Bayesian analysis, probabilities of accidents and the Monty Hall Problem

In my research methods class today we talked about the difference between Classical and Bayesian analysis. In Classical analysis you use available statistics to make inferences about something, while in Bayesian analysis you use other information to interpret available statistics.

Consider the following silly but clarifying example: Suppose I show you a clear bowl with 5 red balls and 5 blue balls, and suppose I ask you to close your eyes and pull out a ball. What is the probability that you will successfully pull out a red ball? A classical statistician will say, correctly, 50 percent, since 5 of the 10 balls are red. However, a Bayesian may say something different if he knows something about who put the game together. For instance, if the Bayesian knows that I am a jokester and have a history of gluing red balls to bowls, then the Bayesian will say the probability that you can “successfully” pull out a red ball will be much less than 50 percent.

Here is another perhaps more relevant example: Suppose 60 percent of all vehicle accidents involve drivers using cell phones. Can we conclude that there is a greater than 50 percent chance that someone using a cell phone while driving will be in an accident? A Classical thinker may conclude “yes,” since more than half of all accidents involve cell phones. Politicians think this way, too, because they promote laws that restrict our ability to use cell phones while driving using these kinds of numbers. However, a Bayesian will want to consider other information, such as the percent of all drivers in accidents and the percent of cell phone use by drivers not in accidents.

For example, if 5 percent of drivers are in an accident on any given day and if the percent of non-accident drivers who use cell phones while driving is 30, then what is the probability that someone will be in an accident given that they are using a cell phone? It turns out to be a lot less than 60 percent–about 9.5 percent. (The formula is (0.05)(0.6)/[(0.05)(0.6)+(0.95)(0.3)] for anyone who wants to check my math.) Of course, to make the point that one should not use cell phones while driving, we should calculate the probability that someone will be in an accident given that they are not using a cell phone. This is less than 3 percent. (The formula is (0.05)(0.4)/[(0.05)(0.4)+(0.95)(0.7)].) So, using a cell phone while driving almost doubles the chance of being in an accident, while not using a cell phone decreases the likelihood of being in an accident by about 40 percent. Clearly one is better off not using a cell phone while driving. Wikipedia has a useful discussion of the math behind the analysis here.

These numbers are hypothetical. I do not have actual data on the percent of cars in accidents and the percent of drivers using cell phones, etc. The point is that we can obtain a better analysis by considering all relevant information carefully. That is, it is not always correct to draw conclusions from data we have presented to us. Moreover, biases can impair our ability to understand what is going on around us, unless we are careful in how we draw conclusions. We see the wisdom in this from observing how people behave during presidential elections. A person’s bias in favor of a particular candidate seems to make him or her impervious to evidence that the candidate is a lying and immoral buffoon.

This type of analysis is also helpful when considering medical tests. If 2 percent of the population has a disease and the doctor gives you a diagnosis that you have the disease, then what is the probability you really have it given that the doctor said you did? The answer depends on how accurate the medical test is. For example, if the medical test is accurate 95 percent of the time, then the chance you have the disease is only about 30 percent. In contrast, if the medical test is accurate only 80 percent of the time, then the chance you have the disease is really less than 8 percent. In either case, I would get a second opinion.

85-doorsWe had fun with this example in class: Suppose you are on the game show “Let’s Make a Deal.” Monty Hall, the show’s host, shows you three doors, A, B and C. Behind one is a new car, behind the other two are goats. You are asked to pick a door. You pick door A. Monty opens door B to reveal a goat and then offers to allow you to switch to door C or stay with your choice of door A. Should you switch or stay? Someone asked Marilyn vos Savant, a woman listed in the Guinness Book of World Records as having the highest IQ, this question. She gave her answer in 1990 in a Parade magazine column. It generated thousands of letters, many from PhDs saying she was wrong. Her column and responses are here. It’s funny to read the reactions of so-called academics. Answer the “stay or switch” question first before reading her response. To play the game to convince yourself that she was right, see this online app here. Play it many times by staying and see how often you win. Then play it many times by switching each time to see how often you win. You’ll find that the probability of winning the car doubles from one-third to two-thirds by switching. There’s also an official “Let’s Make a Deal” website.

Given the choice between watching “Let’s Make a Deal” and presidential candidates debate, I’ll place my odds on the game show.


Nobel Prize in Economic Sciences and ethical insights from contracting theory

kva_logo_09The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 2016, awarded by the Royal Swedish Academy, was given to Oliver Hart, a professor at Harvard University, and Bengt Holmström, who is at the Massachusetts Institute of Technology. They were awarded the prize “for their contributions to contract theory.” While both made significant and award-worthy contributions that changed the way we think about contracting and the organization of firms, their insights also have ethical implications.

