Risk compensation is not a serious problem

Nov 28 JDN 2459547

Risk compensation. It’s one of those simple but counter-intuitive ideas that economists love, and it has been a major consideration in regulatory policy since the 1970s.

The idea is this: The risk we face in our actions is partly under our control. It requires effort to reduce risk, and effort is costly. So when an external source, such as a government regulation, reduces our risk, we will compensate by reducing the effort we expend, and thus our risk will decrease less, or maybe not at all. Indeed, perhaps we’ll even overcompensate and make our risk worse!

It’s often used as an argument against various kinds of safety efforts: Airbags will make people drive worse! Masks will make people go out and get infected!

The basic theory here is sound: Effort to reduce risk is costly, and people try to reduce costly things.

Indeed, it’s theoretically possible that risk compensation could yield the exact same risk, or even more risk than before—or at least, I wasn’t able to prove that for any possible risk profile and cost function it couldn’t happen.

But I wasn’t able to find any actual risk profiles or cost functions that would yield this result, even for a quite general form. Here, let me show you.

Let’s say there’s some possible harm H. There is also some probability that it will occur, which you can mitigate with some choice x. For simplicity let’s say that it’s one-to-one, so that your risk of H occurring is precisely 1-x. Since probabilities must be between 0 and 1, thus so must x.

Reducing that risk costs effort. I won’t say much about that cost, except to call it c(x) and assume the following:

(1) It is increasing: More effort reduces risk more and costs more than less effort.

(2) It is convex: Reducing risk from a high level to a low level (e.g. 0.9 to 0.8) costs less than reducing it from a low level to an even lower level (e.g. 0.2 to 0.1).

These both seem like eminently plausible—indeed, nigh-unassailable—assumptions. And they result in the following total expected cost (the opposite of your expected utility):

(1-x)H + c(x)

Now let’s suppose there’s some policy which will reduce your risk by a factor r, which must be between 0 and 1. Your cost then becomes:

r(1-x)H + c(x)

Minimizing this yields the following result:

rH = c'(x)

where c'(x) is the derivative of c(x). Since c(x) is increasing and convex, c'(x) is positive and increasing.

Thus, if I make r smaller—an external source of less risk—then I will reduce the optimal choice of x. This is risk compensation.

But have I reduced or increased the amount of risk?

The total risk is r(1-x); since r decreased and so did x, it’s not clear whether this went up or down. Indeed, it’s theoretically possible to have cost functions that would make it go up—but I’ve never seen one.

For instance, suppose we assume that c(x) = axb, where a and b are constants. This seems like a pretty general form, doesn’t it? To maintain the assumption that c(x) is increasing and convex, I need a > 0 and b > 1. (If 0 < b < 1, you get a function that’s increasing but concave. If b=1, you get a linear function and some weird corner solutions where you either expend no effort at all or all possible effort.)

Then I’m trying to minimize:

r(1-x)H + axb

This results in a closed-form solution for x:

x = (rH/ab)^(1/(b-1))

Since b>1, 1/(b-1) > 0.

Thus, the optimal choice of x is increasing in rH and decreasing in ab. That is, reducing the harm H or the overall risk r will make me put in less effort, while reducing the cost of effort (via either a or b) will make me put in more effort. These all make sense.

Can I ever increase the overall risk by reducing r? Let’s see.

My total risk r(1-x) is therefore:

r(1-x) = r[1-(rH/ab)^(1/(b-1))]

Can making r smaller ever make this larger?

Well, let’s compare it against the case when r=1. We want to see if there’s a case where it’s actually larger.

r[1-(rH/ab)^(1/(b-1))] > [1-(H/ab)^(1/(b-1))]

r – r^(1/(b-1)) (H/ab)^(1/(b-1)) > 1 – (H/ab)^(1/(b-1))

For this to be true, we would need r > 1, which would mean we didn’t reduce risk at all. Thus, reducing risk externally reduces total risk even after compensation.

Now, to be fair, this isn’t a fully general model. I had to assume some specific functional forms. But I didn’t assume much, did I?

Indeed, there is a fully general argument that externally reduced risk will never harm you. It’s quite simple.

There are three states to consider: In state A, you have your original level of risk and your original level of effort to reduce it. In state B, you have an externally reduced level of risk and your original level of effort. In state C, you have an externally reduced level of risk, and you compensate by reducing your effort.

