## The Lindy effect

Saturday, March 23rd, 2013

The Lindy effect sounds like a short-lived fad from the 1930s, when really it describes such fads — or, rather, their lifetimes:

The longer a technology has been around, the longer it’s likely to stay around. This is a consequence of the Lindy effect. Nassim Taleb describes this effect in Antifragile but doesn’t provide much mathematical detail. Here I’ll fill in some detail.

Taleb, following Mandelbrot, says that the lifetimes of intellectual artifacts follow a power law distribution. So assume the survival time of a particular technology is a random variable X with a Pareto distribution. That is, X has a probability density of the form

f(t) = c/tc+1

for t ? 1 and for some c > 0. This is called a power law because the density is proportional to a power of t.

If c > 1, the expected value of X exists and equals c/(c-1). The conditional expectation of X given that X has survived for at least time k is ck/(c-1). This says that the expected additional life X is ck/(c-1) – k = k/(c-1), and so the expected additional life of X is proportional to the amount of life seen so far. The proportionality constant 1/(c-1) depends on the power c that controls the thickness of the tails. The closer c is to 1, the longer the tail and the larger the proportionality constant. If c = 2, the proportionality constant is 1. That is, the expected additional life equals the life seen so far.

Note that this derivation computed E( X | X > k ), i.e. it only conditions on knowing that X > k. If you have additional information, such as evidence that a technology is in decline, then you need to condition on that information. But if all you know is that a technology has survived a certain amount of time, you can estimate that it will survive about that much longer.

This says that technologies have different survival patterns than people or atoms. The older a person is, the fewer expected years he has left. That is because human lifetimes follow thin-tailed distributions. Atomic decay follows a medium-tailed exponential distribution. The expected additional time to decay is independent of how long an atom has been around. But for technologies follow a thick-tailed distribution.

Another way to look at this is to say that human survival times have an increasing hazard function and atoms have a constant hazard function. The hazard function for a Pareto distribution is c/t and so decreases with time.

The effect applies to many creative artifacts:

The previous post looked at technologies, but the Lindy effect would apply, for example, to books, music, or movies. This suggests the future will be something like a mirror of the present. People have listened to Beethoven for two centuries, the Beatles for about four decades, and Beyoncé for about a decade. So we might expect Beyoncé to fade into obscurity a decade from now, the Beatles four decades from now, and Beethoven a couple centuries from now.

This is in contrast to things that break down:

If you look at a 25 year-old car and a 3 year-old car, you expect the latter to be around longer. The same is true for a 25 year-old accountant and a 3 year-old toddler.