Signaling and the Curse of Knowledge

Jan 3 JDN 2459218

I received several books for Christmas this year, and the one I was most excited to read first was The Sense of Style by Steven Pinker. Pinker is exactly the right person to write such a book: He is both a brilliant linguist and cognitive scientist and also an eloquent and highly successful writer. There are two other books on writing that I rate at the same tier: On Writing by Stephen King, and The Art of Fiction by John Gardner. Don’t bother with style manuals from people who only write style manuals; if you want to learn how to write, learn from people who are actually successful at writing.

Indeed, I knew I’d love The Sense of Style as soon as I read its preface, containing some truly hilarious takedowns of Strunk & White. And honestly Strunk & White are among the best standard style manuals; they at least actually manage to offer some useful advice while also being stuffy, pedantic, and often outright inaccurate. Most style manuals only do the second part.

One of Pinker’s central focuses in The Sense of Style is on The Curse of Knowledge, an all-too-common bias in which knowing things makes us unable to appreciate the fact that other people don’t already know it. I think I succumbed to this failing most greatly in my first book, Special Relativity from the Ground Up, in which my concept of “the ground” was above most people’s ceilings. I was trying to write for high school physics students, and I think the book ended up mostly being read by college physics professors.

The problem is surely a real one: After years of gaining expertise in a subject, we are all liable to forget the difficulty of reaching our current summit and automatically deploy concepts and jargon that only a small group of experts actually understand. But I think Pinker underestimates the difficulty of escaping this problem, because it’s not just a cognitive bias that we all suffer from time to time. It’s also something that our society strongly incentivizes.

Pinker points out that a small but nontrivial proportion of published academic papers are genuinely well written, using this to argue that obscurantist jargon-laden writing isn’t necessary for publication; but he didn’t seem to even consider the fact that nearly all of those well-written papers were published by authors who already had tenure or even distinction in the field. I challenge you to find a single paper written by a lowly grad student that could actually get published without being full of needlessly technical terminology and awkward passive constructions: “A murian model was utilized for the experiment, in an acoustically sealed environment” rather than “I tested using mice and rats in a quiet room”. This is not because grad students are more thoroughly entrenched in the jargon than tenured professors (quite the contrary), nor that grad students are worse writers in general (that one could really go either way), but because grad students have more to prove. We need to signal our membership in the tribe, whereas once you’ve got tenure—or especially once you’ve got an endowed chair or something—you have already proven yourself.

Pinker seems to briefly touch this insight (p. 69), without fully appreciating its significance: “Even when we have an inlkling that we are speaking in a specialized lingo, we may be reluctant to slip back into plain speech. It could betray to our peers the awful truth that we are still greenhorns, tenderfoots, newbies. And if our readers do know the lingo, we might be insulting their intelligence while spelling it out. We would rather run the risk of confusing them while at least appearing to be soophisticated than take a chance at belaboring the obvious while striking them as naive or condescending.”

What we are dealing with here is a signaling problem. The fact that one can write better once one is well-established is the phenomenon of countersignaling, where one who has already established their status stops investing in signaling.

Here’s a simple model for you. Suppose each person has a level of knowledge x, which they are trying to demonstrate. They know their own level of knowledge, but nobody else does.

Suppose that when we observe someone’s knowledge, we get two pieces of information: We have an imperfect observation of their true knowledge which is x+e, the real value of x plus some amount of error e. Nobody knows exactly what the error is. To keep the model as simple as possible I’ll assume that e is drawn from a uniform distribution between -1 and 1.

Finally, assume that we are trying to select people above a certain threshold: Perhaps we are publishing in a journal, or hiring candidates for a job. Let’s call that threshold z. If x < z-1, then since e can never be larger than 1, we will immediately observe that they are below the threshold and reject them. If x > z+1, then since e can never be smaller than -1, we will immediately observe that they are above the threshold and accept them.

But when z-1 < x < z+1, we may think they are above the threshold when they actually are not (if e is positive), or think they are not above the threshold when they actually are (if e is negative).

So then let’s say that they can invest in signaling by putting some amount of visible work in y (like citing obscure papers or using complex jargon). This additional work may be costly and provide no real value in itself, but it can still be useful so long as one simple condition is met: It’s easier to do if your true knowledge x is high.

In fact, for this very simple model, let’s say that you are strictly limited by the constraint that y <= x. You can’t show off what you don’t know.