Which states make you better off?

Well, clearly state B is better than state A: You get reduced risk at no cost to you.

Furthermore, state C must be better than state B: You voluntarily chose to risk-compensate precisely because it made you better off.

Therefore, as long as your preferences are rational, state C is better than state A.

Externally reduced risk will never make you worse off.

QED. That’s it. That’s the whole proof.

But I’m a behavioral economist, am I not? What if people aren’t being rational? Perhaps there’s some behavioral bias that causes people to overcompensate for reduced risks. That’s ultimately an empirical question.

So, what does the empirical data say? Risk compensation is almost never a serious problem in the real world. Measures designed to increase safety, lo and behold, actually increase safety. Removing safety regulations, astonishingly enough, makes people less safe and worse off.

If we ever do find a case where risk compensation is very large, then I guess we can remove that safety measure, or find some way to get people to stop overcompensating. But in the real world this has basically never happened.

It’s still a fair question whether any given safety measure is worth the cost: Implementing regulations can be expensive, after all. And while many people would like to think that “no amount of money is worth a human life”, nobody does—or should, or even can—act like that in the real world. You wouldn’t drive to work or get out of bed in the morning if you honestly believed that.

If it would cost $4 billion to save one expected life, it’s definitely not worth it. Indeed, you should still be able to see that even if you don’t think lives can be compared with other things—because $4 billion could save an awful lot of lives if you spent it more efficiently. (Probablyover a million, in fact, as current estimates of the marginal cost to save one life are about $2,300.) Inefficient safety interventions don’t just cost money—they prevent us from doing other, more efficient safety interventions.

And as for airbags and wearing masks to prevent COVID? Yes, definitely 100% worth it, as both interventions have already saved tens if not hundreds of thousands of lives.

How can we fix medical residency?

Nov 21 JDN 459540

Most medical residents work 60 or more hours per week, and nearly 20% work 80 or more hours. 66% of medical residents report sleeping 6 hours or less each night, and 20% report sleeping 5 hours or less.

It’s not as if sleep deprivation is a minor thing: Worldwide, across all jobs, nearly 750,000 deaths annually are attributable to long working hours, most of these due to sleep deprivation.

By some estimates, medical errors account for as many as 250,000 deaths per year in the US alone. Even the most conservative estimates say that at least 25,000 deaths per year in the US are attributable to medical errors. It seems quite likely that long working hours increase the rate of dangerous errors (though it has been difficult to determine precisely how much).

Indeed, the more we study stress and sleep deprivation, the more we learn how incredibly damaging they are to health and well-being. Yet we seem to have set up a system almost intentionally designed to maximize the stress and sleep deprivation of our medical professionals. Some of them simply burn out and leave the profession (about 18% of surgical residents quit); surely an even larger number of people never enter medicine in the first place because they know they would burn out.

Even once a doctor makes it through residency and has learned to cope with absurd hours, this most likely distorts their whole attitude toward stress and sleep deprivation. They are likely to not consider them “real problems”, because they were able to “tough it out”—and they are likely to assume that their patients can do the same. One of the primary functions of a doctor is to reduce pain and suffering, and by putting doctors through unnecessary pain and suffering as part of their training, we are teaching them that pain and suffering aren’t really so bad and you should just grin and bear it.

We are also systematically selecting against doctors who have disabilities that would make it difficult to work these double-time hours—which means that the doctors who are most likely to sympathize with disabled patients are being systematically excluded from the profession.

There have been some attempts to regulate the working hours of residents, but they have generally not been effective. I think this is for three reasons:

1. They weren’t actually trying hard enough. A cap of 80 hours per week is still 40 hours too high, and looks to me like you are trying to get better PR without fixing the actual problem.

2. Their enforcement mechanisms left too much opportunity to cheat the system, and in fact most medical residents simply became pressured to continue over-working and under-report their hours.

3. They don’t seem to have considered how to effect the transition in a way that won’t reduce the total number of resident-hours, and so residents got less training and hospitals were less served.

The solution to problem 1 is obvious: The cap needs to be lower. Much lower.

The solution to problem 2 is trickier: What sort of enforcement mechanism would prevent hospitals from gaming the system?

