Concave, Convex, and Nonlinear Fragility

Nassim Nicholas Taleb’s book, “Antifragile,” is a wealth of information. I’ve returned to it often since first reading it several years ago. My latest revisit has been to better understand his ideas about representing the nonlinear and asymmetric aspects of fragile/antifragile in terms of “concave” and “convex.” My first read of this left me a bit confused, but I got the gist of it and moved on. Taleb is a very smart guy so I need to understand this.

The first thing I needed to sort out on this revisit was Taleb’s use of language. The fragile/antifragile comparison is variously described in his book as:

  • Concave/Convex
  • Slumped solicitor/Humped solicitor
  • Curves inward/Curves outward
  • Frown/Smile
  • Negative convexity effects/Positive convexity effects
  • Pain more than gain/Gain more than pain
  • Doesn’t “like” volatility (presumable)/”Likes” volatility

Tracking his descriptions is made a little more challenging by reversals in reference when writing of both together (concave and convex then convex and concave) and mis-matches between the text and illustrations. For example:

Nonlinearity comes in two kinds: concave (curves inward), as in the case of the king and the stone, or its opposite, convex (curves outward). And of course, mixed, with concave and convex sections. (note the order: concave / convex) Figures 10 and 11 show the following simplifications of nonlinearity: the convex and the concave resemble a smile and a frown, respectively. (note the order: convex / concave)

Figure 10 shows:

So, “convex, curves outward” is illustrated as an upward curve and “concave, curves inward” is illustrated as a downward curve. Outward is upward and inward is downward. It reads like a yoga pose instruction or a play-by-play call for a game of a Twister.

After this presentation, Taleb simplifies the ideas:

I use the term “convexity effect” for both, in order to simplify the vocabulary, saying “positive convexity effects” and “negative convexity effects.”

This was helpful. The big gain is when Taleb gets to the math and graphs what he’s talking about. Maybe the presentation to this point is helpful to non-math thinkers, but for me it was more obfuscating than illuminating. My adaptation of the graphs presented by Taleb:

With this picture, it’s easier for me to understand the non-linear relationship between a variable’s volatility and fragility vs antifragility. The rest of the chapter is easier to understand with this picture of the relationships in mind.

Team Composition

When a potter begins to throw a pot, she picks up a lump of clay, shapes it into a rough sphere, and throws it onto the spinning potter’s wheel. It may land off-center, and she must carefully begin to shape it until, it is a smooth cylinder. Then she works the clay, stretching and compressing it as it turns. First it is a tower, then it is like a squat mushroom. Only after bringing it up and down several times does she slowly squeeze the revolving clay until its walls rise from the wheel. She cannot go on too long, for the clay will begin to “tire” and then sag. She gives it the form she imagines, then sets it aside. The next day, the clay will be leather hard, and she can turn it over to shape the foot. Some decoration may be scratched into the surface. Eventually, the bowl will be fired, and then the only options are the colors applied to it; its shape cannot be changed.

This is how we shape all the situations in our lives. We must give them rough shape and then throw them down into the center of our lives. We must stretch and compress, testing the nature of things. As we shape the situation, we must be aware of what form we want things to take. The closer something comes to completion, the harder and more definite it becomes. Our options become fewer, until the full impact of our creation is all that there is. Beauty or ugliness, utility or failure, comes from the process of shaping.Deng Ming-Dao, '365 Tao - Daily Meditations'

Building a high-performance team from scratch is just as difficult as turning a low-performing team into a high-performing team. However, there are very different reasons why each of these scenarios are difficult.

Like the potter beginning with a lump of clay, when forming a new team we must understand what we have to work with and have a clear idea of the outcomes we want. As we shape the team, we have to be mindful of how the individuals on the team are changing – or not – and whether those changes are moving toward the outcome. If not, we either need to change the desired outcome or alter the material we have to work with, that is, change out one or more people so that the shape of the team is better suited to reaching the desired outcome. It is also important to monitor the speed at which the team is formed or shaped. Too fast, and the team may not coalesce in a way that is healthy or productive. Too slow and they may not coalesce at all, they may “tire” of the slow pace and disengage.

With existing teams, we may have a limited range of options to change the roster. This is more like the an existing piece of pottery that has been fully set.