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

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

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

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

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

CHAPTER 6 Decision Making Under Uncertainty

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6-1 Introduction    2 4 3

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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6-2 elements of Decision analysis    2 4 5

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

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

We now discuss these elements in some generality.1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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6-3 eMV and Decision trees    2 4 7

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

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

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

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

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

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

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

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

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

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

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

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

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

5SUMPRODUCT(A3:A5,B3:B5)

Figure 6.1 EMV Calculations in Excel

1 2 3 4 5 6 7

A B C D E F G H Decision 1

Payoff/Cost $50,000 $10,000 –$5,000

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

0.1 $5,000 −$1,000

0.6 $3,000 1 0.40.2

0.7

Probability Payoff/Cost Probability Probability Decision 2

Payoff/Cost Decision 3

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