George J. Borjas and Kirk B. Doran find that the productivity of mathematicians who win the Fields Medal, the mathematics “Nobel” awarded to mathematicians under the age of 40, declines after they win. Borjas and Doran look at productivity on a number of margins including papers, citations, and graduate students mentored. At right is a graph of average number of papers compared to a contender group. Productivity falls by a statistically significant ~1 paper per year (in the regression that I think the strongest, in a few variants the decline is a bit more.)

Not all of the decline is due to resting on laurels. Borjas and Doran also find that mathematicians who win the Fields tend to branch out into other areas and this branching out requires them to learn new material which takes time. Steve Smale did work in economics and biology, for example, Rene Thom developed catastrophe theory and David Mumford works in vision and pattern theory. Exploring new topics can also lead to breakthroughs so branching out is not necessarily a negative effect. Borjas and Doran estimate that about half of the productivity effect is resting on laurels and half greater exploration.

(FYI, Borjas and Doran speak of prizes but I prefer to call them awards because awards such as the Fields or Clark Medal are quite different from prizes for purpose, such as the XPrizes, the H Prize or the historically important Orteig Prize, that I discuss in Launching as alternatives to patents.)

Winning the Fields is presumably good for the winner but even taking into account the enhanced incentive to explore it’s not obviously good for mathematics. The explicit purpose of the Fields was to increase not decrease achievement. What can be done?

The Fields Medal may be too important for its own good. In economics the closest thing to the Fields is the John Bates Clark award, given to that American economist under the age of 40 judged to have made the most significant contributions. Chan et al. (2013) find that recipients of the Clark award increase their productivity after winning. But the Clark award is widely seen as a future portent of the Nobel, thus it may have a more stimulative effect as the winner realizes that the next big award is within reach (see the final sentence). In a tournament, it’s important to tier the awards for multiple levels of ability.

In thinking about whether awards increase or decrease productivity on net. the precise counter-factual is important. Let us accept as Truth that winners of the Fields Medal decrease in productivity, even so that doesn’t mean that eliminating the Fields Medal would increase productivity let alone that eliminating all awards would increase productivity (remember, the productivity of the contenders may be more important than the productivity of the winners). Perhaps the most justifiable policy recommendation is that one shouldn’t give awards to young people, a Fields Medal for lifetime achievement, much as the economics Nobel is given, might encourage more achievement.

Paul Samuelson wrote of Chasing the Bitch Goddess of Success:

Scientists are as avaricious and competitive as Smithian businessmen. The coin they seek is not apples, nuts, and yachts; nor is it the coin itself, or power as that term is ordinarily used. Scholars seek fame.

But, paraphrasing Tyler, what price early fame?

“even taking into account the enhanced incentive to explore it’s not obviously good for mathematics……..Perhaps the most justifiable policy recommendation is that one shouldn’t give awards to young people”Nope. I don’t think your conclusions are logical.

What if the desire to win a Fields is what causes the curves to be as high as where they are? In both contenders & Medalists. Your counterfactual of no-Fields or late-life-Fields may very well shift both curves downwards.

There’s reasonable reason to believe people work harder for a near-term reward than a far-horizon reward. In any case, isn’t it true that Math is a game for young minds? Post 40 productivity may not be as important.

Right. The joke at the dog races is that the dog that manages to catch the mechanical rabbit never wants to race again. It’s still great for motivating the dogs, though.

Yeah, this seems kinda obvious…

Yeah, so obvious that Alex discusses this point in the post! See paragraph beginning “In thinking…

A tiered award system, with a prestigious post-Field award, might indeed incentivize post-40 productivity, assuming that we even want these people to stay in mathematics rather than branching out into some other area. (Maybe, it’s not such a bad thing that, once someone has reached the pinnacle in one area, they try to accomplish something else in some other area.)

