Data Analysis & Decision Making

1363025 - Cengage Learning ©

amework for doing all of the EMV calculations. They allow you to use the following folding-back procedure to find the EMVs and the best decision.

Folding-Back Procedure

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

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

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

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

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

Level A

1. Several decision criteria besides EMV are suggested in the section. For each of the following criteria, rank

1363025 - Cengage Learning ©

all three decisions in Figure 6.1 f rom 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 f rom 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 f rom the best outcome, the value in cell A3, vary f rom $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 f rom 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

1363025 - Cengage Learning ©

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.)

6-4 One-Stage Decision Problems

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

EXAMPLE 6.1 NEW PRODUCT DECISIONS AT ACME

The Acme Company must decide whether to market a new product. As in many new-product situations, there is considerable uncertainty about the eventual success of the product. The product is currently part way through the development process, and some fixed development costs have already been incurred. If the company decides to continue development and then market the product, there will be additional fixed costs, and they are estimated to be $6 million. If the product is marketed, its unit margin (selling price minus variable cost) will be $18. Acme classifies the possible market results as “great,” “fair,” and “awful,” and it estimates the probabilities of these outcomes to be 0.45, 0.35, and 0.20, respectively. Finally, the company estimates that the corresponding sales volumes (in thousands of units sold) f rom these three outcomes are 600, 300, and 90, respectively. Assuming that Acme is an EMV maximizer, should it finish development and then market the product, or should it stop development at this point and abandon the product?4

Objective

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

Where Do the Numbers Come From?

1363025 - Cengage Learning ©

Acme’s cost accountants should be able to estimate the monetary inputs: the fixed costs and the unit margin. (Any fixed costs already incurred are sunk and therefore have no relevance to the current decision.) The uncertain sales volume is really a 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 files New Product Decisions - Single-Stage 1a Finished.xlsx and New Product Decisions - Single-Stage 1b Finished.xlsx.) If the company decides to stop development and abandon the product, there are no payoffs, costs, or uncertainties; the EMV is $0. (Actually, this isn’t really an expected value; it is a sure $0.) On the other hand, if the company proceeds with the product, it incurs the fixed cost and receives $18 for every unit it sells. The probability distribution of sales volume given in the problem statement appears in columns A to C, and each sales volume is multiplied by the unit margin to obtain the net revenues in column D. Finally, the formula for the EMV in cell B12 is

= 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

As before, a decision tree is probably overkill for this problem, but it is shown in Figure 6.5. (All monetary and

1363025 - Cengage Learning ©

sales volumes are shown in thousands.) This tree indicates one of at least two equivalent ways to show the EMV calculations. The values at the end nodes ignore the fixed cost, which is instead shown under the decision branch as a negative number. Therefore, the 7074 value above the probability node is the expected net revenue, not including the fixed cost. Then the fixed cost is subtracted f rom this to obtain the 1074 value above the decision node.

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

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

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

■ Figure 6.5 Decision Tree for New Product Model

1363025 - Cengage Learning ©

■ Figure 6.6 Equivalent Decision Tree

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

■ Figure 6.7 Sensitivity Analysis

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

1363025 - Cengage Learning ©

Problems

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

Level A

4. The fixed cost of $6 million in the Acme problem is evidently not large enough to make Acme abandon the product at the current time. How large would the fixed cost need to be to make the abandon option the best option? Explain how the decision tree, especially the version in Figure 6.5, answers this question easily.

5. Perform a sensitivity analysis on the probability of a great market. To do this, enter formulas in cells B9 and B10 (see Figure 6.4) to ensure that the probabilities of “fair” and “awful” remain in the same ratio, 35 to 20, and that all three probabilities continue to sum to 1. Then let the probability of “great” vary f rom 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” f rom 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

1363025 - Cengage Learning ©

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.

6-5 The PrecisionTree Add-In

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

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

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

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

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

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

1363025 - Cengage Learning ©

■ Figure 6.8 PrecisionTree Welcome Screen

■ Figure 6.9 PrecisionTree Ribbon

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

1363025 - Cengage Learning ©

■ Figure 6.10 PrecisionTree Quick Start Buttons

single-stage_decision_ tree video..

node for whether to market the product 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 - 1c Finished.xlsx.) It is essentially a mixture of the trees in Figures 6.4 and 6.5, and it is equivalent to each of them. As in Figure 6.4, the fixed cost is entered as a negative number below the decision branch, and the net revenues are entered below the probability branches. Then PrecisionTree calculates the net value—the sum of the monetary values on any path through the tree—to the right of the 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).5 Then the 1074 value to the right of the decision node is calculated as the maximum of 1074 and 0.

