The Fundamental Growth Curve (Part 2)

by radimentary

[Followup to the previous post. I will likely collect and reorganize these shortforms once the month is over.]

The narrative from the previous post outlines my basic model for how growth works. For a typical skill, be it running cross country or solving Rubik’s cubes or engaging in mathematical research, return on investment follows a curve like this:

In the first stage, growth is fast and cheap. The core fundamentals are the easiest to pick up and best documented. Adepts willing and able to mentor novices are plenty. As the weakest one of the group, you’re no threat to anybody’s status (excepting, perhaps, the second weakest one, but they’re in no position to do anything to you).

This is the regime where the 80/20 principle holds: 80% of the absolute value in the activity is picked up with only 20% of the effort, applied with discernment. Most members in any community are novices somewhere in this first stage, enjoying the immediate gratification of visible gains while lacking the stomach for serious investment.

The gains here are absolute in nature. You might not garner any attention for picking up jogging, but your sleep apnea clears up. You might not win any money playing a game, but you become competent enough to enjoy it and be able to appreciate professional play. You might not prove any new theorems taking undergraduate math classes, but you finally understand how to calculate expected value and stop getting duped by slot-machines.

In the second stage, there is a long plateau, a “wall” that most people hit when the newbie gains fall off and serious commitment or exceptional talent seem necessary to progress.

Past the fundamentals, there exist conflicting and ambiguous schools of thought on how to progress to experthood. The athlete is told to follow three different diets by three different nutritionists. Long debates are held about which chapters of Hartshorne are mandatory to truly learn algebraic geometry. The single-digit-kyu Go player is told to focus on opening, on fighting, on life-and-death problems, on endgame, and each is apparently vitally important and the one true path to greatness.

Months of physical time are spent to see small improvements that have no practical impact on your life: whether you run a 9 minute mile or a 6 minute mile, the only thing you’ll win is a participation ribbon. You blog for years only to see your readership jump from dozens to hundreds. You practice your craft doggedly and jump from the worst surgeon in the hospital to the second-best, but when it comes time for their heart transplants, the billionaires still pass you by to wait in line for that star surgeon.

In the third stage, absolute improvements get even slower, but you finally hit a level of mastery where positional, or relative, gains kick in. And my, do they kick in fast.

You become one of the top players in your cohort, and people start to notice you. Coaches give you special treatment, you win minor awards, get sent to training camps, participate in more rarified cohorts.

You have a renewed and enormous motivation to improve: every tiny absolute improvement could move you up one giant discrete ranking. Jumping from the 11th best-selling author to the 10th doubles your book sales. Shaving a couple seconds off your personal best means a new state record and a full ride to college. Writing one additional paper in graduate school edges out the next candidate to land you the fancy fellowship that keeps your academia dreams alive.

And look, the divide between the second and third stages is not just a shallow artifact of human status regulation. It’s an essential feature of the collaborative optimization problems we face all the time. As long as people work in teams and specialize according to their relative advantage, an individual hardly contributes in a given dimension unless they are the best or nearly so.

Look at it this way: if eight college friends are taking a road trip through gravelly mountain roads, the best driver is going to drive the van most of the time, and the second best driver might help pick up some slack. How well these two drive is dreadfully important, but as for the other six – it makes not a wit of difference if they even have licenses. Everyone in that van is incentivized to pour their resources into helping that best driver become even more skilled.

To be continued…