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.

The usual story about such independent discoveries is that they both deserve recognition. And this makes sense if the point of such prizes is to identify and validate *genius*. But if we awarded prizes instead to create *incentives *for discovery, we should reward neither discoverer. 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.

Yes, independent replications can aid progress in experiments and data analysis. But much less so for math, and few think later replications deserve anywhere near the same reward – so why should simultaneous replications get more? Yes, rewarding both might reduce their risk in seeking the reward, but we already accept an awful lot of risk in such situations. The slight additional risk from rewarding only substantial marginal contributions would create important incentives for researchers to coordinate on their research topics.

Independent simultaneous discovery is a waste, not a triumph. The fact that we seem to feel otherwise is to me further evidence that academia’s main local function is to validate impressive people. Relative to the main goals of most participants, research progress is only a side effect.

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.