On the other hand, I’m not sure that one can just establish an award and declare it to be prestigious. In professional golf, they just finished their FedEx Cup Playoffs this weekend, which has by far the highest cash prize, much more than any of the four so-called “Major” tournaments (Masters, US Open, British Open, and PGA Championship). The FedEx Cup is a season-long competition, while the four Majors are single, standalone tournaments. Yet, due to tradition and history, the four Majors are the most prestigious, each equivalent to the Super Bowl or the World Series. There is a kind of network effect to prizes: the most prestigious ones are the ones that everyone else wants to win.

Upon further review, there is something called the Abel Prize. Is that prize less prestigious than the Fields Medal? See [http://gonitsora.com/mathematicians’-nobel-prize-fields-medal-or-abel-prize/].

It’s interesting that J.C. Fields, the Canadian mathematician that donated the funds to establish the Medals, wanted the awards to recognize both existing work and the promise of future achievement. Apparently, that motivated the restriction that the prize only by given to those under 40. Obviously, Fields was not an economist. (Aside: Those that awarded the Nobel Peace Prize to Barack Obama might also want to review how incentives work.)

Yes, the Abel prize is in a sense more prestigious than the Fields medal; the Abel prize is meant as a career/lifetime achievement award while the Fields medal was always meant specifically to award young mathematicians (although generally those who win it are so successful that it most often something like an “early” career achievement award rather than something you get for one big work). The medalists I know still produce a lot more than most of their peers….However, the reason “productivity” trails off in the metric is obvious: doing work capable of earning a medal requires 12 hrs a day 7 days a week dedicated to extremely difficult and competitive problems – very few people are be able to do this for any length of time at all, much less the number of years required in order to reach the level of success we’re talking about…

I agree for different reasons that the Fields Medal shouldn’t be as big a deal as it is in mathematics. The focus on young research doesn’t seem necessarily all that reasonable.

But this blog post uses number of publications as pretty much the sole measure of productivity, and that I very much disagree with. Maybe a better measure would be number of citations, but even with that the whole idea of finding a purely objective measure of mathematical contribution is insane.

As Alex says, the paper also looks at citations and mentoring.

The most annoying thing about MR commenters is their tendency to look at every *other* commenter as an idiot. I did read the post, you don’t have to check my reading comprehension.

The question is, why should I care about that chart that appears on the right? Shouldn’t I care more about the citations anyway? But even citations–it’s that really what it’s all about? How do we ultimately measure mathematical contribution? There has to be a subjective element that just isn’t captured by these statistics.

Speaking of annoying, I get annoyed by arguments that keep bringing up subjectivity. Sure there’s subjectivity; but shall we give up on all quantitative measurement then?

I think it is perfectly fine to criticize a

particularobjective metric but nihilistic to suggest thatallobjective metrics are doomed just because none is perfect. Or that the quest to measure contribution is “insane”.You can, of course, suggest a better or more encompassing metric or point flaws with the current one but that’s different from complaining

“You can’t measure contribution because it is subjective!”Papers published != Productivity

Generating paper is not the same as contributing to one’s field.

Agreed. The typical pattern of a research mathematician is to make lots of insignificant publications of lemmas or not very impressive preliminary findings to show that you are out there trying to solve a big hard problem, followed by a publication of a “big paper” announcing not just a preliminary result but a major proof. But, one big paper may be worth dozens of preliminary ones to the field.

You get a Field Medal because you have published a “big paper” making a major conclusion that renders more or less obsolete your previous dozen preliminary papers documenting your hunt for the big prize at time when you didn’t fully understand the problem. Once you make that accomplishment, it may take years to be so hot on the hunt of some new problem replacing the one that you’ve solved (for all eternity), and to start publishing preliminary results again and increasing your publication count.

The system forces them to seek fame.

Maybe an economist can publish a paper on how economists are like professional athletes: both motivated by a combination of money, peer respect, and a shot at fame. Often money is the least important of these.

Does it account for regression to the mean? Maybe you’re just more likely to win the Fields medal after a flurry of activity, after which you’re likely to revert back to normal.

