My aunt’s husband, Field Medal winner Atle Selberg, died a few weeks ago. The Washington Post: The Fields Medal was awarded to Dr. Selberg for his work on proving a challenging theory about the distribution of prime numbers. For years, mathematicians had believed that it could be proved only by the laborious application of ponderous techniques. In 1949, Dr. Selberg and another celebrated mathematician, Paul Erdos, achieved a proof through techniques of startling simplicity. Each, it was reported, was to report on his own contribution in the same issue of the same mathematics journal. Because of what has been described as a misunderstanding that led to hurt feelings, Dr. Selberg published first. His Fields Medal, recognizing him for a variety of accomplishments, followed.

I admit that I do, and if I had I would probably have seen the natureof your argument sooner and not made a muddled answer that bothaccepted and denied the premiss that one discovery equals oneunit.

You are arguing, if I understand correctly now, along the lines thatthe prize-giver is a consumer of discoveries or solutions and hasbought more units than he needs, and ought to reduce his bid at leastto the point where he receives only one solution.

I think that argument doesn't work. The case of two simultaneousdiscoveries is compatible with the prize amount being just right oreven too small, as well as too large.

My argument: I see no reason to assume that the expected distributionof number of solutions is a point. ISTM the number of solutions isaffected by both the prize amount and by random factors beyond thecontrol of both prize-giver and contestants. I expect the randomfactors to be significant, because the contestants do. If theydidn't, they would co-ordinate so that only one contestant incurredthe cost of pursuing the prize.

Since there's a significant random factor, for no combination of setprize amount and expected difficulty is the expected distribution ofnumber of solutions exactly one. The distribution might both zero andone, both one and two or more, or all three.

So reducing the prize amount to the point where there is no risk oftwo solutions increases the risk of no solution. If the random factoris large enough, the risk of no solution can matter even at the sametime that there is a significant risk of two or more solutions.

Nature has in effect packaged solutions in units of unpredictablesize. Both the prize-giver and the contestants would like to make theunit size predictable, so that one solution equals one transaction. Theprize-giver would avoid the situation at hand and the contestantswould avoid the risk being the second discoverer. But they can't. Ifthey could, they would have already done so, and the situation wouldalready be more like wages or purchases than prizes.

Stuart, cryonics, a trust fund for oneself, and a published request to future generation to compensate you (via your trust fund) for your work for them.

After all, the marginal contribution to progress of each simultaneous independent discovery is near zero - without that discovery progress would have been nearly the same, since the other sources were available.

Sorry, Robin, I think that's funny counting.

Let's say arguendo that there are no side benefits from discovery.If discovery A's marginal contribution to progress is near zero, since there is no quarrel about the total value of discoveries A+B, that means that discovery B contributes nearly the entire value of A+B. Similarly, if we fix B's value near zero, A contributes nearly A+B. So ISTM you can't make the marginal value argument for both A and B simultaneously.

You might treat the fact of simultaneity as a proxy measure for how easy it was to make the discovery. Others have pointed out problems with that.

Stuart Armstrong: the Fields Medal is a prize with little money attached. But that doesn't mean it's cheap. It probably required lots of other resources to establish it as a serious open-ended award. I think this is relevant to Constant's comment; this was only the second time the Fields Medals were awarded and it probably wasn't well-established.

In case it wasn't obvious, I meant to hold constant the total amount of money awarded as prizes. So if in some cases we awarded less, in other cases we would award more.

Prizes bestowing fame and prestige could work with very little money. "Genius" prizes seem more suited for this than "genuine achievement" ones.

Ah, what would we not give for a time machine, so that future generations could reward researchers today...

Scott, the part that might not be obvious to some is that those paying for research are included in the "participants" for whom research progress is only a side effect.

Anony, yes, co-discoverers must split credit, but that is still much more than being given near zero credit.

Math and Constant, if one wants to reward new interesting proofs of old results, one can simply use that language when defining "discoveries."

Conchis, if the average prize awarded is held constant, I don't see why people would avoid the research topic.

Relative to the main goals of most participants, research progress is only a side effect.

Robin, this strikes me as both true and obvious. "It is not from the benevolence of the physicist, the chemist, or the mathematician that we expect our journal articles, but from their regard to their own glory." I thought economists liked mechanisms that induce selfish agents to work toward socially desirable outcomes... :-)

In practice, simultaneous co-discoverers do not each receive as much credit as a single person would have: most people downgrade their estimate of the contribution when they hear several people did it independently and at the same time. Furthermore, when prizes are involved, they may be split, or they may be awarded to the one person who is judged to have made the largest contribution, but I've never heard of the total prize money being increased so that each co-discoverer gets a full share. So I'm not sure what Robin is complaining about.