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

1363025 - Cengage Learning ©

PrecisionTree Tip

Formatting Numbers

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

■ Figure 6.11 Decision Tree from PrecisionTree

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.

Problems

1363025 - Cengage Learning ©

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

Level A

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

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.

6-6 Multistage Decision Problems

Many real-world decision problems evolve through time in stages. A company first makes a decision. Then it observes an uncertain outcome that provides some 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 f rom 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

1363025 - Cengage Learning ©

the product, but if the news is good, the company might decide to continue with the product. This is the essence of a contingency plan: it specifies what do for each possible uncertain outcome.

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

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

EXAMPLE 6.2 NEW PRODUCT DECISIONS AT ACME WITH TECHNOLOGICAL UNCERTAINTY

In this version of the example, we assume as before that the new product is still in the development stage. However, we now assume that there is a chance that the product will be a failure for technological reasons, such as a new drug that fails to meet FDA approval. At this point in the development process, Acme assesses the probability of technological

1363025 - Cengage Learning ©

failure to be 0.2. The $6 million fixed cost f rom 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 f rom now. (The same issue is relevant in Example 6.1, although we didn’t discuss it there.) This a difficult assessment, and it is an obvious candidate for an eventual sensitivity analysis.6

Solution

The reason this is a two-stage decision problem is that Acme can decide right away to stop development and abandon the 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 Finished.xlsx.) The first decision is whether to continue development. If “Yes,” the fixed development cost is incurred, so it is entered on this branch. Then there is a probability node for the technological success or failure. If

1363025 - Cengage Learning ©

it’s a failure, there are no further costs, but the fixed development cost is lost. If it’s a success, Acme must decide whether to market the product. From this point, the tree is exactly like the single-stage tree, except that the fixed development cost has been incurred.

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

■ Figure 6.12 Decision Tree with Possible Technological Failure

1363025 - Cengage Learning ©

■ Figure 6.13 Risk Profile from Best Strategy

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 f rom 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.

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

1363025 - Cengage Learning ©

■ Figure 6.14 Decision Tree with Larger Probability of Failure

Modeling Issues

We return to the probability distribution of eventual sales volume. The interpretation here is that at the time of the first decision, Acme has assessed what the market might look like after the development stage, which could be quite a while f rom 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

1363025 - Cengage Learning ©

the uncertainty about the eventual sales volume. If the company decides to market the product, it will incur fixed marketing costs of $4 million. To keep the model simple, we now assume that there are only two possible market outcomes, good or bad. The sales volumes for these two possible outcomes are 600,000 units and 100,000 units, and Acme assesses that their probabilities are 0.4 and 0.6. However, before making the ultimate decision, Acme has the option to hire a well-respected marketing research firm for $150,000. If Acme decides to use this option, the result will be a prediction of good or bad. That is, the marketing research firm will predict that either “We think the market for this product will be good” or “We think the market for this product will be bad.” Acme has used this firm before, so it has a sense of the prediction accuracy, as indicated in Table 6.1. Each row in this table indicates the actual market outcome, and each column indicates the prediction. If the actual market is good, the prediction will be good with probability 0.8 and bad with probability 0.2. If the actual market is bad, the prediction will be bad with probability 0.7 and good with probability 0.3. What should Acme do to maximize its EMV?

Table 6.1 Prediction Accuracy of Marketing Research Firm

Actual/Predicted Good Bad Good 0.8 0.2 Bad 0.3 0.7

Objective

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

Where Do the Numbers Come From?

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

1363025 - Cengage Learning ©

Solution

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

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

multistage_ decision_tree video.