+1

But:

Now your talking statistics. Everyone knows statisticians aren’t “real mathematicians”!

If it were only regression to the mean, then any one lucky enough to have high early productivity would see a decline later, including the contenders.

The contenders were selected for late awards, ie, late productivity.

Yes, that’s what makes me still skeptical of this result: that contender group looks biased, with mathematicians who had high late- (or mid-) career productivity in addition to early-career productivity.

Actually, the contenders who got the four “area” awards got them before age 40. We discuss this in the paper. In addition to these four early awards, there are also two late awards as well (Abel and Wolf), but if you drop all people who got late awards from the sample, then the results look similar, as we mention in the paper.

That’s reasonable.

If you’re going to include a long caption on a figure, shouldn’t it be accurate? A short caption can require reading the paper to understand, but someone flipping through your paper who sees that graph thinks that since it claims to define “contender” that it actually does so. (Not to mention that graphs get cut out and put on blogs.)

Um no actually.

Contenders and Medalists need to ‘advertise’ by writing papers in order to win a Medal. Once a Medalist, there is less need to advertise (alternatively, the Medal is a superior advertisement of achievement), so less time spent on papers, more time spent on real research. Contenders are still trying to win the Medal; the Medalists are not. The two groups have slightly different goals.

I’d like to see the same data laid over date rangers when said academics have sons and daughters in high school.

I don’t know if this is true of the Field medal, but winning the Nobel in Economics can lead to lucrative speaking opportunities, which take time away from research.

Field medalists, fairly or not, are not a very hot commodity on the speaking circuit and get paid far less than economists to retell the story of their greatness and offer insights to the next generation,

Also, the prize for the Field medal much less lucrative than that for the Nobel in Economics.

Bill Thurston’s article “On proof and progress in mathematics” utterly demolishes the metric used here.

http://arxiv.org/abs/math.HO/9404236

a gross oversimplification would be that you should measure careers/publications of their grad students instead of the mathematicians themselves, but, read the whole thing.

Actually, we look at the career publications of the grad students as well.

productivity of mathematicians who win the Fields Medal, the mathematics “Nobel” awarded to mathematicians under the age of 40, declines after they winSports Illustrated cover jinx.

Several decades ago H. C. Brown used to show a similar graph for chemistry Nobel prize winners. He then went on to say that he would not be part of the trend. Some people were suspicious that his remarks were just a way to remind the audience that he had a Nobel prize. But, everybody who worked for him seemed to love him.

What rubbish. Self-selection at work. Of course people who are not motivated by money go into science these days, since there’s no real money in science (compared to other fields). Hence that these people are motivated by fame is not surprising. As for the Field prize disincentive, it works out to 1.5 papers a year, big deal. Perhaps the quality of the papers by the losers are inferior to the winners.

A more plausible explanation is that the winners are older than the runners-up, as suggested by inference at Wikipedia: http://en.wikipedia.org/wiki/Fields_Medal (note the youngest winner was 27, and it’s well known in math that brilliant discoveries happen before age 30, so it could well be that the Fields Medal is given to “past their intellectual prime” winners(speculative but I would not be surprised)However, it could indeed well be that the winners rest on their laurels, or branch into other fields (like Nobelian Paul Krugman has in another context), or, it could also be the contenders are trying to win again hence publish more, and, as AlexT says, the incentive still motivates the runners-up.

In any event I agree with Alex that we need more such prizes, which avoid the patent toll-booth (alleged) problem since the math discoveries are royalty-free to all of society.

Actually, the Fields Medalists’ average age at first publication was 23.1 years; the Contenders’ average age at first publication was 24 years. The median age at winning the Fields Medal is about 36. The median age of maximum eligibility for the fields medal (the “zero” point for the contenders) is about 38. So, the Fields Medalists’ careers are distributed across the age profile similarly to those of the contenders, and are definitely not older on average when they reach the “zero” line in the x-axis above. And, in any case, we control for any age differences that remain in the regressions. Check out Table 1 of the paper as well as the data and results sections for more details.

