I admit that I do, and if I had I would probably have seen the nature
of your argument sooner and not made a muddled answer that both
accepted and denied the premiss that one discovery equals one
unit.

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

I think that argument doesn’t work. The case of two simultaneous
discoveries is compatible with the prize amount being just right or
even too small, as well as too large.

My argument: I see no reason to assume that the expected distribution
of number of solutions is a point. ISTM the number of solutions is
affected by both the prize amount and by random factors beyond the
control of both prize-giver and contestants. I expect the random
factors to be significant, because the contestants do. If they
didn’t, they would co-ordinate so that only one contestant incurred
the cost of pursuing the prize.

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

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

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

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.

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…

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.

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…

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