Book review: The Black Swan: the impact of the highly improbable (Part I)


How do we interpret and prepare for very rare events? Most of us tend to focus on the outcomes most likely to occur. We might think that those are the only possible outcomes. Or, we might briefly consider some others, only to ignore them because they are too outlandish. But should we perhaps take rare events more seriously?

To Nassim Nicholas Taleb, author of The Black Swan: the impact of the highly improbable (unrelated to a recent movie with a similar title), we would be wise to take them far more seriously. In fact, according to Taleb (or NNT, as he refers to himself throughout the book), highly improbable events are generally the ones that are actually important. Furthermore, by ignoring them, we’re creating heavily inaccurate representations of what the future ought to look like.

The title deserves some explanation. A Black Swan, in Taleb’s parlance, is an event that is completely unpredictable in advance, but that becomes explainable, and seemingly inevitable, when we change our stories in the future to reflect our more accurate hindsight. Once upon a time, it was believed in certain parts of the world that all swans were white. A black swan sighting was seen as an impossibility — until someone found one. Our stories then changed as the existence of black swans became part of our collective body of knowledge, so now we’re less likely to be surprised if we see one. The facts didn’t change, but our way of understanding them did.

Where do Black Swans occur? Taleb distinguishes between two classes of possible events: Mediocristan and Extremistan. In Mediocristan, we can count on unlikely events not occurring. But, in Extremistan, events perceived as unlikely can and do happen regularly. Moreover, in Extremistan, the world is dominated by these rare events; they’re the ones that shape history.

Let’s look at an example that Taleb discusses, supposedly a model example of the difference between events from Mediocristan and events from Extremistan. Suppose we were to lead 1000 people at random to a room. We could weigh each person in order to compute the average weight of these 1000 people. We suspect that this average weight is representative of the average weight of all people in the world, in the sense that if we search out the heaviest person in the world and average em in with the other 1000, the average changes quite little.

On the other hand, if we run the same experiment with net wealth, we see a very different picture. Most likely, the 1000 selected people will not include any billionaires. If we average in Bill Gates with the rest of the people, the average net wealth will change dramatically, since his wealth will completely trump the rest of the group.

So, according to Taleb, weight is an event from Mediocristan, whereas net wealth is an event from Extremistan. The former is well-modeled by the Gaussian bell curve, whereas the latter is not.

Furthermore, by picking a sample of 1000 people, we can develop a good estimate of the maximum weight of a person, whereas we cannot develop a good estimate of the maximum net wealth. One person could be so far off the chart in wealth that we can’t estimate the maximum until we select a sample that contains most of the population of the world.

This was the first thing I read in this book that make me angry. Taken purely naïvely, it’s clearly true that weight fits better on a bell curve than does net wealth. But that’s just because net wealth is the wrong number to measure! Let me explain why.

Suppose two people, one much heavier than the other, each wish to gain 10 pounds. How do they do it? They each have to eat roughly the same number of excess calories in order to gain the weight. In other words, it requires pretty much the same amount of work for a 500-pound person to gain 10 pounds as it does for a 100-pound person to gain 10 pounds. This means that weight is fairly well-measured on a linear scale.

But the story is completely different for net wealth. Suppose two people each wish to gain $10000. One person currently has $50000, and the other has $50 million. Which will have an easier time of it?

The second one, by far. If nothing else, if the two people each make the same (proportional) investments, they earn very different amounts: if the first person invests $50000 and earns an extra $10000 through eir investments, then the second person, investing in the same way, will earn not $10000 but $10 million.

In other words, money accumulation is exponential. Therefore, if we’re going to say anything sensible at all about the distribution of net wealth, we first need to take logarithms. So, we should expect that the logarithm of net wealth is normally distributed.

Sadly, Taleb doesn’t mention that. Much later on, he claims that net wealth (not normalized) is distributed nicely for a while (he doesn’t like words like “Gaussian,” ones that have actual meanings), but then its tail is distributed by a power law. And, it does appear to be the case that the tail of the distribution satisfies a power law, although I don’t understand it. But there’s no explanation for why, and that seems highly suspect to me. Why should it change at some point? My explanation isn’t some mathematical magic; it should be obvious to anyone who knows even a little bit about probability distributions. I didn’t have to collect any data to see what the right quantity to measure is. Sadly, the fact that net wealth is best measured logarithmically, whereas spending is best measured linearly (we all pay pretty much the same amount of money for rice, regardless of how much money we have), is the cause of a large amount of social inequity.

In fact, even if we look at weight, we’ll find that the Gaussian distribution on a linear scale is not a particularly good fit: if the mean weight is (say) 150 pounds (the actual number isn’t particularly relevant), then it’s clear that there are more people who weigh 350 pounds than those who weigh -50 pounds. We ought to correct our scale here too, by noting that it is probably a bit easier for heavier people to gain weight, statistically speaking.

So, to Taleb, the existence of really rich people is a Black Swan. To me, it’s not: it’s completely inevitable and predictable once we’ve found the right quantity to measure.

(To be continued…)

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About Simon

Hi. I'm Simon Rubinstein-Salzedo. I'm a mathematics postdoc at Dartmouth College. I'm also a musician; I play piano and cello, and I also sometimes compose music and study musicology. I also like to play chess and write calligraphy. This blog is a catalogue of some of my thoughts. I write them down so that I understand them better. But sometimes other people find them interesting as well, so I happily share them with my small corner of the world.
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4 Responses to Book review: The Black Swan: the impact of the highly improbable (Part I)

  1. Sergey says:

    I saw a talk about income distribution which may be relevant, wherein it was asserted that there are actually two separate distributions for income. The majority of people have an exponential (Boltzmann-Gibbs) income distribution, which is a consequence of most monetary transactions being basically random exchanges (like energy exchanges when particles collide in a gas). However, the richest people have a power law (Pareto “tail”) income distribution, as a consequence of investing. Economists are mostly interested in the “tail”, and have thus ignored the majority distribution (which would also force them to abandon the assumption that economic decisions are completely rational). Please check out http://arxiv.org/PS_cache/cond-mat/pdf/0103/0103544v2.pdf

  2. Pingback: Book review: The Black Swan: the impact of the highly improbable (Part II) | Quasi-Coherent

  3. Pingback: Book review: The Black Swan: the impact of the highly improbable (Part III) | Quasi-Coherent

  4. Pingback: Swans are White, I Swear! « Sophronismos

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