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.