Hart’s work is based on the idea that contracts are inherently incomplete. That is, it is not possible to design a contract that specifies outcomes for all possible contingencies and states of the world, since there will be some (or perhaps many) situations that are unforeseeable or so complicated that they cannot be reasonably accounted for in contracts. In other words, all contracts contain gaps or missing provisions. Because contracts are incomplete, someone has to be charged with making decisions and reaping the consequences of those decisions (whether good or bad) in those cases not dictated by contract. This is another way of saying that someone has to have authority. Hart’s theories help us understand what authority means, particularly in terms of ownership and decision-making rights.

Holmström’s work focuses on the idea that information is often hidden or not available to everyone. This is, it is not possible for everyone to have all relevant information when they are making important decisions or for bosses to know perfectly what each of their employees is doing in the firm. The implication is that people will have to rely on others for information or, more generally, to do things on their behalf. One person relying on another can create what is known as the Principal-Agent Problem. An example is when employers have to rely on employees for reports on how the business is operating or to take actions on behalf of the firm, such as negotiate with customers. Holmström’s theories provide insights into how contracts should be structured so as to provide the necessary incentives for people to provide accurate information or to act in the interest of their employer or the organization to which they belong.

Ethical problems arise when there is a conflict of interests, values or rights. Interests are things we care about; values are ends in themselves or ideals to which we aspire; and rights are entitlements to things we obtain or do. Hart and Holmström’s ideas provide clarity about where and why these conflicts can arise in the business world. For example, the rights of firm owners are in conflict with the interests of workers; firm owners want workers to work hard and generate increased profits for the owners, but doing so is costly for workers, who have an interest in relaxing and taking long lunch breaks. Employees have an interest in receiving higher wages and better benefits, but these conflict with the interests of employers who pay for them. Interestingly, Holmström also demonstrated that there is a conflict between the interests of firm owners to maximize profits and their interest in running the firm efficiently. In other words, it is not possible for a firm owner to provide an efficient incentive system for workers and at the same time generate the highest profits possible. To have one the owner must give up on the other.

Contracting theory that builds on Hart and Holmström’s work is based on the assumption that people will try to take advantage of their situation. In other words, people will try to lie, cheat and steal unless they have an incentives not to, which incentives are governed by contracts. This raises the question of whether contracts would be necessary if everyone were perfectly ethical, honest and forthright. It’s an interesting academic question only, because we don’t live in a world where everyone is perfectly ethical, honest and forthright. Yet. Until then, we have to use the best theories for organizing ourselves. Fortunately, Hart and Holmström have given us a solid foundation on which to do this.

When is an increase in price fair?

In my previous post I wrote about the fairness of drug companies that dramatically increase the price of their products. I suggested, and public reaction confirmed, that these price increases are considered unfair.

When is a price increase fair? When is it unfair?

The economic principle of profit maximization tells us that firms ought to increase prices when there is an increase in demand or a decrease in supply, regardless of whether the change is short-term or permanent. So if a hurricane is closing in on the East coast of the U.S., then suppliers of lumber, bottled water, gasoline and other supplies that people will need should and will increase the prices of these things … a lot. But is that fair?

Daniel Kahneman, Jack L. Knetsch and Richard Thaler published a paper on this topic 30 years ago in the American Economic Review. They argued that people’s perceptions of fairness will (or ought to) constrain impulses to take advantage of short-run increases in demand or other reasons to increase prices, under certain conditions.

Their discussion entails two elements. One has to do with reference points and the other has to do with reasons for gaining at the expense of others.

First, a reference point is the basis upon which people create expectations about the fairness of a transaction or a change in prices. If transactions or price changes are consistent with the reference point, then people will judge the transactions or price changes as fair. In other words, reference transactions and reference profits are considered fair. If there is a deviation from the reference point, then assessments of fairness will be made based on whether good reasons exist for the deviations.

The authors define the reference points as follows:

Market prices, posted prices, and the history of previous transactions between a firm and a transactor can serve as reference transactions. When there is a history of similar transaction between firm and transactor, the most recent price, wage, or rent will be adopted for reference unless the terms of the previous transaction were explicitly temporary. For new transactions, prevailing competitive prices or wages provide the natural reference.