If your true value x > z, then you should choose y = x. Then, upon observing your signal, we know immediately that you must be above the threshold.

But if your true value x < z, then you should choose y = 0, because there’s no point in signaling that you were almost at the threshold. You’ll still get rejected.

Yet remember before that only those with z-1 < x < z+1 actually need to bother signaling at all. Those with x > z+1 can actually countersignal, by also choosing y = 0. Since you already have tenure, nobody doubts that you belong in the club.

This means we’ll end up with three groups: Those with x < z, who don’t signal and don’t get accepted; those with z < x < z+1, who signal and get accepted; and those with x > z+1, who don’t signal but get accepted. Then life will be hardest for those who are just above the threshold, who have to spend enormous effort signaling in order to get accepted—and that sure does sound like grad school.

You can make the model more sophisticated if you like: Perhaps the error isn’t uniformly distributed, but some other distribution with wider support (like a normal distribution, or a logistic distribution); perhaps the signalling isn’t perfect, but itself has some error; and so on. With such additions, you can get a result where the least-qualified still signal a little bit so they get some chance, and the most-qualified still signal a little bit to avoid a small risk of being rejected. But it’s a fairly general phenomenon that those closest to the threshold will be the ones who have to spend the most effort in signaling.

This reveals a disturbing overlap between the Curse of Knowledge and Impostor Syndrome: We write in impenetrable obfuscationist jargon because we are trying to conceal our own insecurity about our knowledge and our status in the profession. We’d rather you not know what we’re talking about than have you realize that we don’t know what we’re talking about.

For the truth is, we don’t know what we’re talking about. And neither do you, and neither does anyone else. This is the agonizing truth of research that nearly everyone doing research knows, but one must be either very brave, very foolish, or very well-established to admit out loud: It is in the nature of doing research on the frontier of human knowledge that there is always far more that we don’t understand about our subject than that we do understand.

I would like to be more open about that. I would like to write papers saying things like “I have no idea why it turned out this way; it doesn’t make sense to me; I can’t explain it.” But to say that the profession disincentivizes speaking this way would be a grave understatement. It’s more accurate to say that the profession punishes speaking this way to the full extent of its power. You’re supposed to have a theory, and it’s supposed to work. If it doesn’t actually work, well, maybe you can massage the numbers until it seems to, or maybe you can retroactively change the theory into something that does work. Or maybe you can just not publish that paper and write a different one.

Here is a graph of one million published z-scores in academic journals:

It looks like a bell curve, except that almost all the values between -2 and 2 are mysteriously missing.

If we were actually publishing all the good science that gets done, it would in fact be a very nice bell curve. All those missing values are papers that never got published, or results that were excluded from papers, or statistical analyses that were massaged, in order to get a p-value less than the magical threshold for publication of 0.05. (For the statistically uninitiated, a z-score less than -2 or greater than +2 generally corresponds to a p-value less than 0.05, so these are effectively the same constraint.)

I have literally never read a single paper published in an academic journal in the last 50 years that said in plain language, “I have no idea what’s going on here.” And yet I have read many papers—probably most of them, in fact—where that would have been an appropriate thing to say. It’s actually quite a rare paper, at least in the social sciences, that actually has a theory good enough to really precisely fit the data and not require any special pleading or retroactive changes. (Often the bar for a theory’s success is lowered to “the effect is usually in the right direction”.) Typically results from behavioral experiments are bizarre and baffling, because people are a little screwy. It’s just that nobody is willing to stake their career on being that honest about the depth of our ignorance.

This is a deep shame, for the greatest advances in human knowledge have almost always come from people recognizing the depth of their ignorance. Paradigms never shift until people recognize that the one they are using is defective.

This is why it’s so hard to beat the Curse of Knowledge: You need to signal that you know what you’re talking about, and the truth is you probably don’t, because nobody does. So you need to sound like you know what you’re talking about in order to get people to listen to you. You may be doing nothing more than educated guesses based on extremely limited data, but that’s actually the best anyone can do; those other people saying they have it all figured out are either doing the same thing, or they’re doing something even less reliable than that. So you’d better sound like you have it all figured out, and that’s a lot more convincing when you “utilize a murian model” than when you “use rats and mice”.

