inputs as uncertain and the less sensitive inputs as certain at their base values. He also calculated the risk profile from developing the drug. This indicates the probability distribution of NPV, taking all sources of uncertainty into account. Although this risk profile was not exactly optimistic (90% chance of losing money using the conservative probabilities of success, 67% chance of losing money with the more optimistic product-specific probabilities of success), this risk profile compared favorably with Bayer's other potential projects. This evaluation, plus the rigor and defensibility of the study, led Bayer management to give the go-ahead on preclinical testing. ■
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 economic 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 implemented in Excel with a very powerful and flexible add-in from Palisade called PrecisionTree.
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 completing 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, after all, 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 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 revenue because of insufficient capacity. If they do expand and demand for their products is lower than expected, they will be stuck with expensive underutilized capacity. Of course, in today's global economy, 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 stability, competition from local businesses, and others.
■ Banks must continually make decisions on whether to grant loans to businesses or individuals. As we all know, 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 companies liable—for huge settlements—when accidents occurred, even though the companies followed all existing regulations. Therefore, when utility companies decide, say, 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 professional sports these days. Management and coaches typically do not make important decisions by gut feeling. 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 of 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.
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
6-2a Identifying the Problem
This is the same issue that was discussed in Section 1.4 of Chapter 1. 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 undergraduate degree (the trigger), and you want to choose the Business School where you should get your MBA degree. You have then framed the problem as “Which MBA program?”, but maybe you should frame it more generally as “What should I do next, now that I have my 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 “Which MBA program?” to “What do I do next?”, you change the decision problem in a fundamental way.
6-2b Possible Decisions
The possible decisions depend on the previous step: how the problem is specified. But after the problem identification, 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. Note some problems are of a multistage nature, as discussed in Section 6.5. 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 decisions and observed outcomes. These ideas are clarified in Section 6.5.
6-2c 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, then we typically choose a small set of representative outcomes from this continuum.
6-2d 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. Note that 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 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 However, 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 assessments for demand of the new product will necessarily contain a heavy subjective component—your best guesses based on your experience and possibly the inputs of the marketing experts in your company. In fact, we would venture to say that the 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 Cleveland Cavaliers will win the NBA championship this year. Will this assessment change if you hear later that LeBron James 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 6.5, we show how probabilities can be updated in a formal way by using an important law of probabilities called Bayes’ rule.
6-2e 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. Obviously, nonmonetary consequences can be very difficult to quantify, but an attempt must be made to do so. Otherwise, it is impossible to make meaningful trade-offs.
6-2f Decision Criterion
Once all of these elements of a decision problem have been specified, it is time to 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 probabilities sum to 1.) Another decision might lead to the following: a payoff of $5000 with probability 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 best (or 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 company that wants to limit its losses. (Actually, any percentile, not just the 5th, could be chosen.)
■ 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 minimizing risk but, unfortunately, 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 many 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 criterion 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 businesses). It provides a rational way of making decisions, at least when the monetary payoffs and costs are of “moderate” size relative to the decision maker's wealth. (Section 6.6 will present 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 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 = 50000(0.1) + 10000(0.3) + (−5000)(0.6) = $3500
■ Decision 2: EMV = 5000(0.6) + (−1000)(0.4) = $2600
Therefore, according to the EMV criterion, you should choose decision 1.
6-2g More about the EMV Criterion
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, suppose 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.
Third, the EMV criterion is easy to operationalize in a spreadsheet. This is shown in Figure 6.1. (See the file Simple Decision Problem.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
=SUMPRODUCT(A3:A5,B3:B5)
Figure 6.1 EMV Calculations in Excel
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Of course, 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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Finally, 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 (although they might not be 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 criterion provides a rational way to make good decisions, even for one-shot deals. Therefore, we use this criterion in most of this chapter.
FUNDAMENTAL INSIGHT
What It Means to Be an EMV Maximizer
An EMV maximizer, by definition, is indifferent when faced with the choice between entering a gamble that has a certain 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 −$250, and paying a sure $250 to avoid the gamble. (This latter scenario is the basis of insurance.)
6-2h Decision Trees
A decision problem evolves through time. A decision is made, then an uncertain outcome 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 of these elements of the decision problem, including 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.
The decision tree for the simple decision problem discussed earlier appears in Figure 6.3.
Figure 6.3 Simple Decision Tree
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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. First, it is important to realize that decision trees such as this one have been used for over 50 years. They all use the following 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.)
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 preferred branch. 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.