I believe the answer is very steep overtime pay requirements, coupled with regular and intensive auditing. Every hour a medical resident goes over their cap, they should have to be paid triple time. Audits should be performed frequently, randomly and without notice. And if a hospital is caught falsifying their records, they should be required to pay all missing hours to all medical residents at quintuple time. And Medicare and Medicaid should not be allowed to reimburse these additional payments—they must come directly out of the hospital’s budget.

Under the current system, the “punishment” is usually a threat of losing accreditation, which is too extreme and too harmful to the residents. Precisely because this is such a drastic measure, it almost never happens. The punishment needs to be small enough that we will actually enforce it; and it needs to hurt the hospital, not the residents—overtime pay would do precisely that.

That brings me to problem 3: How can we ensure that we don’t reduce the total number of resident-hours?

This is important for two reasons: Each resident needs a certain number of hours of training to become a skilled doctor, and residents provide a significant proportion of hospital services. Of the roughly 1 million doctors in the US, about 140,000 are medical residents.

The answer is threefold:

1. Increase the number of residency slots (we have a global doctor shortage anyway).

2. Extend the duration of residency so that each resident gets the same number of total work hours.

3. Gradually phase in so that neither increase needs to be too fast.

Currently a typical residency is about 4 years. 4 years of 80-hour weeks is equivalent to 8 years of 40-hour weeks. The goal is for each resident to get 320 hour-years of training.

With 140,000 current residents averaging 4 years, a typical cohort is about 35,000. So the goal is to each year have at least (35,000 residents per cohort)(4 cohorts)(80 hours per week) = 11 million resident-hours per week.

In cohort 1, we reduce the cap to 70 hours, and increase the number of accepted residents to 40,000. Residents in cohort 1 will continue their residency for 4 years, 7 months. This gives each one 321 hour-years of training.

In cohort 2, we reduce the cap to 60 hours, and increase the number of accepted residents to 46,000.

Residents in cohort 2 will continue their residency for 5 years, 4 months. This gives each one 320 hour-years of training.

In cohort 3, we reduce the cap to 55 hours, and increase the number of accepted residents to 50,000.

Residents in cohort 3 will continue their residency for 6 years. This gives each one 330 hour-years of training.

In cohort 4, we reduce the cap to 50 hours, and increase the number of accepted residents to 56,000. Residents in cohort 4 will continue their residency for 6 years, 6 months. This gives each one 325 hour-years of training.

In cohort 5, we reduce the cap to 45 hours, and increase the number of accepted residents to 60,000. Residents in cohort 5 will continue their residency for 7 years, 2 months. This gives each one 322 hour-years of training.

In cohort 6, we reduce the cap to 40 hours, and increase the number of accepted residents to 65,000. Residents in cohort 6 will continue their residency for 8 years. This gives each one 320 hour-years of training.

In cohort 7, we keep the cap to 40 hours, and increase the number of accepted residents to 70,000. This is now the new standard, with 8-year residencies with 40 hour weeks.

I’ve made a graph here of what this does to the available number of resident-hours each year. There is a brief 5% dip in year 4, but by the time we reach year 14 we’ve actually doubled the total number of available resident-hours at any given time—without increasing the total amount of work each resident does, simply keeping them longer and working them less intensively each year. Given that quality of work is reduced by working longer hours, it’s likely that even this brief reduction in hours would not result in any reduced quality of care for patients.


I have thus managed to increase the number of available resident-hours, ensure that each resident gets the same amount of training as before, and still radically reduce the work hours from 80 per week to 40 per week. The additional recruitment each year is never more than 6,000 new residents or 15% of the current number of residents.

It takes several years to effect this transition. This is unavoidable if we are trying to avoid massive increases in recruitment, though if we were prepared to simply double the number of admitted residents each year we could immediately transition to 40-hour work weeks in a single cohort and the available resident-hours would then strictly increase every year.

This plan is likely not the optimal one; I don’t know enough about the details of how costly it would be to admit more residents, and it’s possible that some residents might actually prefer a briefer, more intense residency rather than a longer, less stressful one. (Though it’s worth noting that most people greatly underestimate the harms of stress and sleep deprivation, and doctors don’t seem to be any better in this regard.)

But this plan does prove one thing: There are solutions to this problem. It can be done. If our medical system isn’t solving this problem, it is not because solutions do not exist—it is because they are choosing not to take them.

What’s wrong with police unions?

Nov 14 JDN 2459531

In a previous post I talked about why unions, even though they are collusive, are generally a good thing. But there is one very important exception to this rule: Police unions are almost always harmful.