Perhaps we can at least push a little bit toward plainer language. It helps to be addressing a broader audience: it is both blessing and curse that whatever I put on this blog is what you will read, without any gatekeepers in my path. I can use plainer language here if I so choose, because no one can stop me. But of course there’s a signaling risk here as well: The Internet is a public place, and potential employers can read this as well, and perhaps decide they don’t like me speaking so plainly about the deep flaws in the academic system. Maybe I’d be better off keeping my mouth shut, at least for awhile. I’ve never been very good at keeping my mouth shut.

Once we get established in the system, perhaps we can switch to countersignaling, though even this doesn’t always happen. I think there are two reasons this can fail: First, you can almost always try to climb higher. Once you have tenure, aim for an endowed chair. Once you have that, try to win a Nobel. Second, once you’ve spent years of your life learning to write in a particular stilted, obscurantist, jargon-ridden way, it can be very difficult to change that habit. People have been rewarding you all your life for writing in ways that make your work unreadable; why would you want to take the risk of suddenly making it readable?

I don’t have a simple solution to this problem, because it is so deeply embedded. It’s not something that one person or even a small number of people can really fix. Ultimately we will need to, as a society, start actually rewarding people for speaking plainly about what they don’t know. Admitting that you have no clue will need to be seen as a sign of wisdom and honesty rather than a sign of foolishness and ignorance. And perhaps even that won’t be enough: Because the fact will still remain that knowing what you know that other people don’t know is a very difficult thing to do.

The sausage of statistics being made


Nov 11 JDN 2458434

“Laws, like sausages, cease to inspire respect in proportion as we know how they are made.”

~ John Godfrey Saxe, not Otto von Bismark

Statistics are a bit like laws and sausages. There are a lot of things in statistical practice that don’t align with statistical theory. The most obvious examples are the fact that many results in statistics are asymptotic: they only strictly apply for infinitely large samples, and in any finite sample they will be some sort of approximation (we often don’t even know how good an approximation).

But the problem runs deeper than this: The whole idea of a p-value was originally supposed to be used to assess one single hypothesis that is the only one you test in your entire study.

That’s frankly a ludicrous expectation: Why would you write a whole paper just to test one parameter?

This is why I don’t actually think this so-called multiple comparisons problem is a problem with researchers doing too many hypothesis tests; I think it’s a problem with statisticians being fundamentally unreasonable about what statistics is useful for. We have to do multiple comparisons, so you should be telling us how to do it correctly.

Statisticians have this beautiful pure mathematics that generates all these lovely asymptotic results… and then they stop, as if they were done. But we aren’t dealing with infinite or even “sufficiently large” samples; we need to know what happens when your sample is 100, not when your sample is 10^29. We can’t assume that our variables are independently identically distributed; we don’t know their distribution, and we’re pretty sure they’re going to be somewhat dependent.

Even in an experimental context where we can randomly and independently assign some treatments, we can’t do that with lots of variables that are likely to matter, like age, gender, nationality, or field of study. And applied econometricians are in an even tighter bind; they often can’t randomize anything. They have to rely upon “instrumental variables” that they hope are “close enough to randomized” relative to whatever they want to study.

In practice what we tend to do is… fudge it. We use the formal statistical methods, and then we step back and apply a series of informal norms to see if the result actually makes sense to us. This is why almost no psychologists were actually convinced by Daryl Bem’s precognition experiments, despite his standard experimental methodology and perfect p < 0.05 results; he couldn’t pass any of the informal tests, particularly the most basic one of not violating any known fundamental laws of physics. We knew he had somehow cherry-picked the data, even before looking at it; nothing else was possible.

This is actually part of where the “hierarchy of sciences” notion is useful: One of the norms is that you’re not allowed to break the rules of the sciences above you, but you can break the rules of the sciences below you. So psychology has to obey physics, but physics doesn’t have to obey psychology. I think this is also part of why there’s so much enmity between economists and anthropologists; really we should be on the same level, cognizant of each other’s rules, but economists want to be above anthropologists so we can ignore culture, and anthropologists want to be above economists so they can ignore incentives.

Another informal norm is the “robustness check”, in which the researcher runs a dozen different regressions approaching the same basic question from different angles. “What if we control for this? What if we interact those two variables? What if we use a different instrument?” In terms of statistical theory, this doesn’t actually make a lot of sense; the probability distributions f(y|x) of y conditional on x and f(y|x, z) of y conditional on x and z are not the same thing, and wouldn’t in general be closely tied, depending on the distribution f(x|z) of x conditional on z. But in practice, most real-world phenomena are going to continue to show up even as you run a bunch of different regressions, and so we can be more confident that something is a real phenomenon insofar as that happens. If an effect drops out when you switch out a couple of control variables, it may have been a statistical artifact. But if it keeps appearing no matter what you do to try to make it go away, then it’s probably a real thing.

