I think this account of marginal contribution is wrong. Here’s a handwavy model to explain why.
Suppose there are N people in the world working on X, and you’re the Mth best. And suppose (laughably) that every organization doing X hires exactly one person, the best person it can get. And (also laughably) that everyone works for the best organization they can, and that that’s the one doing the most valuable work in X. Write A(n) for the importance of the nth-best organization’s work and B(n) for the quality of the nth-best person’s work.
OK. So the total utility we get is the sum of A(n) B(n). Now, suppose you weren’t there. The organization employing you gets the next-best person instead, and then the next-best organization gets the next-next-best, etc. In other words, instead of A(1)B(1) + A(2)B(2) + … + A(N)B(N) we get A(1)B(1) + … + A(M-1)B(M-1) + A(M)B(M+1) + A(M+1)B(M+2) + etc. The total utility loss is therefore A(M)(B(M)-B(M+1)) + A(M+1)(B(M+1)-B(M+2)) + etc. The first term here is what James_Miller describes, but there are all the others too.
(Another way for the scenario to play out: Everyone’s already employed; then you drop out and your employer has to hire … whom? Not the next-best candidate, because s/he is already working for someone else. They’ll get the (N+1)th-best candidate, not the (M+1)th-best. The most likely actual outcome is something intermediate between the one I described above and this one.)
Suppose, for instance, that the As and Bs obey a Zipf-like law: A(n) = 1/n, B(n) = 1/n. Then the utility loss is sum {M..N} of 1/n (1/n − 1/(n+1)) = sum {M..N} of 1/n^2(n+1) ~= 1⁄2 (1/M^2 − 1/N^2), whereas James_Miller’s account gives about 1/M^3. If M is much smaller than N—i.e., if you’d be one of the best in the field—then James’s figure for the utility loss is too small by a factor on the order of M. If M is comparable to N—i.e., if you’d be towards the bottom of the pack—then James’s figure is too small by a factor on the order of N-M+1. In between, some slightly funny things happen. Other than right at the endpoints, it’s a pretty good approximation to say that James’s figure is too small by a factor of M(N-M)/N.
This applies well to small organizations or departments, but large organizations, especially universities, could hire researches to work in a field, rather than on a specific task. Researchers working on important problems can work on things that no one would do in their absence.
“X” was meant to be the name of a field rather than of a specific task. I don’t think that makes much difference. But yes, there are contexts in which if you weren’t available your work simply wouldn’t be done. In that case, your marginal contribution equals your contribution, and once again, the “your contribution minus the next-best person’s” calculation gives much too small an answer.
If A(n)=1 (i.e., more attractive employers aren’t actually doing more useful work) then the Miller marginal-utility loss is 1/M(M+1) and the gjm marginal utility loss is 1/M-1/N, for much the same ratio as before.
If A(n) or B(n) or both decrease really quickly with n—A(n) = 2^-n, say—then the error is smaller.
The super-naive approach of pretending that the marginal utility loss equals the utility your work would have done is a much better approximation than “replace self with next-best candidate and change nothing else” is.
I think this account of marginal contribution is wrong. Here’s a handwavy model to explain why.
Suppose there are N people in the world working on X, and you’re the Mth best. And suppose (laughably) that every organization doing X hires exactly one person, the best person it can get. And (also laughably) that everyone works for the best organization they can, and that that’s the one doing the most valuable work in X. Write A(n) for the importance of the nth-best organization’s work and B(n) for the quality of the nth-best person’s work.
OK. So the total utility we get is the sum of A(n) B(n). Now, suppose you weren’t there. The organization employing you gets the next-best person instead, and then the next-best organization gets the next-next-best, etc. In other words, instead of A(1)B(1) + A(2)B(2) + … + A(N)B(N) we get A(1)B(1) + … + A(M-1)B(M-1) + A(M)B(M+1) + A(M+1)B(M+2) + etc. The total utility loss is therefore A(M)(B(M)-B(M+1)) + A(M+1)(B(M+1)-B(M+2)) + etc. The first term here is what James_Miller describes, but there are all the others too.
(Another way for the scenario to play out: Everyone’s already employed; then you drop out and your employer has to hire … whom? Not the next-best candidate, because s/he is already working for someone else. They’ll get the (N+1)th-best candidate, not the (M+1)th-best. The most likely actual outcome is something intermediate between the one I described above and this one.)
Suppose, for instance, that the As and Bs obey a Zipf-like law: A(n) = 1/n, B(n) = 1/n. Then the utility loss is sum {M..N} of 1/n (1/n − 1/(n+1)) = sum {M..N} of 1/n^2(n+1) ~= 1⁄2 (1/M^2 − 1/N^2), whereas James_Miller’s account gives about 1/M^3. If M is much smaller than N—i.e., if you’d be one of the best in the field—then James’s figure for the utility loss is too small by a factor on the order of M. If M is comparable to N—i.e., if you’d be towards the bottom of the pack—then James’s figure is too small by a factor on the order of N-M+1. In between, some slightly funny things happen. Other than right at the endpoints, it’s a pretty good approximation to say that James’s figure is too small by a factor of M(N-M)/N.
This applies well to small organizations or departments, but large organizations, especially universities, could hire researches to work in a field, rather than on a specific task. Researchers working on important problems can work on things that no one would do in their absence.
“X” was meant to be the name of a field rather than of a specific task. I don’t think that makes much difference. But yes, there are contexts in which if you weren’t available your work simply wouldn’t be done. In that case, your marginal contribution equals your contribution, and once again, the “your contribution minus the next-best person’s” calculation gives much too small an answer.
A few other remarks.
If A(n)=1 (i.e., more attractive employers aren’t actually doing more useful work) then the Miller marginal-utility loss is 1/M(M+1) and the gjm marginal utility loss is 1/M-1/N, for much the same ratio as before.
If A(n) or B(n) or both decrease really quickly with n—A(n) = 2^-n, say—then the error is smaller.
The super-naive approach of pretending that the marginal utility loss equals the utility your work would have done is a much better approximation than “replace self with next-best candidate and change nothing else” is.