For example, if a gas station has been selling gas for $1.99 a gallon for several weeks, then the reference price is $1.99 a gallon, and people will expect that to be the price the next time they get gas. If the price of gasoline increases, say to $2.09 a gallon, then people will likely consider the price increase unfair unless they understand there to be a good reason for the change. Good reasons have reference points, too. For example, raising prices at the expected rate of inflation is not considered unfair, since inflation-based price increases can be a reference point. We have historical experience with that, so there is no basis for claiming unfairness. The same about price changes in gasoline. If the reference point for price changes is 5 cents a gallon, then people won’t be bothered by finding the price of gasoline is higher by 5 cents the next time they buy gas. But if prices increase by more than 5 cents, then they might believe or expect something unfair is happening at their expense.

Understanding reference points for transactions, pricing and profits can help us understand why unfairness might be claimed when businesses increase prices. For example, if employees had previous experiences of getting pay increases when competitors raised their employees’ wages, then a reference point is created for employees. But if the employer has not historically increased wages when their competitors have, then the reference point for the employer will differ from that of the employees, resulting in disagreements about fairness.

Second, an important principle of fairness is that one should not gain by imposing a cost or harm on others.

Raising the price of lumber, bottled water, gasoline and other supplies that people will need when a hurricane is imminent can be considered imposing a harm on others to acquire a short-term gain. It doesn’t matter that economics dictates that price increases are needed to resolve shortages that will arise when there is a sudden increase in demand. There is usually no viable reference point for such behavior.

Raising prices because a business experiences an increase in costs is different. Such behavior is not considered as imposing a harm on others merely to benefit at their expense. Businesses are not gaining substantially, if at all, when they increase prices to cover their rising costs.

The implication is that people are willing to accept as fair an increase in price to cover rising costs. They also consider it fair for businesses to maintain prices when costs decline. But people consider raising prices when demand increases or without explanation or justification to be unfair.

Most people have no reference point for drug prices rising 5000 percent or even 500 percent. Such behavior is very unfair.

Because they can, but should they?

Mylan is the company that produces the EpiPen, a device that injects a measured dose of epinephrine when someone has a severe allergic reaction. Mylan didn’t invent the drug or device. The company acquired it in 2007 from Merck, which bought the rights for the drug years earlier from another company. By some estimates, Mylan’s EpiPen controls roughly 90 percent of the market for epinephrine injection devices. When Mylan purchased the rights to the EpiPen in 2007, the drug cost about $100 for a two-pen set. It currently retails for more than $600.

Mylan isn’t the only company to buy a drug and then dramatically increase its price. Earlier this year, the Wall Street Journal (as well as other news outlets) reported on pharmaceutical companies that buy rival’s drugs and then jack up the prices. A noteworthy example is Martin Shkreli, a hedge fund manager and CEO of Turing Pharmaceuticals, who bought the drug Daraprim and raised its price from about $13 a dose to $750. The drug is used to treat a variety of infections and other diseases. Turing bought the rights to the drug from a company, which bought the rights to the drug from another company, which bought the rights to the drug from another company, …

Why did Mylan increase the price of the EpiPen? Why did Turing increase the price of the drug Daraprin? Because they can. The companies control exclusive rights to the drug and the demand for the drugs are highly inelastic. First, by having exclusive rights over the drugs and with little if any competition, they possess monopoly power. This means they can act as price makers rather than price takers. Second, as I explained to my microeconomics students today, demand for the drugs is inelastic because there are few substitutes to them and they are necessary. This means that consumers will not (or cannot) be very responsive to large changes in prices. If you are subject to severe allergic reactions, you probably will not forgo the drug if its price increases. You will grumble and complain but buy it anyway. Thus, the combination of monopoly power and inelastic demand makes raising prices economically rational.

The drug companies and other commentator point to other reasons for high drug prices. Some argue that the patent system and long and costly regulatory approval processes are to blame. While these affect the initial costs for many drugs, they don’t explain why the price rose so quickly years after the drug has been on the market. If Daraprim was profitable at $13.50 a dose, then patenting and regulatory costs won’t explain the 5,000 percent increase in its price after Turing bought it.

According to an article in today’s Wall Street Journal, drugmakers are pointing a finger at middlemen for rising drug prices. Drug company executives say that the system is to blame. Everyone has to take a cut, such as pharmacy-benefits managers. Drug companies say that they have to offer increasingly larger rebates to pharmacy-benefits managers to induce them to accept their drugs as part of their company’s health plans. These benefits managers, in turn, are blaming drug prescription services and health insurers. While there might be some merit here, it’s hard to believe that middlemen and insurance companies are largely responsible for the dramatic increases in drug prices. I can’t imagine that a benefits manager will say “no” to the only drug that is available to treat severe allergic reactions or some infections. Drug companies won’t have to offer large rebate inducements if their drug prices were not already very high.