Because of the powerful career incentives toward publication and the strange obsession among journals with a p-value less than 0.05, another norm has emerged: Don’t actually trust p-values that are close to 0.05. The vast majority of the time, a p-value of 0.047 was the result of publication bias. Now if you see a p-value of 0.001, maybe then you can trust it—but you’re still relying on a lot of assumptions even then. I’ve seen some researchers argue that because of this, we should tighten our standards for publication to something like p < 0.01, but that’s missing the point; what we need to do is stop publishing based on p-values. If you tighten the threshold, you’re just going to get more rejected papers and then the few papers that do get published will now have even smaller p-values that are still utterly meaningless.

These informal norms protect us from the worst outcomes of bad research. But they are almost certainly not optimal. It’s all very vague and informal, and different researchers will often disagree vehemently over whether a given interpretation is valid. What we need are formal methods for solving these problems, so that we can have the objectivity and replicability that formal methods provide. Right now, our existing formal tools simply are not up to that task.

There are some things we may never be able to formalize: If we had a formal algorithm for coming up with good ideas, the AIs would already rule the world, and this would be either Terminator or The Culture depending on whether we designed the AIs correctly. But I think we should at least be able to formalize the basic question of “Is this statement likely to be true?” that is the fundamental motivation behind statistical hypothesis testing.

I think the answer is likely to be in a broad sense Bayesian, but Bayesians still have a lot of work left to do in order to give us really flexible, reliable statistical methods we can actually apply to the messy world of real data. In particular, tell us how to choose priors please! Prior selection is a fundamental make-or-break problem in Bayesian inference that has nonetheless been greatly neglected by most Bayesian statisticians. So, what do we do? We fall back on informal norms: Try maximum likelihood, which is like using a very flat prior. Try a normally-distributed prior. See if you can construct a prior from past data. If all those give the same thing, that’s a “robustness check” (see previous informal norm).

Informal norms are also inherently harder to teach and learn. I’ve seen a lot of other grad students flail wildly at statistics, not because they don’t know what a p-value means (though maybe that’s also sometimes true), but because they don’t really quite grok the informal underpinnings of good statistical inference. This can be very hard to explain to someone: They feel like they followed all the rules correctly, but you are saying their results are wrong, and now you can’t explain why.

In fact, some of the informal norms that are in wide use are clearly detrimental. In economics, norms have emerged that certain types of models are better simply because they are “more standard”, such as the dynamic stochastic general equilibrium models that can basically be fit to everything and have never actually usefully predicted anything. In fact, the best ones just predict what we already knew from Keynesian models. But without a formal norm for testing the validity of models, it’s been “DSGE or GTFO”. At present, it is considered “nonstandard” (read: “bad”) not to assume that your agents are either a single unitary “representative agent” or a continuum of infinitely-many agents—modeling the actual fact of finitely-many agents is just not done. Yet it’s hard for me to imagine any formal criterion that wouldn’t at least give you some points for correctly including the fact that there is more than one but less than infinity people in the world (obviously your model could still be bad in other ways).

I don’t know what these new statistical methods would look like. Maybe it’s as simple as formally justifying some of the norms we already use; maybe it’s as complicated as taking a fundamentally new approach to statistical inference. But we have to start somewhere.

What’s wrong with academic publishing?

JDN 2457257 EDT 14:23.

I just finished expanding my master’s thesis into a research paper that is, I hope, suitable for publication in an economics journal. As part of this process I’ve been looking into the process of submitting articles for publication in academic journals… and I’ve found has been disgusting and horrifying. It is astonishingly bad, and my biggest question is why researchers put up with it.

Thus, the subject of this post is what’s wrong with the system—and what we might do instead.

Before I get into it, let me say that I don’t actually disagree with “publish or perish” in principle—as SMBC points out, it’s a lot like “do your job or get fired”. Researchers should publish in peer-reviewed journals; that’s a big part of what doing research means. The problem is how most peer-reviewed journals are currently operated.

