My drive to the train station on local roads takes ~10 minutes at 45 mph. Slowing to 35 costs me 3 mins, while speeding up to an unsafe 55 gives back less than 2. On longer distances, this asymmetry is more extreme. The human mind is built to think in straight lines, and that usually works, since any smooth function looks linear up close. That is the whole basis of a Taylor approximation. It breaks down when the second term is significant. In real life, this happens often. Here are some convex and concave function examples:

Convex. Flat, then steep.

  • 30 investments, 25 write-offs, and the returns come from one.
  • A thousand falls from one foot leave you fine; one fall from a thousand feet does not.
  • Many emergent behaviors: knowledge accumulates and does nothing for years until it produces the brilliant idea.

Concave. Steep, then flat.

  • The first $100k is life changing, the tenth doesn't change your lifestyle much (but still great!).
  • Year 1 of practice makes you competent and is fun because you notice the change. Year 10 is grueling and makes you only slightly better than year 9.
  • A reputation takes years to build and one event to destroy. The second event barely registers.

I first heard the word convexity from a bond trader when I started my career, and it sounded like a cheat code: a bond with positive convexity gains more on a rally than it loses on an equivalent selloff. In 2013, I worked through Boyd's online lectures and his text Convex Optimization. Taleb's Antifragile is another great reference on the same idea.

Curvature distorts forecasting locally, and it breaks summary statistics globally. Jensen's inequality is the formal version: the average of a convex function is not the function of the average. Up 50% then down 50% averages zero but leaves you at 75. A river 4ft deep on average still drowns people.

Pressing the accelerator feels like it should pay, and we assume the payoff scales with the sensation. That instinct can be misleading. It can also work in your favor: find places where the cost of many attempts are small and fixed, but the payoff is large.