Heavy Tail: Our World Obeys Extreme Values

There is a major discrepancy between the perception of the average person and the true face of the world. Understanding this difference can elevate your perspective on life.

Typically, we grow up living in various “averaged” small worlds. If you have siblings, good food at home is surely divided equally. In primary school, most of your classmates come from your neighborhood, with living conditions roughly similar to yours; you use the same textbooks and each person has one school uniform. Even in university or at work, although your companions may come from all over, since you are in the same university or workplace, your abilities and income levels are likely comparable.

Living in these small worlds long-term, your physical sensation is that differences between people shouldn’t be too large, and any good thing is best distributed equally. You silently believe that “average” is synonymous with “justice.”

But occasionally leaving these small worlds, you’ll find the world is not average at all. When I first took a train journey as a child, my greatest impression wasn’t how large the outside world was, but how unique my hometown of Harbin was: it turned out that more of China’s land was fields and villages, and cities were like isolated islands.

Yet those islands carry a decisive weight.

Most of the population in industrialized countries live in cities, and a few large cities account for a disproportionate amount of population, economy, and innovation.

People are even less average. A reality you surely don’t feel in your daily life is that the richest 1% of the world’s population owns more wealth than the remaining 99% combined; the world’s 8 richest people—just these 8 individuals—own more wealth than the bottom half of the global population combined [1].

In fact, it’s the same in every field: a few companies occupy most of the industry’s business, a few stars have most of the fans, a few bestsellers and blockbuster movies generate most of the sales and box office…

This is what we often call the “80/20 Rule,” which perhaps originated with the Italian economist Vilfredo Pareto: he first noticed that 20% of the population in Italy owned 80% of the land, then found this pattern everywhere: 20% of star products generate 80% of revenue, 20% of core customers contribute 80% of turnover, and a few top employees produce most of a company’s performance… Of course, the specific ratio isn’t necessarily 20 to 80; it could be 10 to 90 or 1 to 99. The key is that an extreme minority holds a particularly large share.

You don’t see it around you, but the truth of the world is that it is not average at all. It tends toward extremes.

You might be angry about this: Why? If we confiscated the assets of those 8 richest people, wouldn’t it immediately double the net worth of half the people in the world? Why don’t we knock off the top influencers to let “everything else thrive”?

Or perhaps, you want to join that extreme minority.

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This phenomenon of “extreme minority occupying a large share” is called a “heavy-tailed distribution” in statistics, and sometimes a “fat tail.” Mathematically, it has more than one implementation form, the most typical being the power law distribution, but we need not dwell on the details here. Simply put, a heavy tail means the tail of the distribution curve is relatively thick and large.

For example, human height is not a heavy-tailed distribution, but a so-called “normal distribution,” a symmetrical bell-shaped curve: most people are concentrated near the average, with few people at either tail, and the tails are very short—the shortest person is over a meter tall, and the tallest won’t exceed three meters.

If human wealth were distributed like height, particularly rich people would be very rare, and they wouldn’t be that rich; after all, no one’s height can be ten times yours. However, wealth is a heavy-tailed distribution: there are many people whose wealth exceeds yours by ten times, and some exceed you by a hundred or a thousand times…

Talking about averages for a heavy-tailed distribution is meaningless because the impact of the extreme minority is simply too great. You can certainly look for a spouse based on the average height of Beijing residents, but if you insist that your “other half’s” income must exceed the average, you will eliminate far more than half the people.

Height, life expectancy, and IQ can all be discussed in terms of averages, and we are fortunate that human innate conditions do not differ too much. But generally speaking, the world does not obey the average.

The world obeys extreme values.

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Then you might ask, since human height and IQ are similar, how can wealth differ so much? There are many causes of heavy-tailed distributions, and they all relate to some kind of “positive feedback process.” Positive feedback, simply put, is “the larger it is, the easier it is to get larger,” also known as the “Matthew Effect.”

For example, suppose there’s a financial product with 10% annual interest. If you have 1 million yuan, you can earn 100,000 yuan in a year; but if you only have 10,000 yuan, you can only earn 1,000 yuan. Your investment actions and adventurous spirit haven’t differed, but your income can differ by that much. This is the most fundamental reason for the widening gap between rich and poor. The rich aren’t necessarily more diligent than the poor; they get richer because they are already rich.

For another example, in 2025, under the AI wave, large companies like Oracle and AMD could see their stock prices soar by 30% to 40% overnight just by announcing cooperation with OpenAI! How can such large companies still have such huge growth potential? In economics, this is called “Gibrat’s law,” meaning a firm’s growth rate is approximately independent of its size. If both large and small companies can grow by 10% a year, their size gap will inevitably widen.

Fame is also positive feedback. An already popular star, because of high traffic, finds it easier than an ordinary star to obtain the highest quality resources—the audience pays more attention to her, the platforms inevitably value her more, and thus her follower growth rate further accelerates.

Even in academia, it’s the same. A few star scientists have the greatest impact on scientific progress, publishing far more papers than ordinary scientists [2]. Why? Because higher prestige brings more research funding and better collaboration opportunities, allowing them to pick the best research topics, which in turn makes them a bit smarter… You say wealth is unfair, but actually, even scientific research is unfair.

The reason wealth, opportunity, and prestige can form positive feedback is that they all belong to a “multiplicative world”:

Your Incremental Gain = Your Current Action × Your Current Assets.

The larger the assets, the larger the incremental gain. Simply put, there is “compound interest.” However, many people in the world live in an “additive world”: one day’s labor for one day’s pay, where the current income has nothing to do with previous accumulation—no compound interest.