First of all, in case you didn’t know, most scientific journals are owned by for-profit corporations. The largest corporation Elsevier, owns The Lancet and all of ScienceDirect, and has net income of over 1 billion Euros a year. Then there’s Springer and Wiley-Blackwell; between the three of them, these publishers account for over 40% of all scientific publications. These for-profit publishers retain the full copyright to most of the papers they publish, and tightly control access with paywalls; the cost to get through these paywalls is generally thousands of dollars a year for individuals and millions of dollars a year for universities. Their monopoly power is so great it “makes Rupert Murdoch look like a socialist.”

For-profit journals do often offer an “open-access” option in which you basically buy back your own copyright, but the price is high—the most common I’ve seen are $1800 or $3000 per paper—and very few researchers do this, for obvious financial reasons. In fact I think for a full-time tenured faculty researcher it’s probably worth it, given the alternatives. (Then again, full-time tenured faculty are becoming an endangered species lately; what might be worth it in the long run can still be very difficult for a cash-strapped adjunct to afford.) Open-access means people can actually read your paper and potentially cite your paper. Closed-access means it may languish in obscurity.

And of course it isn’t just about the benefits for the individual researcher. The scientific community as a whole depends upon the free flow of information; the reason we publish in the first place is that we want people to read papers, discuss them, replicate them, challenge them. Publication isn’t the finish line; it’s at best a checkpoint. Actually one thing that does seem to be wrong with “publish or perish” is that there is so much pressure for publication that we publish too many pointless papers and nobody has time to read the genuinely important ones.

These prices might be justifiable if the for-profit corporations actually did anything. But in fact they are basically just aggregators. They don’t do the peer-review, they farm it out to other academic researchers. They don’t even pay those other researchers; they just expect them to do it. (And they do! Like I said, why do they put up with this?) They don’t pay the authors who have their work published (on the contrary, they often charge submission fees—about $100 seems to be typical—simply to look at them). It’s been called “the world’s worst restaurant”, where you pay to get in, bring your own ingredients and recipes, cook your own food, serve other people’s food while they serve yours, and then have to pay again if you actually want to be allowed to eat.

They pay for the printing of paper copies of the journal, which basically no one reads; and they pay for the electronic servers that host the digital copies that everyone actually reads. They also provide some basic copyediting services (copyediting APA style is a job people advertise on Craigslist—so you can guess how much they must be paying).

And even supposing that they actually provided some valuable and expensive service, the fact would remain that we are making for-profit corporations the gatekeepers of the scientific community. Entities that exist only to make money for their owners are given direct control over the future of human knowledge. If you look at Cracked’s “reasons why we can’t trust science anymore”, all of them have to do with the for-profit publishing system. p-hacking might still happen in a better system, but publishers that really had the best interests of science in mind would be more motivated to fight it than publishers that are simply trying to raise revenue by getting people to buy access to their papers.

Then there’s the fact that most journals do not allow authors to submit to multiple journals at once, yet take 30 to 90 days to respond and only publish a fraction of what is submitted—it’s almost impossible to find good figures on acceptance rates (which is itself a major problem!), but the highest figures I’ve seen are 30% acceptance, a more typical figure seems to be 10%, and some top journals go as low as 3%. In the worst-case scenario you are locked into a journal for 90 days with only a 3% chance of it actually publishing your work. At that rate publishing an article could take years.

Is open-access the solution? Yes… well, part of it, anyway.

There are a large number of open-access journals, some of which do not charge submission fees, but very few of them are prestigious, and many are outright predatory. Predatory journals charge exorbitant fees, often after accepting papers for publication; many do little or no real peer review. There are almost seven hundred known predatory open-access journals; over one hundred have even been caught publishing hoax papers. These predatory journals are corrupting the process of science.

There are a few reputable open-access journals, such as BMC Biology and PLOSOne. Though not actually a journal, ArXiv serves a similar role. These will be part of the solution, most definitely. Yet even legitimate open-access journals often charge each author over $1000 to publish an article. There is a small but significant positive correlation between publication fees and journal impact factor.

We need to found more open-access journals which are funded by either governments or universities, so that neither author nor reader ever pays a cent. Science is a public good and should be funded as such. Even if copyright makes sense for other forms of content (I’m not so sure about that), it most certainly does not make sense for scientific knowledge, which by its very nature is only doing its job if it is shared with the world.

These journals should be specifically structured to be method-sensitive but results-blind. (It’s a very good thing that medical trials are usually registered before they are completed, so that publication is assured even if the results are negative—the same should be done with other sciences. Unfortunately, even in medicine there is significant publication bias.) If you could sum up the scientific method in one phrase, it might just be that: Method-sensitive but results-blind. If you think you know what you’re going to find beforehand, you may not be doing science. If you are certain what you’re going to find beforehand, you’re definitely not doing science.

