The Lindy effect

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.

Update: Beethoven, Beatles, and Beyoncé: more on the Lindy effect

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10 comments on “The Lindy effect
  1. Sinai says:

    It’s not just mathematical, it’s simple Copernican common sense….

  2. John says:

    By the way, there are people who see power law distributions everywhere. In reaction, some people are quick to point out that few things really follow power laws.

    That argument doesn’t matter here. Assuming a Pareto distribution simplifies the calculations, but the conclusion would remain qualitatively the same if we used any distribution with a decreasing hazard function.

  3. Dave Tate says:

    The propensity of individuals in a large population to attribute power law distributions to phenomena around them follows a power law distribution.

  4. Interestingly, though, while life expectancy doesn’t grow linearly, it does grow, and for the same reasons — life expectancy at birth has to account for younger people dying. Looking at this UK-based calculator, female life expectancy at birth is ~81 years, by age 20 it’s ~82y, and if you live to retirement (65) you’re up to ~85y.

  5. John says:

    Andrew: True, but a British woman’s expected additional life is 81 years at birth, 62 years at age 21, and 20 years at age 65.

  6. If you’re ever at a loss for a topic to blog about I’d love to read your thoughts on insights gained by using the hazard function properly. It’s one of those great metrics that I know I should use more but don’t.

  7. Fred says:

    Wonder what it means in terms of the global population (of technologies).
    Is the average amount of technologies ever increasing (of course it depends on the “birth rate” of technologies).

  8. John says:

    It’s a simplification to speak of technologies dying. They grow and decline in use. I suppose you could say a technology is “dead” when it falls below a certain level of use. In a sense, technologies hardly ever die.

    I talked to someone about the C++ standard a few months ago. He said that much as they tried, they were unable to remove triglyphs from the language because people are using them. These were added to support pre-ASCII character sets! (Some programmers were unable to type symbols like > because these were not in their character sets, and so C++ provided a workaround.) Just when you think surely something has vanished from the earth, you might be surprised.

  9. gwern says:

    Do we really need to invoke anything like power laws? This sounds like it might be a version of the http://en.wikipedia.org/wiki/Doomsday_argument

  10. Bruce says:

    Cf. waiting for a taxi vs. waiting for a bus.

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