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

1363025 - Cengage Learning ©

■ Figure 6.15 Skeleton of Decision Tree with Option to Buy Information

6.6a Bayes’ rule

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

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

■ Figure 6.16 Bayesian Updating Process

1363025 - Cengage Learning ©

Bayes’ rule: Frequency approach

Consider the following situation. During a routine physical exam, a middle-aged man named Joe tests positive for a certain disease. There were previously no indications that Joe had this disease, but the positive test sends him into a panic. He “knows” now that he has the disease. Or does he? Suppose that only 1% of all undiagnosed middle-aged men have this disease. Also, suppose the test Joe took is not entirely accurate. For middle-aged men who don’t have the disease, there is a 10% chance they will test positive (the false positive rate). For middle-aged men who have the disease, there is a 5% chance they will test negative (the false negative rate). So with a positive test result, what is the probability that Joe has the disease?

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

■ Figure 6.17 Frequency Approach for Updating Probabilities

We suspect that you are surprised by this low posterior probability. If so, you’re not alone. Highly trained physicians,

1363025 - Cengage Learning ©

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

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

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

This f requency approach is quite straightforward, even though it often yields surprising and unintuitive results. You are asked to apply it to several other scenarios in the problems. For now, we apply it to Acme’s problem in Example 6.3. The prior probabilities of good or bad markets are 40% and 60%. The probabilities for the accuracy of the predictions are given in Table 6.1. From these, we can follow 1000 similar products and predictions, exactly as we did with Joe’s disease. (We use 1000 rather than 10,000 for variety. The number chosen doesn’t influence the results at all.) The diagram appears in Figure 6.18. The probability of a good market increases f rom 40% to 64% with a good prediction, and it decreases to 16% with a bad prediction. Also, the probability that the prediction will be good is 50%.

1363025 - Cengage Learning ©

■ Figure 6.18 Frequencies for Updating Probabilities in Acme Problem

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

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

Bayes’ rule: Probability approach

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

The typical situation is that there are several outcomes such as “good market” and “bad market.” In general, denote these outcomes as O1 to On, assuming there are n possibilities. Then we start with n prior probabilities, P(O1) to P(On), that sum to 1. Next, we observe new information, I, such as a market prediction, and we want the n updated posterior probabilities, P(O1|I) to P(On|I), that sum to 1. We assume the “opposite”

1363025 - Cengage Learning ©

conditional probabilities, P(I|O1) to P(I|On), are given. In Bayesian terminology, these are called likelihoods. These likelihoods are the accuracy probabilities in Table 6.1. For example, one of these is the probability of seeing a good prediction, given that the market will be good.

However, these likelihoods are not what we need in the decision tree. Because time goes f rom left to right, we first need the (unconditional) probabilities of possible predictions and then we need the posterior probabilities of market outcomes, given the predictions (see Figure 6.15). So Bayes’ rule is a formal rule for turning these conditional probabilities around. It is given in Equation (6.1) for any i f rom 1 to n.

Bayes’ Rule

(6.1)

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

Denominator of Bayes’ Rule (Law of Total Probability)

(6.2)

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

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

Bayes’ Rule for Two Outcomes

(6.3)

1363025 - Cengage Learning ©

Implementing Bayes’ rule for acme

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

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

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

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

=B14*$B9/G$7

bayesian_revision video.

■ Figure 6.19 Bayes’ Rule Calculations for Acme Problem

1363025 - Cengage Learning ©

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

Completing the acme decision tree

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

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

Fundamental Insight

Making Sequential Decisions

Whenever you have a chance to make several sequential decisions and you will learn useful information between decision points, the decision you make initially depends on the decisions you plan to make in the future, and these depend on the information you will learn in the meantime.

1363025 - Cengage Learning ©

In other words, when you decide what to do initially, you should look ahead to see what your future options will be, and what your decision will be under each option. Such a contingency plan is typically superior to a myopic (short- sighted) plan that doesn’t take into account future options in the initial decision making.

■ Figure 6.20 Completed Decision Tree for Acme Problem

6-6b The value of information

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

1363025 - Cengage Learning ©

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

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

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

■ Figure 6.21 Completed Tree with a Smaller Fixed Marketing Cost

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

Equation for EVI

(6.4)

1363025 - Cengage Learning ©

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

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

Although the calculation of EVI is straightforward once the decision tree has been created, the decision tree itself requires a lot of probability assessments and Bayes’ rule calculations. These can be difficult, depending on the type of information available. Therefore, it is sometimes useful to ask how much any information could be worth, regardless of its form or accuracy. The result is called the expected value of perfect information, or EVPI, and it is given by Equation (6.5).

