Convexity Bias and Optionality

NOW THE PHILOSOPHER'S STONE

Much of medieval thinking went into finding the philosopher’s stone. It is always good to be reminded that chemistry is the child of alchemy, much of which consisted of looking into the chemical powers of substances. The main efforts went into creating value by transforming metals into gold by the method of transmutation. The necessary substance was called the philosopher’s stone—lapis philosophorum. Many people fell for it, a list that includes such scholars as Albertus Magnus, Isaac Newton, and Roger Bacon and great thinkers who were not quite scholars, such as Paracelsus.

It is a matter of no small import that the operation of transmutation was called the Magnus Opus—the great(est) work. I truly believe that the operation I will discuss—based on some properties of optionality—is about as close as we can get to the philosopher’s stone.

The following note would allow us to understand:

(a) The severity of the problem of conflation (mistaking the price of oil for geopolitics, or mistaking a profitable bet for good forecasting—not convexity of payoff and optionality).

(b) Why anything with optionality has a long-term advantage—and how to measure it.

(c) An additional subtle property called Jensen’s inequality.

Recall from our traffic example in [Chapter 18] that 90,000 cars for an hour, then 110,000 cars for the next one, for an average of 100,000, and traffic will be horrendous. On the other hand, assume we have 100,000 cars for two hours, and traffic will be smooth and time in traffic short.

The number of cars is the something, a variable; traffic time is the function of something. The behavior of the function is such that it is, as we said, “not the same thing.” We can see here that the function of something becomes different from the something under nonlinearities.

(a) The more nonlinear, the more the function of something divorces itself from the something. If traffic were linear, then there would be no difference in traffic time between the two following situations: 90,000, then 110,000 cars on the one hand, or 100,000 cars on the other.

(b) The more volatile the something—the more uncertainty—the more the function divorces itself from the something. Let us consider the average number of cars again. The function (travel time) depends more on the volatility around the average. Things degrade if there is unevenness of distribution. For the same average you prefer to have 100,000 cars for both time periods; 80,000 then 120,000, would be even worse than 90,000 and 110,000.

(c) If the function is convex (antifragile), then the average of the function of something is going to be higher than the function of the average of something. And the reverse when the function is concave (fragile).

As an example for (c), which is a more complicated version of the bias, assume that the function under question is the squaring function (multiply a number by itself). This is a convex function. Take a conventional die (six sides) and consider a payoff equal to the number it lands on, that is, you get paid a number equivalent to what the die shows—1 if it lands on 1, 2 if it lands on 2, up to 6 if it lands on 6. The square of the expected (average) payoff is then (1+2+3+4+5+6 divided by 6)2, equals 3.52, here 12.25. So the function of the average equals 12.25.

But the average of the function is as follows. Take the square of every payoff, 12+22+32+42+52+62 divided by 6, that is, the average square payoff, and you can see that the average of the function equals 15.17.

So, since squaring is a convex function, the average of the square payoff is higher than the square of the average payoff. The difference here between 15.17 and 12.25 is what I call the hidden benefit of antifragility—here, a 24 percent “edge.”

There are two biases: one elementary convexity effect, leading to mistaking the properties of the average of something (here 3.5) and those of a (convex) function of something (here 15.17), and the second, more involved, in mistaking an average of a function for the function of an average, here 15.17 for 12.25. The latter represents optionality.

Someone with a linear payoff needs to be right more than 50 percent of the time. Someone with a convex payoff, much less. The hidden benefit of antifragility is that you can guess worse than random and still end up outperforming. Here lies the power of optionality—your function of something is very convex, so you can be wrong and still do fine—the more uncertainty, the better.

This explains my statement that you can be dumb and antifragile and still do very well.

This hidden “convexity bias” comes from a mathematical property called Jensen’s inequality. This is what the common discourse on innovation is missing. If you ignore the convexity bias, you are missing a chunk of what makes the nonlinear world go round. And it is a fact that such an idea is missing from the discourse. Sorry.