Being the (Pareto) Best in the World

The gen­er­al­ized effi­cient mar­kets (GEM) prin­ci­ple says, roughly, that things which would give you a big wind­fall of money and/​or sta­tus, will not be easy. If such an op­por­tu­nity were available, some­one else would have already taken it. You will never find a $100 bill on the floor of Grand Cen­tral Sta­tion at rush hour, be­cause some­one would have picked it up already.

One way to cir­cum­vent GEM is to be the best in the world at some rele­vant skill. A su­per­hu­man with hawk-like eye­sight and the speed of the Flash might very well be able to snag $100 bills off the floor of Grand Cen­tral. More re­al­is­ti­cally, even though fi­nan­cial mar­kets are the ur-ex­am­ple of effi­ciency, a hand­ful of firms do make im­pres­sive amounts of money by be­ing faster than any­one else in their mar­ket. I’m un­likely to ever find a proof of the Rie­mann Hy­poth­e­sis, but Terry Tao might. Etc.

But be­ing the best in the world, in a sense suffi­cient to cir­cum­vent GEM, is not as hard as it might seem at first glance (though that doesn’t ex­actly make it easy). The trick is to ex­ploit di­men­sion­al­ity.

Con­sider: be­com­ing one of the world’s top ex­perts in pro­teomics is hard. Be­com­ing one of the world’s top ex­perts in macroe­co­nomic mod­el­ling is hard. But how hard is it to be­come suffi­ciently ex­pert in pro­teomics and macroe­co­nomic mod­el­ling that no­body is bet­ter than you at both si­mul­ta­neously? In other words, how hard is it to reach the Pareto fron­tier?

Hav­ing reached that Pareto fron­tier, you will have cir­cum­vented the GEM: you will be the sin­gle best-qual­ified per­son in the world for (some) prob­lems which ap­ply macroe­co­nomic mod­el­ling to pro­teomic data. You will have a re­al­is­tic shot at a big money/​sta­tus wind­fall, with rel­a­tively lit­tle effort.

(Ob­vi­ously we’re over­sim­plify­ing a lot by putting things like “macroe­co­nomic mod­el­ling skill” on a sin­gle axis, and break­ing it out onto mul­ti­ple axes would strengthen the main point of this post. On the other hand, it would com­pli­cate the ex­pla­na­tion; I’m keep­ing it sim­ple for now.)

Let’s dig into a few de­tails of this ap­proach…

Elbow Room

There are many table ten­nis play­ers, but only one best player in the world. This is a side effect of rank­ing peo­ple on one di­men­sion: there’s only go­ing to be one point fur­thest to the right (ab­sent a tie).

Pareto op­ti­mal­ity pushes us into more di­men­sions. There’s only one best table ten­nis player, and only one best 100-me­ter sprinter, but there can be an un­limited num­ber of Pareto-op­ti­mal table ten­nis/​sprint­ers.

Prob­lem is, for GEM pur­poses, elbow room mat­ters. Maybe I’m the on the pareto fron­tier of Bayesian statis­tics and geron­tol­ogy, but if there’s one per­son just lit­tle bit bet­ter at statis­tics and worse at geron­tol­ogy than me, and an­other per­son just a lit­tle bit bet­ter at geron­tol­ogy and worse at statis­tics, then GEM only gives me the ad­van­tage over a tiny lit­tle chunk of the skill-space.

This brings up an­other as­pect…

Prob­lem Density

Claiming a spot on a Pareto fron­tier gives you some chunk of the skill-space to call your own. But that’s only use­ful to the ex­tent that your ter­ri­tory con­tains use­ful prob­lems.

Two pieces fac­tor in here. First, how large a ter­ri­tory can you claim? This is about elbow room, as in the di­a­gram above. Se­cond, what’s the den­sity of use­ful prob­lems within this re­gion of skill-space? The table ten­nis/​sprint­ing space doesn’t have a whole lot go­ing on. Statis­tics and geron­tol­ogy sounds more promis­ing. Cryp­tog­ra­phy and mon­e­tary eco­nomics is prob­a­bly a par­tic­u­larly rich Pareto fron­tier these days. (And of course, we don’t need to stop at two di­men­sions—but we’re go­ing to stop there in this post in or­der to keep things sim­ple.)

Dimensionality

One prob­lem with this whole GEM-vs-Pareto con­cept: if chas­ing a Pareto fron­tier makes it eas­ier to cir­cum­vent GEM and gain a big wind­fall, then why doesn’t ev­ery­one chase a Pareto fron­tier? Ap­ply GEM to the en­tire sys­tem: why haven’t peo­ple already picked up the op­por­tu­ni­ties ly­ing on all these Pareto fron­tiers?

An­swer: di­men­sion­al­ity. If there’s 100 differ­ent spe­cialties, then there’s only 100 peo­ple who are the best within their spe­cialty. But there’s 10k pairs of spe­cialties (e.g. statis­tics/​geron­tol­ogy), 1M triples (e.g. statis­tics/​geron­tol­ogy/​macroe­co­nomics), and some­thing like 10^30 com­bi­na­tions of spe­cialties. And each of those pareto fron­tiers has room for more than one per­son, even al­low­ing for elbow room. Even if only a small frac­tion of those com­bi­na­tions are use­ful, there’s still a lot of space to stake out a ter­ri­tory.

And to a large ex­tent, peo­ple do pur­sue those fron­tiers. It’s no se­cret that an aca­demic can eas­ily find fer­tile fields by work­ing with some­one in a differ­ent de­part­ment. “In­ter­dis­ci­plinary” work has a rep­u­ta­tion for be­ing un­usu­ally high-yield. Similarly, car­ry­ing sci­en­tific work from lab to mar­ket has a rep­u­ta­tion for high yields. Thanks to the “curse” of di­men­sion­al­ity, these gold­mines are not in any dan­ger of ex­haust­ing.