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

Equation for EVPI

(6.5)

To calculate EVPI for the Acme problem, forget about the marketing research firm and its possible prediction. All you have is the bottom of the tree in Figure 6.20, which uses Acme’s prior probabilities of market outcomes. So Acme’s EMV with no information is $1.4 million. To find the EMV with f ree perfect

1363025 - Cengage Learning ©

information, imagine that Acme will be told, truthfully, whether the market will be good or bad before the marketing decision has to be made. With probability 0.4, the prior probability, it will be told “good,” and with probability 0.6, it will be told “bad.” If it knows the market will be good, it should market the product because the net payoff is $10.8 million f rom sales minus $4 million for the fixed cost, or $6.8 million, a positive value. On the other hand, if it knows the market will be bad, it should not market the product because this would lead to a net payoff of $1.8 million f rom sales minus $4 million for the fixed cost, which is less than the 0 value it could get f rom not marketing the product. Therefore, according to Equation (6.5), EVPI is

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

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

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

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

1363025 - Cengage Learning ©

Presumably, other types of imperfect information could have the same effect and hence be worth something, but they can’t be worth more than $0.12 million.

Fundamental Insight

The Value of Information

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

6-6c Sensitivity analysis

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

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

The strategy region graph in Figure 6.22 shows how Acme’s EMVs f rom hiring and not hiring the firm vary as the prior

1363025 - Cengage Learning ©

probability of a good market varies f rom 0.2 to 0.6. Acme wants the largest EMV, which corresponds to the Yes line for small probabilities (up to about 0.48) and the No line for large probabilities. Why would Acme hire the firm only when the prior probability of a good market is small? The reason is that this is when the firm’s predictions are most helpful. On the right side of the graph, Acme is already fairly sure that the market is good, so the marketing research firm’s predictions in this case are less useful. Of course, the value of a sensitivity analysis such as this is that your intuition might not be so good. It tells you what you might not have been able to figure out on your own.

■ Figure 6.22 Sensitivity Analysis on Prior Probability of Good Market

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

1363025 - Cengage Learning ©

■ Figure 6.23 Sensitivity Analysis with a Data Table

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

■ Figure 6.24 Sensitivity Analysis for Marketing Decision after a Bad Prediction

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

1363025 - Cengage Learning ©

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

■ Figure 6.25 Two-Way Sensitivity Analysis

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

1363025 - Cengage Learning ©

Problems

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

Level A

11. In Example 6.2, Acme’s probability of technological success, 0.8, is evidently large enough to make “continue development” the best decision. How low would this probability have to be to make the opposite decision best?

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

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

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

15. In using Bayes’ rule for the presence of a disease (see Figure 6.17 and the file Bayes Rule for Disease.xlsx), we assumed that there are only two test results, positive or negative. Suppose there is another possible test result, “maybe.” The 2 × 2 range B9:C10 in the file should now be replaced by a 2 × 3 range, B9:D10, for positive, maybe, and negative (in that

1363025 - Cengage Learning ©

order). Let the corresponding probabilities in row 9 be 0.85, 0.10, and 0.05, and let those in row 10 be 0.05, 0.15, and 0.80. Redo the Bayes’ rule calculations with both the f requency approach and the probability approach.

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

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

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

19. Suppose you are a heterosexual white male and are going to be tested to see if you are HIV positive. Assume that if you are HIV positive, your test will always come back positive. Assume that if you are not HIV positive, there is still a 0.001 chance that your test will indicate that you are HIV positive. In reality, 1 of 10,000 heterosexual white males is HIV positive. Your doctor calls and says that you have tested HIV positive. He is sorry but there is a 99.9% (1 − 0.001) chance that you have HIV. Is he correct? What is the actual probability that you are HIV positive?

1363025 - Cengage Learning ©

Level B

20. If you examine the decision tree in Figure 6.12 (or any other decision trees f rom PrecisionTree), you will see two numbers (in blue font) to the right of each end node. The bottom number is the combined monetary value f rom following the corresponding path through the tree. The top number is the probability that this path will be followed, given that the best strategy is used. With this in mind, explain (1) how the positive probabilities following the end nodes are calculated, (2) why some of the probabilities following the end nodes are 0, and (3) why the sum of the probabilities following the end nodes is necessarily 1.

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

a. For which values of D should Acme fix the product and then market it, given that the first level of technological failure occurs?

