Productivity as a function of ability in theoretical fields

I ar­gued in this post that the differ­ences in ca­pa­bil­ity be­tween differ­ent re­searchers are vast (Kaj So­tala pro­vided me with some in­ter­est­ing em­piri­cal ev­i­dence that backs up this claim). Ein­stein’s con­tri­bu­tions to physics or John von Neu­mann’s con­tri­bu­tions to math­e­mat­ics (and a num­ber of other dis­ci­plines) are ar­guably at least hun­dreds of times greater than that of an av­er­age physi­cist or math­e­mat­i­cian.

At the same time, Yud­kowsky ar­gues that “in the space of brain de­signs” the differ­ence be­tween the village idiot and Ein­stein is tiny. Their brains are ex­tremely similar, with the ex­cep­tion of some “minor ge­netic tweaks”. Hence we get the fol­low­ing pic­ture:

The pic­ture I am paint­ing is rather some­thing like this:
It would seem that these pic­tures are in­com­pat­i­ble—some­thing that would be a prob­lem for my pic­ture, since I think that Yud­kowsky’s pic­ture is right. So how can they both be true? The an­swer is, ob­vi­ously, that they are mea­sur­ing differ­ent things. The first is mea­sur­ing some­thing like differ­ence in brain de­sign that is rele­vant for in­tel­li­gence. The sec­ond is rather mea­sur­ing the differ­ence in ca­pa­bil­ity to come up with phys­i­cal the­o­ries that are of use for mankind. Here the village idiot is on par with the chimp and the mouse—all of whom have no such ca­pa­bil­ity whatsover. The av­er­age physi­cist has some such ca­pa­bil­ity, but it’s just a frac­tion of Ein­stein’s.
Why is this? Well it is not be­cause the village idiot has no ca­pa­bil­ity at all to come up with phys­i­cal the­o­ries. In fact, a prim­i­tive phys­i­cal the­ory that is quite use­ful is hard-wired into our brains. Rather, the rea­son is that the village idiot has no ca­pa­bil­ity to come up with a phys­i­cal the­ory that is not already well-known.
Prob­lems in the­o­ret­i­cal physics and math­e­mat­ics are typ­i­cally prob­lems that are so com­plex that they are hard to solve for some of the world’s smartest peo­ple. This means that un­less you’re quite smart, your chances of con­tribut­ing any­thing at at all to these dis­ci­plines is very slim. But, if you are but a tiny bit smarter than ev­ery­one else, you’ll be able to spot solu­tions to prob­lem af­ter prob­lem that oth­ers have strug­gled with—these prob­lems be­ing prob­lems pre­cisely be­cause they were hard to solve for peo­ple with a cer­tain level of in­tel­li­gence. Thus we get some­thing like the fol­low­ing re­la­tion­ship be­tween cog­ni­tive abil­ity, in Yud­kowsky’s sense, and abil­ity to come up with use­ful phys­i­cal the­o­ries, i.e. pro­duc­tivity—what I’m talk­ing about:

It is for this rea­son that peo­ple like von Neu­mann and Ein­stein are so vastly much more pro­duc­tive than the av­er­age math­e­mat­i­cian/​physi­cist. The differ­ence in in­tel­li­gence is tiny on Yud­kowsky’s scale—ob­vi­ously much smaller than that be­tween Ein­stein and the village idiot—but this tiny differ­ence al­lowed von Neu­mann and Ein­stein to solve lots of prob­lems that were just too hard for other math­e­mat­i­ci­ans/​physi­cists. (It fol­lows that an ar­tifi­cial in­tel­li­gence just a tiny bit smarter than Ein­stein and von Neu­mann would be as much more pro­duc­tive than them as they are in re­la­tion to other math­e­mat­i­cian/​physi­cists).
(Ob­vi­ously other char­ac­ter­is­tics be­sides in­tel­li­gence are very im­por­tant in these fields—e.g. work ethic. I put that com­pli­ca­tion aside here, though.)
The same pat­tern holds in many other fields—e.g. sports. In a sense, the differ­ence in abil­ity be­tween Ra­fael Nadal and no 300 on the ATP rank­ing is very small—e.g. they are hit­ting the ball roughly as hard, are roughly as good at, say, shoot­ing the ball within half a me­tre of the base-line when not un­der pres­sure, etc—but this small differ­ence in abil­ity makes for a huge differ­ence in pro­duc­tivity (in the sense that lots of peo­ple want to watch Nadal—which means that his games gen­er­ate a lot of util­ity—but few peo­ple want to watch no 300).
But there are also fields where you have an en­tirely differ­ent pat­tern. The differ­ence in pro­duc­tivity be­tween the world’s best cleaner and the av­er­age cleaner is, I’d guess, tiny. Similarly, if Peter is twice as strong as Paul, he will be able to fetch as much as wa­ter as needed in half the time Paul needs—nei­ther more, nor less. In other words, the re­la­tion­ship be­tween abil­ity and pro­duc­tivity in these fields is lin­ear:
You get ap­prox­i­mately this lin­ear pat­tern in many phys­i­cal jobs, but also in some in­tel­lec­tual jobs. As­sume, for in­stance, that there is an in­tel­lec­tual field where the only thing that de­ter­mines your pro­duc­tivity is your abil­ity to ac­quire and mem­o­rize fac­tual in­for­ma­tion. Say also that this field is neatly sep­a­rated into small prob­lems, so that your knowl­edge of one prob­lem doesn’t af­fect your abil­ity to solve other prob­lems. In this case, a twice as good ca­pac­ity to ac­quire and mem­o­rize fac­tual in­for­ma­tion will mean that you’ll be able to solve twice as many of these prob­lems—nei­ther more nor less. Now there is ob­vi­ously no in­tel­lec­tual field where you have ex­actly this pat­tern, but there are fields—the more “de­scrip­tive”, as op­posed to the­o­ret­i­cal, so­cial sci­ences come to mind—which at least ap­proach it, and where the differ­ences in pro­duc­tivity hence are much smaller than they are in the­o­ret­i­cal physics or math­e­mat­ics. (Of course, there are other pat­terns be­sides these; for in­stance, in some jobs, what’s im­por­tant is that you meet some min­i­mum level of abil­ity, be­yond which more abil­ity trans­lates in very lit­tle ad­di­tional pro­duc­tivity.)
Due to the fact that differ­ent aca­demic dis­ci­plines have more or less the same pay struc­ture, and are gov­erned by similar rules and so­cial in­sti­tu­tions, these large differ­ences be­tween them are, how­ever, sel­dom noted. This con­tributes to our in­abil­ity to see how huge the differ­ences in pro­duc­tivity be­tween differ­ent sci­en­tists are in some dis­ci­plines.
The differ­ence be­tween these two pat­terns is due to the fact that the first kinds of jobs are more “so­cial” than the lat­ter kinds in a par­tic­u­lar way. The use­ful­ness of your work in the­o­ret­i­cal physics is de­pen­dent on how good oth­ers are at the­o­ret­i­cal physics in a way the use­ful­ness of your wa­ter fetch­ing isn’t. Even if you’re weak, you’ll still con­tribute some­thing by car­ry­ing a small amount of wa­ter to put out a fire, but if you’re not above a cer­tain level of cog­ni­tive abil­ity, your work in the­o­ret­i­cal physics will have no value what­so­ever.
I sup­pose that economists must have writ­ten on this phe­nomenon—what I term other-de­pen­dent pro­duc­tivity. If so I’d be in­ter­ested in that and in adopt­ing their ter­minol­ogy.
I think one rea­son why peo­ple have trou­ble ac­cept­ing Yud­kowsky’s pic­ture is that they note how vastly much more pro­duc­tive Ein­stein was than an av­er­age physi­cist (let alone the village idiot...) and then in­fer that this differ­ence must be due to a vast differ­ence in in­tel­li­gence. Hence point­ing out that the differ­ence in pro­duc­tivity could be vast even though the differ­ence in in­tel­li­gence is not, due to the fact that pro­duc­tivity in the­o­ret­i­cal physics is strongly other-de­pen­dent, should make peo­ple more dis­posed to ac­cept Yud­kowsky’s pic­ture.
It would be in­ter­est­ing to dis­cuss what the re­la­tion­ship be­tween abil­ity and pro­duc­tivity is in differ­ent jobs and in­tel­lec­tual fields. I leave that for later, though. Ob­vi­ously, the ques­tion of how abil­ity is to be defined is rele­vant here. This ques­tion was ex­ten­sively dis­cussed in the com­ments to Yud­kowsky’s post but I have avoided to dis­cuss it for two rea­sons: firstly, be­cause I think it is pos­si­ble to get an in­tu­itive grasp of the phe­nom­ena I’m dis­cussing with­out a pre­cise defi­ni­tion of abil­ity, and, sec­ondly, be­cause an ex­ten­sive dis­cus­sion of this no­tion would have made the post far too long and com­pli­cated.
Edit: Here is a rele­vant ar­ti­cle I just found on Marginal Revolu­tion on “win­ner-take-all economies” where “small differ­ences in skills can mean large differ­ences in re­turns”. It also has some use­ful tips for fur­ther read­ing.