Most recently, police unions have been leading the charge to fight vaccine mandates. This despite the fact that COVID-19 now kills more police officers than any other cause. They threatened that huge numbers of officers would leave if the mandates were imposed—but it didn’t happen.

But there is a much broader pattern than this: Police unions systematically take the side of individual police offers over the interests of public safety. Even the most incompetent, negligent, or outright murderous behavior by police officers will typically be defended by police unions. (One encouraging development is that lately even some police unions have been reluctant to defend the most outrageous killings by police officers—but this very much the exception, not the rule.)

Police unions are also unusual among unions in their political ties. Conservatives generally oppose unions, but are much friendlier toward police unions. At the other end of the spectrum, socialists normally love unions, but have distanced themselves from police unions for a long time. (The argument in that article that this is because “no other job involves killing people” is a bit weird: Ostensibly, the circumstances in which police are allowed to kill people are not all that different from the circumstances in which private citizens are. Just like us, they’re only supposed to use deadly force to prevent death or grievous bodily harm to themselves or others. The main thing that police are allowed to do that we aren’t is imprison people. Killing isn’t supposed to be a major part of the job.)

Police union also have some other weird features. The total membership of all police unions exceeds the total number of police officers in the United States, because a single officer is often affiliated with multiple unions—normally not at all how unions work. Police unions are also especially powerful and well-organized among unions. They are especially well-funded, and their members are especially loyal.

If we were to adopt a categorical view that unions are always good or always bad—as many people seem to want to—it’s difficult to see why police unions should be different from teachers’ unions or factory workers’ unions. But my argument was very careful not to make such categorical statements. Unions aren’t always or inherently good; they are usually good, because of how they are correcting a power imbalance between workers and corporations.

But when it comes to police, the situation is quite different. Police unions give more bargaining power to government officers against… what? Public accountability? The democratic system? Corporate CEOs are accountable only to their shareholders, but the mayors and city councils who decide police policy are elected (in most of the UK, even police commissioners are directly elected). It’s not clear that there was an imbalance in bargaining power here we would want to correct.

A similar case could be made against all public-sector unions, and indeed that case often is extended to teachers’ unions. If we must sacrifice teachers’ unions in order to destroy police unions, I’d be prepared to bite that bullet. But there are vital differences here as well. Teachers are not responsible for imprisoning people, and bad teachers almost never kill people. (In the rare cases in which teachers have committed murder, they have been charged to the full extent of the law, just as they would be in any other profession.) There surely is some misconduct by teachers that some unions may be protecting, but the harm caused by that misconduct is far lower than the harm caused by police misconduct. Teacher unions also provide a layer of protection for teachers to exercise autonomy, promoting academic freedom.

The form of teacher misconduct I would be most concerned about is sexual abuse of students. And while I’ve seen many essays claiming that teacher unions protect sexual abusers, the only concrete evidence I could find on the subject was a teachers’ union publicly complaining that the government had failed to pass stricter laws against sexual abuse by teachers. The research on teacher misconduct mainly focuses on other casual factors aside from union representation.

Even this Fox News article cherry-picking the worst examples of unions protecting abusive teachers includes line after line like “he was ultimately fired”, “he was pressured to resign”, and “his license was suspended”. So their complaint seems to be that it wasn’t done fast enough? But a fair justice system is necessarily slow. False accusations are rare, but they do happen—we can’t just take someone’s word for it. Ensuring that you don’t get fired until the district mounts strong evidence of misconduct against you is exactly what unions should be doing.

Whether unions are good or bad in a particular industry is ultimately an empirical question. So let’s look at the data, shall we? Teacher unions are positively correlated with school performance. But police unions are positively correlated with increased violent misconduct. There you have it: Teacher unions are good, but police unions are bad.

Does power corrupt?

Nov 7 JDN 2459526

It’s a familiar saying, originally attributed to the Lord Acton: “Power tends to corrupt, and absolute power corrupts absolutely. Great men are nearly always bad men.”

I think this saying is not only wrong, but in fact dangerous. We can all observe plenty of corrupt people in power, that much is true. But if it’s simply the power that corrupts them, and they started as good people, then there’s really nothing to be done. We may try to limit the amount of power any one person can have, but in any large, complex society there will be power, and so, if the saying is right, there will also be corruption.