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

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

22. The model in Example 6.3 has only two market outcomes, good and bad, and two corresponding predictions, good and bad. Modify the decision tree by allowing three outcomes and three predictions, good, fair, and bad. You can change the inputs to the model (monetary values and probabilities) in any reasonable way you like. Then you will also have to modify the Bayes’ rule calculations. You can decide

1363025 - Cengage Learning ©

whether it is easier to modify the existing tree or start f rom scratch with a new tree.

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

a. Use the posterior probabilities as prior probabilities in a second Bayes’ rule calculation. (Now prior means prior to the second test.) If Joe also tests positive in this second test, what is the posterior probability that he has the disease?

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

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

6-7 The Role of Risk Aversion

Rational decision makers are sometimes willing to violate the EMV maximization criterion when large amounts of money are at stake. These decision makers are willing to sacrifice some EMV to reduce risk. Are you ever willing to do so personally? Consider the following scenarios.

1363025 - Cengage Learning ©

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

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

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

Fundamental Insight

Risk Aversion

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

1363025 - Cengage Learning ©

sure dollar amount to the gamble. That is, you are willing to accept a sure dollar amount that is somewhat less than the EMV of the gamble, just to avoid risk. The more EMV you are willing to give up, the more risk averse you are.

6-7a Utility functions

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

■ Figure 6.26 Risk-Averse Utility Function

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

1363025 - Cengage Learning ©

branches and take maximums of expected utilities at decision branches. We will look at a numerical example later in this section.

6-7b Exponential utility

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

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

An exponential utility function has the following form:

Exponential utility

(6.6)

Here x is a monetary value (a payoff if positive, a cost if negative), U(x) is the utility of this value, and R > 0 is the risk tolerance. As the name suggests, the risk tolerance measures how much risk the decision maker will accept. The larger the

1363025 - Cengage Learning ©

value of R, the less risk averse the decision maker is. That is, a person with a large value of R is more willing to take risks than a person with a small value of R. In the limit, a person with an extremely large value of R is an EMV maximizer.

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

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

• Option 1: Receive no payoff at all. • Option 2: Receive a payoff of R dollars or incur a cost

of R/2 dollars, depending on the flip of a fair coin.

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

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

We illustrate the use of the expected utility criterion, and exponential utility in particular, in the following version of the

1363025 - Cengage Learning ©

Acme decision problem.

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

EXAMPLE 6.4 NEW PRODUCT DECISIONS WITH RISK AVERSION

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

Objective

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

Where Do the Numbers Come From?

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

Solution

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

1363025 - Cengage Learning ©

done here, you will see utility values rather than monetary values. The Certainty Equivalent option will be discussed shortly.

■ Figure 6.27 Risk Tolerance as an Extra Input

■ Figure 6.28 PrecisionTree Utility Function Dialog Box

The resulting tree that displays expected utilities appears in Figure 6.29. If you compare this tree to the EMV tree in Figure 6.11, you will see that Acme’s best strategy has been reversed. As an EMV maximizer, Acme should market the product. As a risk-averse expected utility maximizer, it should not market the product. The expected utility f rom marketing the product is negative, − 0.047, and the utility f rom not marketing the product is 0. (We formatted these as decimal numbers in the tree.

They are not expressed in dollars.) Neither of these expected utilities is meaningful in an absolute sense; only their relative

1363025 - Cengage Learning ©

values are important. The larger corresponds to the best decision.

■ Figure 6.29 Decision Tree Displaying Expected Utilities

PrecisionTree Tip

Calculations with utilities

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

6-7c Certainty equivalents

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

Recall f rom Figure 6.11 that the EMV for marketing the product is $1.074 million. However, this is not how the risk-averse Acme sees the decision to market the product. Acme now sees this gamble as equivalent to a loss of approximately $230,000. In

1363025 - Cengage Learning ©

other words, Acme would pay up to $230,000 to avoid this gamble. Of course, it doesn’t have to pay anything. The company can simply decide to abandon the product and receive a sure payoff of $0; hence, TRUE on the bottom branch indicates the best decision.

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

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

■ Figure 6.30 Decision Tree Displaying Certainty Equivalents

6-7d Is expected utility maximization used?

