My drive to the train station takes about ten minutes at 45 mph. Slowing to 35 costs me 3 mins. Speeding up to 55 gives back less than 2. A maniacal 65 buys barely a minute on top of that. The human mind is built to think in straight lines, and usually that works, since any smooth function looks linear if you zoom in far enough. That is the whole basis of a Taylor approximation. The trouble starts when the second term is the one that matters and this happens often, here are some convex and concave examples:

Convex. Flat, then steep. The last unit does more than the first.

  • Thirty investments, twenty-five 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 idea.

Concave. Steep, then flat. The first unit does most of the work.

  • The first hundred thousand of income is life changing, the tenth doesn't change your lifestyle much (but still great!).
  • Year one of practice makes you competent and is fun because you notice the change. Year ten is grueling and makes you only slightly better than year nine.
  • Adding people to an overdue project. The first hire helps, the fifth adds coordination cost, and past that you are slowing down.
  • A reputation takes years to build and one event to destroy. The second event barely registers.

I first heard the word 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, so the move in your favor is worth more than the move against you. In 2013, I worked through Boyd's online lectures and his text Convex Optimization and was hooked.

Curvature distorts forecasting in the small, and it breaks summary statistics in the large. Jensen's inequality is the formal version: the average of a convex function is not the function of the average. Up fifty percent then down fifty percent averages zero and leaves you at seventy-five. A river four feet deep on average drowns people.

You feel the difference when you speed up and assume the payoff scales with the feeling. That instinct costs you in higher stakes environments. The same curvature works for you once you look for it. Make the cost of each attempt small and fixed, then take enough attempts that one of them can be large. Most of them will come to nothing, which is the design working rather than failing.