Incidentally, the Erdos-Selberg story is quite a bit more subtle than the Washington Post story indicates. See http://www.math.columbia.ed... for the details.

As Constant said, the significance of one's work in mathematics is judged not only by one's results but by the methods developed therein. Techniques used in a proof often stimulate additional research, sometimes creating entirely new fields of mathematics. Thus a subsequent proof may be more significant than the original one.

The case of math may be somewhat different from the case of science generally. In math, the result is not the only thing that matters. The proof is also important. Finding new proofs of the same result is a worthwhile endeavor. It's not just the result that is discovered, but the proof as well, and so a new proof of an old result is a genuine discovery.

Of course, if Erdos and Selberg actually came up with the same proof, then perhaps it was, in hindsight, a waste of effort. However, we do not approach our tasks with hindsight.

Incidentally, Paul Erdos seems to me quite famous, more so than most any Fields winner. At least, I have often seen him mentioned. If it's fame people want - and isn't that a large part of it - then Erdos did well without the Fields medal.

Okay, I think my previous post responded to the wrong point. I was referring to awards for specific discoveries (i.e. dis/proving P=NP, etc.) rather then general "greatness" awards. But I think my point still stands: if there is such a sharp cutoff for "me-too"s, *and* if the prize money is the driving factor for the researcher, the risk/reward profile is skewed to the point that they may just devote their brain to making money outside of academia.

I agree that the money *can't* have much of an incentive effect because of the low probability of getting it (even if you do good research) and how long it will take to be recognized, so the prize is more effective at "validation".

In my opinion, awards currently encourage researchers to spend more time on marketing their research and social networking to make sure the research they work on will be widely known and popular enough in the academic community to be nominated for an award. Awards miss scientists who produced the original work in favor of those who had more academic influence (due to their skills, position, or prior awards) to successfully popularize and market that work. I have no doubt that popularizing is also an important skill and might be a great contribution all by itself worth to be awarded if the prior work is properly cited (ironically, not citing prior work greatly increases the chances of getting an award).

In case it wasn't obvious, I meant to hold constant the total amount of money awarded as prizes. So if in some cases we awarded less, in other cases we would award more.

## The Function of Prizes

Tom, you need to learn some economics.

I admit that I do, and if I had I would probably have seen the natureof your argument sooner and not made a muddled answer that bothaccepted and denied the premiss that one discovery equals oneunit.

You are arguing, if I understand correctly now, along the lines thatthe prize-giver is a consumer of discoveries or solutions and hasbought more units than he needs, and ought to reduce his bid at leastto the point where he receives only one solution.

I think that argument doesn't work. The case of two simultaneousdiscoveries is compatible with the prize amount being just right oreven too small, as well as too large.

My argument: I see no reason to assume that the expected distributionof number of solutions is a point. ISTM the number of solutions isaffected by both the prize amount and by random factors beyond thecontrol of both prize-giver and contestants. I expect the randomfactors to be significant, because the contestants do. If theydidn't, they would co-ordinate so that only one contestant incurredthe cost of pursuing the prize.

Since there's a significant random factor, for no combination of setprize amount and expected difficulty is the expected distribution ofnumber of solutions exactly one. The distribution might both zero andone, both one and two or more, or all three.

So reducing the prize amount to the point where there is no risk oftwo solutions increases the risk of no solution. If the random factoris large enough, the risk of no solution can matter even at the sametime that there is a significant risk of two or more solutions.

Nature has in effect packaged solutions in units of unpredictablesize. Both the prize-giver and the contestants would like to make theunit size predictable, so that one solution equals one transaction. Theprize-giver would avoid the situation at hand and the contestantswould avoid the risk being the second discoverer. But they can't. Ifthey could, they would have already done so, and the situation wouldalready be more like wages or purchases than prizes.

Stuart, cryonics, a trust fund for oneself, and a published request to future generation to compensate you (via your trust fund) for your work for them.

Tom, you need to learn some economics.

After all, the marginal contribution to progress of each simultaneous independent discovery is near zero - without that discovery progress would have been nearly the same, since the other sources were available.

Sorry, Robin, I think that's funny counting.