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 outcomes. 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 decision 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, the 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 6.5, decision trees provide a useful view of more complex problems. In addition, decision trees provides a framework for doing all of the EMV calculations. Specifically, they allow you to use the following folding-back procedure to find the EMVs and the best decision.
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.
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. 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 calculating EMVs in a decision tree and thereby identifying the best decision strategy.
PROBLEMS
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.
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 decision 2? (There is no single correct answer.)
6-3 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 decision tree in
EXAMPLE
6.1 NEW PRODUCT DECISIONS AT ACME
The Acme Company is trying to 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 estimates that the corresponding sales volumes (in thousands of units sold) of 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?3
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. (Of course, any fixed costs already incurred are sunk and therefore have no relevance to the current decision.) The uncertain sales volume is really a continuous 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 file New Product Decisions - Single-Stage.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
=SUMPRODUCT(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
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As before, a decision tree is probably overkill for this problem, but it is shown in Figure 6.5. 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. Therefore, 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 6.5 Decision Tree for New Product Model
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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. Either tree is perfectly acceptable. However, this 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.)
Figure 6.6 Equivalent Decision Tree
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Given 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 K3, the EMV from marketing is calculated in cell K4 with the formula
=(1-K3)*SUMPRODUCT(D8:D10,B8:B10)-B4
Then a data table is used in the usual way, with cell K3 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 6.7 Sensitivity Analysis
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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 power of decision trees will become evident.
PROBLEMS
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 probability 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 probabilities 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?
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, uncertainty can sometimes be broken down into a sequence of uncertain outcomes. Here is a typical example. A company 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 competing 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 completing the project minus its cost of preparing the bid, where these costs are assumed to be known.
6-4 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 depict the decision tree). Fortunately, Palisade has developed a powerful add-in to Excel 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.
Figure 6.8 PrecisionTree Welcome Screen
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Figure 6.9 PrecisionTree Ribbon
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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.
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 (developed by Albright). To do this, you can bring up the Welcome screen in Figure 6.8 at any time through the PrecisionTree 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 6.10 PrecisionTree Quick Start Buttons
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It is instructive to examine PrecisionTree's decision tree for Acme's single stage problem. The completed tree appears in Figure 6.11. (See the file New Product Decisions - Single-Stage with PrecisionTree.xlsx.) It is essentially a mixture of the trees in Figures 6.4 and 6.5, but it is completely 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 corresponding triangle end nodes. For the folding back process, it uses these net values. Specifically, the 1074 value to the right of the probability node is calculated (automatically) as (4800)(0.45) +(−600)(0.35) + (−4380)(0.20).4 Then the 1074 value to the right of the decision node is calculated as the maximum of 1074 and 0.
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single-stage_decision_tree video.
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 does the folding-back calculations automatically, and (3) it permits quick sensitivity analyses on any of the model inputs. Besides, as discussed in Quick Start video, 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.
PrecisionTree Tip: Formatting Numbers
If you are fussy 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.
We finish this section with one important reminder discussed in the Quick Start videos. 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.
Figure 6.11 Decision Tree from PrecisionTree
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PROBLEMS
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.
9. Use PrecisionTree's Sensitivity Analysis tools to perform 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.
6-5 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 information. 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 decision 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 information 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, then the company might decide to continue with the product. This is the essence of a contingency plan: it tells you what do for each possible uncertain outcome.
An important aspect of multistage decision problems is that probabilities can change through time. Specifically, 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 change; the probability will decrease. Sometimes this reassessment of probabilities can be done in an informal subjective manner. But whenever possible, it should be done with a powerful probability law called Bayes’ rule. This rule provides a mathematical way of updating probabilities as new information becomes available, and we will explain how it works in this section.
Another important aspect of multistage decision problems is the value of information. 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 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 this 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 products in the past—but it is probably partly subjective, based on how the product's development has proceeded so far. The probability 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.5
Solution
The reason this is a two-stage decision problem is that Acme can decide right away to stop development and abandon the product, 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 finished tree appears in Figure 6.12. (See the file New Product Decisions - Technological Uncertainty.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 is gone.
Figure 6.12 Decision Tree with Possible Technological Failure
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Figure 6.13 Risk Profile from Best Strategy
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PrecisionTree Tip: 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. (The @RISK add-in discussed in the next two chapters has this same setting.)
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 failure. If you change this probability from 0.2 to 0.25 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.
Figure 6.14 Decision Tree with Larger Probability of Failure
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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. ■
The next example illustrates another possible multistage extension of the Acme decision problem. This example provides an opportunity to introduce two important topics discussed earlier: Bayes’ rule for updating probabilities and the value of information.
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 market the product, given the uncertainty about the eventual sales volume. If the company decides to market