The previous discussion indicates that expected utility maximization is a somewhat involved task. The question, then, is whether the effort is justified. Theoretically, expected utility maximization might be interesting to researchers, but is it really used in the business world? The answer appears to be: not very often. For example, one article on the practice of decision making [see Kirkwood (1992)] quotes Ronald Howard —the same person we quoted previously—as having found risk aversion to be of practical concern in only 5% to 10% of business decision analyses. This same article quotes the president of a Fortune 500 company as saying, “Most of the

1363025 - Cengage Learning ©

decisions we analyze are for a few million dollars. It is adequate to use expected value (EMV) for these.”

Problems

Level A

25. Explain what it means in general when we say a risk- averse decision maker is willing to give up some EMV to avoid risk? How is this apparent in certainty equivalents of gambles?

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

27. You saw in Example 6.4 how Acme prefers to abandon the product when the risk tolerance in cell B12 is $5 million. This is despite the fact that the EMV f rom continuing with the product is well above 0. Using this same risk tolerance, experiment with the sales volume f rom a great market in cell C8 to see approximately how large it has to be for Acme to prefer the “continue with product” decision. (One way to do this is with a data table, using the TRUE/FALSE value in cell B14 as the single output.) With this sales volume, what is the certainty equivalent of the “continue with product” decision? How does it compare to the decision’s EMV? Explain the difference between the two.

Level B

28. Starting with the finished version of Example 6.2, change the decision criterion to “maximize expected utility,” using an exponential utility function with risk tolerance $5 million. Display certainty equivalents on the tree.

a. Keep doubling the risk tolerance until the company’s best strategy is the same as with the EMV criterion—continue with

1363025 - Cengage Learning ©

development and then market if successful. Comment on the implications of this.

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

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

6-8 Conclusion

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

SUMMARY OF KEY TERMS

TERM EXPLANATION EXCEL PAGES EQUATION

1363025 - Cengage Learning ©

Expected monetary value (EMV)

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

  228  

EMV criterion Choose the decision with the maximum EMV

  228  

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

  230  

Folding-back procedure

Calculation method for decision tree; starting at the right, take EMVs at probability nodes, maximums of EMVs at decision nodes

  231  

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

Has its own ribbon

236  

Contingency plan

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

  240  

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

  242  

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

  248 6.1

Law of total probability

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

  248  

Expected value of information (EVI)

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

  252 6.4

Expected value of perfect information (EVPI)

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

  252 6.5

Strategy region graph

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

PrecisionTree 254  

Tornado and spider graphs

Useful for seeing which inputs affect a selected

PrecisionTree 256  

1363025 - Cengage Learning ©

EMV the most Expected utility maximization

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

  258  

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

  258  

Exponential utility function, risk tolerance

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

  259 6.6

Certainty equivalent

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

  262  

PROBLEMS

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

Conceptual Questions

C.1. Your company needs to make an important decision that involves large monetary consequences. You have listed all of the possible outcomes and the monetary payoffs and costs f rom all outcomes and all potential decisions. You want to use the EMV criterion, but you realize that this requires probabilities and you see no way to find the required probabilities. What can you do?

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

C.3. In the previous question, suppose you have the option of receiving a check for $2700 instead of making the risky decision described. Would you make the risky decision, where you could lose $5000,

1363025 - Cengage Learning ©

or would you take the sure $2700? What would influence your decision?

C.4. In a classic oil-drilling example, you are trying to decide whether to drill for oil on a field that might or might not contain any oil. Before making this decision, you have the option of hiring a geologist to perform some seismic tests and then predict whether there is any oil or not. You assess that if there is actually oil, the geologist will predict there is oil with probability 0.85. You also assess that if there is no oil, the geologist will predict there is no oil with probability 0.90. Why will these two probabilities not appear on the decision tree? Which probabilities will be on the decision tree?

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

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

C.7. A potentially huge hurricane is forming in the Caribbean, and there is some chance that it might make a direct hit on Hilton Head Island, South

1363025 - Cengage Learning ©

Carolina, where you are in charge of emergency preparedness. You have made plans for evacuating everyone f rom the island, but such an evacuation is obviously costly and upsetting for all involved, so the decision to evacuate shouldn’t be made lightly. Discuss how you would make such a decision. Is EMV a relevant concept in this situation? How would you evaluate the consequences of uncertain outcomes?