How do I know that this saying is wrong?

First of all, note that corruption varies tremendously, and with very little correlation with most sensible notions of power.

Consider used car salespeople, stockbrokers, drug dealers, and pimps. All of these professions are rather well known for their high level of corruption. Yet are people in these professions powerful? Yes, any manager has some power over their employees; but there’s no particular reason to think that used car dealers have more power over their employees than grocery stores, and yet there’s a very clear sense in which used car dealers are more corrupt.

Even power on a national scale is not inherently tied to corruption. Consider the following individuals: Nelson Mandela, Mahatma Gandhi, Abraham Lincoln, and Franklin Roosevelt.

These men were extremely powerful; each ruled an entire nation.Indeed, during his administration, FDR was probably the most powerful person in the world. And they certainly were not impeccable: Mandela was a good friend of Fidel Castro, Gandhi abused his wife, Lincoln suspended habeas corpus, and of course FDR ordered the internment of Japanese-Americans. Yet overall I think it’s pretty clear that these men were not especially corrupt and had a large positive impact on the world.

Say what you will about Bernie Sanders, Dennis Kucinich, or Alexandria Ocasio-Cortez. Idealistic? Surely. Naive? Perhaps. Unrealistic? Sometimes. Ineffective? Often. But they are equally as powerful as anyone else in the US Congress, and ‘corrupt’ is not a word I’d use to describe them. Mitch McConnell, on the other hand….

There does seem to be a positive correlation between a country’s level of corruption and its level of authoritarianism; the most democratic countries—Scandinavia—are also the least corrupt. Yet India is surely more democratic than China, but is widely rated as about the same level of corruption. Greece is not substantially less democratic than Chile, but it has considerably more corruption. So even at a national level, power is the not the only determinant of corruption.

I’ll even agree to the second clause: “absolute power corrupts absolutely.” Were I somehow granted an absolute dictatorship over the world, one of my first orders of business would be to establish a new democratic world government to replace my dictatorial rule. (Would it be my first order of business, or would I implement some policy reforms first? Now that’s a tougher question. I think I’d want to implement some kind of income redistribution and anti-discrimination laws before I left office, at least.) And I believe that most good people think similarly: We wouldn’t want to have that kind of power over other people. We wouldn’t trust ourselves to never abuse it. Anyone who maintains absolute power is either already corrupt or likely to become so. And anyone who seeks absolute power is precisely the sort of person who should not be trusted with power at all.

It may also be that power is one determinant of corruption—that a given person will generally end up more corrupt if you give them more power. This might help explain why even the best ‘great men’ are still usually bad men. But clearly there are other determinants that are equally important.

And I would like to offer a different hypothesis to explain the correlation between power and corruption, which has profoundly different implications: The corrupt seek power.

Donald Trump didn’t start out a good man and become corrupt by becoming a billionaire or becoming President. Donald Trump was born a narcissistic idiot.

Josef Stalin wasn’t a good man who became corrupted by the unlimited power of ruling the Soviet Union. Josef Stalin was born a psychopath.

Indeed, when you look closely at how corrupt leaders get into power, it often involves manipulating and exploiting others on a grand scale. They are willing to compromise principles that good people wouldn’t. They aren’t corrupt because they got into power; they got into power because they are corrupt.

Let me be clear: I’m not saying we should compromise all of our principles in order to achieve power. If there is a route by which power corrupts, it is surely that. Rather, I am saying that we must maintain constant vigilance against anyone who seems so eager to attain power that they will compromise principles to do it—for those are precisely the people who are likely to be most dangerous if they should achieve their aims.

Moreover, I’m saying that “power corrupts” is actually a very dangerous message. It tells good people not to seek power, because they would be corrupted by it. But in fact what we actually need in order to get good people in power is more good people seeking power, more opportunities to out-compete the corrupt. If Congress were composed entirely of people like Alexandria Ocasio-Cortez, then the left-wing agenda would no longer seem naive and unrealistic; it would simply be what gets done. (Who knows? Maybe it wouldn’t work out so well after all. But it definitely would get done.) Yet how many idealistic left-wing people have heard that phrase ‘power corrupts’ too many times, and decided they didn’t want to risk running for office?

Indeed, the notion that corruption is inherent to the exercise of power may well be the greatest tool we have ever given to those who are corrupt and seeking to hold onto power.