How to calculate emv in decision tree

DECISION MODELING AND ANALYSIS

Read Case 6.3: Electronic Timing System for Olympics on pages 275-276 of the textbook. For this assignment, you will assess and use the correct support tool to develop a decision tree as described in Part “a” of Case 6.3. Analyze and apply the best decision making process to provide answers and brief explanations for parts “a”, “b”, “c”, and “d”. The answers and explanations can be placed in the same Excel document as the decision tree.

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 criterion. 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.
If p2 = 0.8 and p3 = 0.1, what value of p1 makes Chang indifferent between abandoning the project and going ahead with it?
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 = 0.4, p2 = 0.8, and p3 = 0.1
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 = 0.4, p2 = 0.8, and p3 = 0.1.
In your Excel document,

Develop a decision tree using the most appropriate support tool as described in Part a.
Calculate the value of p1 as described in Part b. Show calculations.
Calculate the possible profit using the most appropriate support tool as described in Part c. Show calculations.
Calculate risk tolerance as described in Part d. Show calculations.
CASE 6.3

Sarah Chang is the owner of a small electronics company. In six months, a proposal is due for an electronic timing system for the next Olympic Games. For several years, Chang's company has been developing a new microprocessor, a critical 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 contract. 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.

CHAPTER

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.