The process should still be highly selective, but it should be possible—indeed, expected—to submit to multiple journals at once. If journals want to start paying their authors to entice them to publish in that journal rather than take another offer, that’s fine with me. Researchers are the ones who produce the content; if anyone is getting paid for it, it should be us.

This is not some wild and fanciful idea; it’s already the way that book publishing works. Very few literary agents or book publishers would ever have the audacity to say you can’t submit your work elsewhere; those that try are rapidly outcompeted as authors stop submitting to them. It’s fundamentally unreasonable to expect anyone to hang all their hopes on a particular buyer months in advance—and that is what you are, publishers, you are buyers. You are not sellers, you did not create this content.

But new journals face a fundamental problem: Good researchers will naturally want to publish in journals that are prestigious—that is, journals that are already prestigious. When all of the prestige is in journals that are closed-access and owned by for-profit companies, the best research goes there, and the prestige becomes self-reinforcing. Journals are prestigious because they are prestigious; welcome to tautology club.

Somehow we need to get good researchers to start boycotting for-profit journals and start investing in high-quality open-access journals. If Elsevier and Springer can’t get good researchers to submit to them, they’ll change their ways or wither and die. Research should be funded and published by governments and nonprofit institutions, not by for-profit corporations.

This may in fact highlight a much deeper problem in academia, the very concept of “prestige”. I have no doubt that Harvard is a good university, better university than most; but is it actually the best as it is in most people’s minds? Might Stanford or UC Berkeley be better, or University College London, or even the University of Michigan? How would we tell? Are the students better? Even if they are, might that just be because all the better students went to the schools that had better reputations? Controlling for the quality of the student, more prestigious universities are almost uncorrelated with better outcomes. Those who get accepted to Ivies but attend other schools do just as well in life as those who actually attend Ivies. (Good news for me, getting into Columbia but going to Michigan.) Yet once a university acquires such a high reputation, it can be very difficult for it to lose that reputation, and even more difficult for others to catch up.

Prestige is inherently zero-sum; for me to get more prestige you must lose some. For one university or research journal to rise in rankings, another must fall. Aside from simply feeding on other prestige, the prestige of a university is largely based upon the students it rejects—its “selectivity” score. What does it say about our society that we value educational institutions based upon the number of people they exclude?

Zero-sum ranking is always easier to do than nonzero-sum absolute scoring. Actually that’s a mathematical theorem, and one of the few good arguments against range voting (still not nearly good enough, in my opinion); if you have a list of scores you can always turn them into ranks (potentially with ties); but from a list of ranks there is no way to turn them back into scores.

Yet ultimately it is absolute scores that must drive humanity’s progress. If life were simply a matter of ranking, then progress would be by definition impossible. No matter what we do, there will always be top-ranked and bottom-ranked people.

There is simply no way mathematically for more than 1% of human beings to be in the top 1% of the income distribution. (If you’re curious where exactly that lies today, I highly recommend this interactive chart by the New York Times.) But we could raise the standard of living for the majority of people to a level that only the top 1% once had—and in fact, within the First World we have already done this. We could in fact raise the standard of living for everyone in the First World to a level that only the top 1%—or less—had as recently as the 16th century, by the simple change of implementing a basic income.

There is no way for more than 0.14% of people to have an IQ above 145, because IQ is defined to have a mean of 100 and a standard deviation of 15, regardless of how intelligent people are. People could get dramatically smarter over timeand in fact have—and yet it would still be the case that by definition, only 0.14% can be above 145.

Similarly, there is no way for much more than 1% of people to go to the top 1% of colleges. There is no way for more than 1% of people to be in the highest 1% of their class. But we could increase the number of college degrees (which we have); we could dramatically increase literacy rates (which we have).

We need to find a way to think of science in the same way. I wouldn’t suggest simply using number of papers published or even number of drugs invented; both of those are skyrocketing, but I can’t say that most of the increase is actually meaningful. I don’t have a good idea of what an absolute scale for scientific quality would look like, even at an aggregate level; and it is likely to be much harder still to make one that applies on an individual level.

But I think that ultimately this is the only way, the only escape from the darkness of cutthroat competition. We must stop thinking in terms of zero-sum rankings and start thinking in terms of nonzero-sum absolute scales.