To be sure, the problem is complicated. Should we force drug companies to price their products “reasonably”? What is a reasonable price for a life-saving drug? Who’s to say that $20 or $30 or even $50 is unreasonable for Daraprim? These companies also employ thousands of workers and their stocks are part of savings, retirement and other investment portfolios for many people. If we mandate lower prices for drugs, then what happens to drug company employees and investors?

If drug companies can raise prices, and justify the price increases on economic grounds, then should they? Economic considerations are important, but they are not the only values that matter. Fairness matters too. Is it fair to ask patients to pay 5000 percent more for a lifesaving drug?

Fairness can be more difficult to justify than economic rationale. But we can simplify things. While there are many bases for arguing “fairness,” all of them are grounded in expectations. When our expectations are met, then we have little grounds for arguing unfairness. However, when expectations are not met, people typically feel justified in claiming unfairness. Consumers with a history and experience in paying $13.50 for a drug to treat infections will continue to expect that prices will be about the same the next time they buy the drug. While most people can expect gradual increases in prices, for instance due to inflation, there is no reasonable argument anyone can make that would convince me that consumers would expect a 5000 percent increase for Daraprim or a 500 percent increase for the EpiPen in a relatively short period of time. If drug companies want to increase prices more than what consumers expect, then the companies need to speak directly to consumers and change their expectations. That is fair. But I expect that will be quite a challenge for drug companies.


Are we better off today than our grandparents were in the 1950s?

I finished reading Robert Gordon’s massive book, The Rise and Fall of American Growth: The U.S. Standard of Living Since the Civil War. And it is massive, topping 750 pages including notes and references.

Gordon’s thesis is simple: American growth and the improvement in the standard of living for people living between 1920 and 1970 was better than the improvement in growth and living standards for people living between 1870 and 1920 and for those living between 1970 and today. Stated differently, while my grandparents had it better than their grandparents, the improvements that my grandparents experienced were more substantial than the improvements I have seen.

The reason is that the inventions and innovations that occurred between 1920 and 1950 were more significant and had a greater impact on labor productivity and personal well-being than the inventions and innovations that occurred before and after that time period. Gordon systematically reviews “all” of them (hence the long book). He considers improvements in food, housing, transportation, communication, health, working conditions, financial services.

Here are simple examples: Going from no cars in the late 19th century to cars available to most households in the early 20th century had a greater impact on people and businesses than going from cars then to cars with airbags and rearview cameras today. Similarly, going from no electric lighting to electric lighting had a greater impact than going from electric lighting to more efficient electric lighting. Going from no penicillin to penicillin was more profound than going from penicillin to more powerful penicillin. Going from no telephones to telephones was more important than going from telephones to mobile phones today. The list goes on and on. People living between 1920 and 1970 saw more radical changes in their lives than people living since the 1970s.

What made the book fun was the presentation of stories and historical examples he gave. Even if you are not interested in the economics behind his thesis, you can read the chapters and gain insights about how life improved for people during the early half of the 20th century.

Gordon raises some warnings. It is not likely that we will see the kind of growth that the economy and people experienced from 1920 to 1950–what he refers to as the “Great Leap Forward”–anytime soon. And worse, the widening level of inequality we are seeing will only make things worse. The growth rate of real income for the top 10 percent and for the bottom 90 percent of persons in the U.S. was about the same in the mid 20th century–about 2.5 percent per year. But since the 1970s the top 10 percent of wage earners has seen real wages increase more than 1.4 percent annually while real wages for the bottom 90 percent of workers have decreased during the same time period. This is not good for a healthy, growing and productive economy.

Trash bins, staples and more unintended consequences

Like most university departments and business offices generally, my department has a workstation near the copy machine. It is a large wood table with staplers, tape, paperclips, pens, a cutting board and other items one might need to manage copies, reports, etc. While there have always been the occasional used, bent staple left on the table, I have noticed a substantial increase in the number of discarded staples on the table. Why?

The University of Missouri has initiated a “low waste initiative” (a report about the program in the campus newspaper is here). The intent is to reduce general waste and to promote recycling. Trash cans from office spaces have been removed and replaced with a blue recycling bucket. Attached to the bucket is a small black bin for non-recyclable waste. These black bins are about the size of a large cupholder you might find in a car. Unfortunately, they are not attached to all recycling bins, and there are none near the copy machine workstation or office commons. So what do people do when they remove staples from paper? Ideally they should walk down the hall to a “black bin” to discard the staples. But that is not happening. Since regular trash bins at the workstation are now gone, the staples end up left on the workstation table.