Let's say arguendo that there are no side benefits from discovery.If discovery A's marginal contribution to progress is near zero, since there is no quarrel about the total value of discoveries A+B, that means that discovery B contributes nearly the entire value of A+B. Similarly, if we fix B's value near zero, A contributes nearly A+B. So ISTM you can't make the marginal value argument for both A and B simultaneously.

You might treat the fact of simultaneity as a proxy measure for how easy it was to make the discovery. Others have pointed out problems with that.

Stuart Armstrong: the Fields Medal is a prize with little money attached. But that doesn't mean it's cheap. It probably required lots of other resources to establish it as a serious open-ended award. I think this is relevant to Constant's comment; this was only the second time the Fields Medals were awarded and it probably wasn't well-established.

In case it wasn't obvious, I meant to hold constant the total amount of money awarded as prizes. So if in some cases we awarded less, in other cases we would award more.

Prizes bestowing fame and prestige could work with very little money. "Genius" prizes seem more suited for this than "genuine achievement" ones.

Ah, what would we not give for a time machine, so that future generations could reward researchers today...

Scott, the part that might not be obvious to some is that those paying for research are included in the "participants" for whom research progress is only a side effect.

Anony, yes, co-discoverers must split credit, but that is still much more than being given near zero credit.

Math and Constant, if one wants to reward new interesting proofs of old results, one can simply use that language when defining "discoveries."

Conchis, if the average prize awarded is held constant, I don't see why people would avoid the research topic.

Relative to the main goals of most participants, research progress is only a side effect.

Robin, this strikes me as both true and obvious. "It is not from the benevolence of the physicist, the chemist, or the mathematician that we expect our journal articles, but from their regard to their own glory." I thought economists liked mechanisms that induce selfish agents to work toward socially desirable outcomes... :-)

In practice, simultaneous co-discoverers do not each receive as much credit as a single person would have: most people downgrade their estimate of the contribution when they hear several people did it independently and at the same time. Furthermore, when prizes are involved, they may be split, or they may be awarded to the one person who is judged to have made the largest contribution, but I've never heard of the total prize money being increased so that each co-discoverer gets a full share. So I'm not sure what Robin is complaining about.

Incidentally, the Erdos-Selberg story is quite a bit more subtle than the Washington Post story indicates. See http://www.math.columbia.ed... for the details.

As Constant said, the significance of one's work in mathematics is judged not only by one's results but by the methods developed therein. Techniques used in a proof often stimulate additional research, sometimes creating entirely new fields of mathematics. Thus a subsequent proof may be more significant than the original one.

Here is a relevant article from The New Yorker: http://www.newyorker.com/ar...

The case of math may be somewhat different from the case of science generally. In math, the result is not the only thing that matters. The proof is also important. Finding new proofs of the same result is a worthwhile endeavor. It's not just the result that is discovered, but the proof as well, and so a new proof of an old result is a genuine discovery.

Of course, if Erdos and Selberg actually came up with the same proof, then perhaps it was, in hindsight, a waste of effort. However, we do not approach our tasks with hindsight.

Incidentally, Paul Erdos seems to me quite famous, more so than most any Fields winner. At least, I have often seen him mentioned. If it's fame people want - and isn't that a large part of it - then Erdos did well without the Fields medal.

Okay, I think my previous post responded to the wrong point. I was referring to awards for specific discoveries (i.e. dis/proving P=NP, etc.) rather then general "greatness" awards. But I think my point still stands: if there is such a sharp cutoff for "me-too"s, *and* if the prize money is the driving factor for the researcher, the risk/reward profile is skewed to the point that they may just devote their brain to making money outside of academia.

I agree that the money *can't* have much of an incentive effect because of the low probability of getting it (even if you do good research) and how long it will take to be recognized, so the prize is more effective at "validation".

In my opinion, awards currently encourage researchers to spend more time on marketing their research and social networking to make sure the research they work on will be widely known and popular enough in the academic community to be nominated for an award. Awards miss scientists who produced the original work in favor of those who had more academic influence (due to their skills, position, or prior awards) to successfully popularize and market that work. I have no doubt that popularizing is also an important skill and might be a great contribution all by itself worth to be awarded if the prior work is properly cited (ironically, not citing prior work greatly increases the chances of getting an award).

Happy birthday Robin! Interesting post as always.

In case it wasn't obvious, I meant to hold constant the total amount of money awarded as prizes. So if in some cases we awarded less, in other cases we would award more.