Also, the citation-adjusted quality of the papers by the losers does not decrease after the prize; we report this in the paper.

Introduction of the counterfeit Nobel for economists probably increased the rate at which economists did harm to the world’s economies.

Whereas “the enhanced incentive to explore” for the mathematicians may far outweigh a drop in their measured quasi-productivity. The intellectual curse of academic life is the “Let’s stay mainstream” route to promotion.

Here is the paper, since I can’t find a link in the post.

How much is a statistical artifact of the fact that a highly productive person will be more likely to be marginally less rather than more productive in the future? And once you have solved a big enough problem to get this level of recognition, how easy is it to fond something else of that magnitude?

Below is a part of a speech given by Dr. Richard Hamming, “You and Your Research” in 1986. I think he’s close to the real issue in this segment. Award winners can’t really work on the small problems anymore and so fail to “plant the acorns”. A solution doesn’t seem evident as it isn’t the award that is the problem but the recognition of their early work. Even without the award, some sort of accolades would be given and the expectations to work the big problems would rise.

“Age is another factor which the physicists particularly worry about. They always are saying that you have got to do it when you are young or you will never do it. Einstein did things very early, and all the quantum mechanic fellows were disgustingly young when they did their best work. Most mathematicians, theoretical physicists, and astrophysicists do what we consider their best work when they are young. It is not that they don’t do good work in their old age but what we value most is often what they did early. On the other hand, in music, politics and literature, often what we consider their best work was done late. I don’t know how whatever field you are in fits this scale, but age has some effect.

But let me say why age seems to have the effect it does. In the first place if you do some good work you will find yourself on all kinds of committees and unable to do any more work. You may find yourself as I saw Brattain when he got a Nobel Prize. The day the prize was announced we all assembled in Arnold Auditorium; all three winners got up and made speeches. The third one, Brattain, practically with tears in his eyes, said, “I know about this Nobel-Prize effect and I am not going to let it affect me; I am going to remain good old Walter Brattain.” Well I said to myself, “That is nice.” But in a few weeks I saw it was affecting him. Now he could only work on great problems.

When you are famous it is hard to work on small problems. This is what did Shannon in. After information theory, what do you do for an encore? The great scientists often make this error. They fail to continue to plant the little acorns from which the mighty oak trees grow. They try to get the big thing right off. And that isn’t the way things go. So that is another reason why you find that when you get early recognition it seems to sterilize you. In fact I will give you my favorite quotation of many years. The Institute for Advanced Study in Princeton, in my opinion, has ruined more good scientists than any institution has created, judged by what they did before they came and judged by what they did after. Not that they weren’t good afterwards, but they were superb before they got there and were only good afterwards.”

http://www.cs.virginia.edu/~robins/YouAndYourResearch.html

+1

The shape of the curves is interesting (i.e. they make me a little twitchy).

Each person has basically one opportunity to be seriously considered for the Fields medal, i.e. the last slot before he’s ineligible (median age 36). It almost looks like there’s some strategic behavior going on: there’s a peak at roughly T-6 (i.e getting traction before the T-4 decision) and again at T-2 (getting traction before the T-0 cut-off). Of course, this may also be interacting with a tenure-case peak, say PhD + postdoc + 6 is also mid-30’s, though it varies by country and it’s unlikely that a “Fields medal contender” has much to worry about.

The detail in the contenders’ T+ curves is a little odd, too – even if it’s not contributing to statistical significance — it’s the larger dataset, but the less smooth curve. All of these men would have tenured positions at excellent schools by T+0 or so (, mathematicians don’t need big budget support to keep a lab running…I’m not sure I see what’s in the data…

My productivity is sadly more at risk from MR than from winning the Fields medal, so I haven’t urgently read the paper yet….

maybe this explains why Tyler is the most productive infovore on the planet.

It’s fun to see all the experts in the peanut gallery getting thoroughly shot down.

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