An accumulation of discarded stables on workstation tables is an unintended consequence of the University’s low waste trash program.

Discarded staples are not a threat to world peace and they don’t contribute to climate change, but they are a nuisance and a minor hazard. They are sharp and are not sanitary, so getting inadvertently poked by one could become a painful problem.

There are always consequences for changes in policies, programs and incentives. While we can be careful and thoughtful in considering “all” the ramifications of changes we make to rules and policies, there will often be unintended consequences, both good and bad.

A working paper by National Bureau of Economic Research scholars identifies an interesting but negative unintended consequence to a program designed to promote greater school attendance. In their report, the authors described an experiment in which students were rewarded for meeting an attendance threshold at school. The reward produced the expected results. The reward increased school attendance. However, when the reward program ended, there were unintended effects. As stated by the authors:

Among students with high baseline attendance, the incentive had no effect on attendance after it was discontinued, and test scores were unaffected. Among students with low baseline attendance, the incentive lowered post-incentive attendance, and test scores decreased. For these students, the incentive was also associated with lower interest in school material and lower optimism and confidence about their ability. This suggests incentives might have unintended long-term consequences for the very students they are designed to help the most.

So the introduction of an incentive had the unintended consequence of driving out or reducing intrinsic motivation, a topic I have studied (here).

While we may not be able to anticipate all unintended consequences — that’s why they are “unintended” — we can probably do better than we are. And when we identify them, and if the consequences are significant enough, then we should consider revisions to the programs and policies we have implemented. I don’t know if the University of Missouri will be changing its “low waste program” anytime soon, but it would sure be nice to have a more convenient way of discarding used staples.

Business leadership and the making and punishing of unethical employees

A study published in the current issue of Business Ethics Quarterly links ethical leadership with improved engagement of employees at work, greater employee voice and lower intentions for employees to exit. In other works, when employees perceive or know their leaders to be ethical, they are more likely to feel good about being at work, more willing to communicate their opinions, recommendation, concerns or ideas to their supervisors, and less likely to leave or intend to the leave the business.

In this context, an ethical leader is someone who is a moral person and who models high moral standards at work. The specific indicators of ethical leadership used in the BEQ paper draw from research by scholars at Pennsylvania State University. If valid, the indicators are informative. There are 10 of them. Ethical leaders

  • conduct their personal lives in an ethical manner
  • make fair and balanced decisions
  • can be trusted
  • ask what the right is when making decisions
  • listen to their employees
  • discuss business ethics and values with their employees
  • have the best interest of their employees in mind
  • set an example of behaving ethically at work
  • discipline employees who violate ethical standards
  • define success by the way results are obtained in addition to results.

I would add one more item to the list. When designing and implementing performance measures and incentives, ethical leaders are careful to ensure that they are promoting incentives rather than pressures to perform. The line between incentive and pressure can be thin. Leaders who are not careful may find that their efforts to motivate workers create pressures for them to lie, cheat or steal.

The CEO of Wells Fargo is learning this lesson the hard way. According to the Wall Street Journal’s report of John Stumpf’s testimony during a Senate Banking Committee hearing yesterday (September 21), the Bank is accused “of fostering a culture where low-paid branch employees were pressured to meet impossible sales quotas to keep their jobs, and so signed up customers for products without their knowledge.” Pressure does not create an environment where employees behave ethically. Even well-meaning employees may find the temptation to fudge numbers or behave inappropriately too strong in such an environment. The Bank reported that it fired more than 5,000 employees for wrongdoing.

So, Wells Fargo created unethical employees and then punished them.

Reminds me of the statement by Thomas More in his book, Utopia, made famous by Drew Barrymore’s character Danielle (aka Cinderella) in the movie Ever After. Danielle is arguing with Henry, the Prince of France, for the release of her servant, who is bound with other poor and destitute prisoners for the America’s. Here is the exchange:

Danielle: A servant is not a thief, your Highness, and those who are cannot help themselves.

Henry: Really! Well then by all means, enlighten us.

Danielle (quoting More): If you suffer your people to be ill-educated, and their manners corrupted from infancy, and then punish them for those crimes to which their first education disposed them, what else is to be concluded, sire, but that you first make thieves and then punish them?

Henry: Well, there you have it. Release him.

That’s quite a commentary about one of the nation’s most prominent banks.