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

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

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

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

LEVEL A

30. The SweetTooth Candy Company knows it will need 10 tons of sugar six months f rom now to implement its production plans. The company has essentially two options for acquiring the needed sugar. It can either buy the sugar at the going market price when it is needed, six months f rom now, or it can buy a futures contract now. The contract guarantees

1363025 - Cengage Learning ©

delivery of the sugar in six months but the cost of purchasing it will be based on today’s market price. Assume that possible sugar futures contracts available for purchase are for five tons or ten tons only. No futures contracts can be purchased or sold in the intervening months. Thus, SweetTooth’s possible decisions are to (1) purchase a futures contract for ten tons of sugar now, (2) purchase a futures contract for five tons of sugar now and purchase five tons of sugar in six months, or (3) purchase all ten tons of needed sugar in six months. The price of sugar bought now for delivery in six months is $0.0851 per pound. The transaction costs for five-ton and ten-ton futures contracts are $65 and $110, respectively. Finally, the company has assessed the probability distribution for the possible prices of sugar six months f rom now (in dollars per pound). The file P06_30.xlsx contains these possible prices and their corresponding probabilities.

a. Identify the decision that minimizes SweetTooth’s expected cost of meeting its sugar demand.

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

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

a. Identify the strategy that maximizes this tire manufacturer’s expected profit.

b. Perform a sensitivity analysis on the optimal

1363025 - Cengage Learning ©

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

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

a. Identify the strategy that maximizes the landowner’s expected net earnings f rom this opportunity.

b. Perform a sensitivity analysis on the optimal decision, letting each of the inputs vary one at a time plus or minus 25% f rom its base value, and summarize your findings. Which of the inputs appears to have the largest effect on the best solution?

33. Techware Incorporated is considering the introduction of two new software products to the market. The company has four options regarding these products: introduce neither product, introduce product 1 only, introduce product 2 only, or introduce both products. Research and development costs for products 1 and 2 are $180,000 and $150,000, respectively. Note that the first option entails no costs because research and development efforts have not yet begun. The success of these software products depends on the national economy in the coming year. The company’s revenues, depending

1363025 - Cengage Learning ©

on its decision and the state of the economy, are given in the file P06_33.xlsx. The probabilities of a strong, fair, or weak economy in the coming year are assessed to be 0.30, 0.50, and 0.20, respectively.

a. Identify the strategy that maximizes Techware’s expected net revenue.

b. Perform a sensitivity analysis on the optimal decision, letting each of the inputs vary one at a time plus or minus 25% f rom its base value, and summarize your findings. Which of the inputs appears to have the largest effect on the best solution?

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

a. Identify the strategy that maximizes the investor’s expected one-year earnings.

b. Perform a sensitivity analysis on the optimal decision, letting the amount available to invest and the risk-f ree return both vary, one at a time, plus or minus 100% f rom their base values, and summarize your findings.

35. A buyer for a large department store chain must place orders with an athletic shoe manufacturer six months prior to the time the shoes will be sold in the department stores. The buyer must decide on November 1 how many pairs of the manufacturer’s newest model of tennis shoes to order for sale during the coming summer season. Assume that each pair of this new brand of tennis shoes costs the department store chain $45 per pair. Furthermore, assume that each pair of these shoes can then be

1363025 - Cengage Learning ©

sold to the chain’s customers for $70 per pair. Any pairs of these shoes remaining unsold at the end of the summer season will be sold in a closeout sale next fall for $35 each. The probability distribution of consumer demand for these tennis shoes during the coming summer season has been assessed by market research specialists and is provided in the file P06_35.xlsx. Finally, assume that the department store chain must purchase these tennis shoes f rom the manufacturer in lots of 100 pairs.

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

b. Perform a sensitivity analysis on the optimal decision, letting the three monetary inputs vary one at a time over reasonable ranges, and summarize your findings. Which of the inputs appears to have the largest effect on the best solution?

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

37. You have sued your employer for damages suffered when you recently slipped and fell on an icy surface that should have been treated by your company’s physical plant department. Your injury was sufficiently serious that you, in consultation with

1363025 - Cengage Learning ©

your attorney, decided to sue your company for $500,000. Your company’s insurance provider has offered to settle this suit with you out of court. If you decide to reject the settlement and go to court, your attorney is confident that you will win the cas