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We instinctively feel this is very unfair: Shouldn’t returns be distributed according to labor? How can they be distributed according to assets?

Realizing they are in a heavy-tailed world, many people have an urge to “flatten” the world. You can consider adjustment through taxation, but don’t be too radical, because almost all historical attempts to “rob the rich to help the poor” have ended in economic collapse. Thomas Sowell lists several examples in Social Justice Fallacies [3]. For instance, the Ugandan government found that most of the wealthy in the country were entrepreneurs from Asia while the locals were very poor; feeling it was unfair, they expelled these Asians… As a result, the Ugandan economy collapsed, and the wealth of the locals decreased rather than increased.

The point here is that multiplication isn’t just good for the rich person being multiplied; it’s good for everyone. Your company spent a fortune creating a new product and wants to find an influencer to promote it. If the platform talks to you and says a certain star is already too rich and is crowding out opportunities for ordinary influencers, could you please hire someone less famous for the sake of fairness? You certainly couldn’t agree!

You don’t hire a superstar because you like her—you do it because she can maximize the amplification of your product’s visibility. The multiplicative effect is two-way; both you and she want to be multiplied by a large number. Moreover, consumers want to buy the best products on the channels they follow most, getting the highest percentage discount with the largest number of people.

The alliance of the strong is actually a win-win situation. If you want to start a company, you want super-rich people to invest in you. If you want to do research, you want top scientists to collaborate with you. These people have more resources, broader vision, and stronger capabilities, making it most likely for your project to succeed. Letting superstars perform to their full potential is a plus for collaborators, consumers, and society as a whole.

Of course, the byproduct of “winner-take-all” is the widening gap between rich and poor. But actually, the problem isn’t as serious as people imagine. Stars won’t be popular forever; people get bored with them, they age, and they fail to keep up with the latest changes. Gibrat’s Law can’t keep large companies young forever; the industry they are in will decline, and they will be disrupted because they miss the latest innovation. There have never been rich people who remain so forever in history; the wealth of the rich is often quickly squandered by future generations.

In fact, even within a person’s lifetime, it’s not that the rich are always rich and the poor always poor. Economists have long found [4] that if you look at a cross-section of all Americans’ income in a given year, the gap between rich and poor is indeed huge and increasing year by year—but if you look at longitudinal data, following people over their lifetimes, you’ll find that those who were very poor when young don’t remain poor for a lifetime; most will see significantly higher incomes as adults. Meanwhile, those who were once particularly wealthy don’t stay lucky forever; their ranking is likely to drop at some stage.

Investment failure, intergenerational dilution, divorce and dissolution, taxes, and inflation are all ways the wealth of the rich is reduced. The investments made by the rich are their best and most heroic contribution to society, because an investment that “busts” is nature’s fastest way to regulate the wealth gap.

The wealth gap is indeed a problem, but if it is the price of the multiplicative world, we would rather pay that price than cancel the multiplication. It’s not scary that someone flies high; it’s scary when the entire airflow stops.

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Rather than complaining that the heavy tail is unfair, it’s better to think about how to jump out of the additive world and join the multiplicative world. To do that, you need to change your “averaging” mindset.

If what you do all day is add up a few fixed items, you will instinctively want to raise the average, especially by making up for your shortcomings. For example, the college entrance exam consists of several subjects, each with a maximum score of 150 points. This means even if your math reaches the level of a professional mathematician, you can only get 150 points at most. Thus, for a student strong in math but weak in English, the best strategy is to tutor in English rather than continue to strengthen math.

This is like a taxi driver who only has 30 days a month to earn money, so his primary concern is not to miss work rather than to increase his daily income.

But in the multiplicative world, what you want to be is a multiplier participating in an equation; you only need one very long “long board” (strength). We care about how strong a mathematician’s math is—the stronger the better, with no upper limit—rather than how weak his English is.

As a thinking tools course, we certainly hope you become a cognitive generalist—but in your specific career, you must be able to leverage your “long board” strength.

OpenAI CEO Sam Altman once said:

“Generally, the highest return on investment comes from getting better at the things you’re already good at, rather than improving in your weak areas… Most people focus too much on their weaknesses. You want to be world-class at some narrow intersection, rather than doing a whole bunch of things fairly well.”

The secret to joining the heavy tail is to identify a field that allows you to multiply and then implement positive feedback.

For example, venture capital is a clearly heavy-tailed business: about 10% of investment exits contribute 85% of cash returns [5]. Thus, many people think venture capital is about casting a wide net, investing in many companies, and if one grows large, it’s a win… Actually, that’s not it.

The secret of venture capital is that you can never just invest once and forget about it; you must choose the right time to double down and grow with the company’s positive feedback. One right that investors value most is “pro rata rights” [6], which allows me to invest more in you in the future in a certain proportion, ensuring my stake isn’t diluted. As your shareholder, I can spot your potential earlier than outsiders; once you show extreme potential, I will increase my investment.

The multiplicative world requires you not to treat every partner equally; you must seek to mutually promote and grow with “large multipliers.”

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Positive feedback isn’t always a good thing. The spread of viruses and panic is also positive feedback. Financial storms, hurricanes, and earthquakes are also the result of cascaded positive feedback in complex systems. Nassim Taleb’s signature theory, “Black Swan” events, also occur on the heavy tail. So in a heavy-tailed world, you can’t just think about striking it rich; you must protect your principal first. To do that, you need to understand what is “fragile” and what is “antifragile” [7]…

And nothing—except perhaps the Moore’s Law for computer chips and the scaling law of AI so far—will grow at a fixed ratio forever. When positive feedback reaches a certain point, diminishing marginal returns occur [8], and you’ll have to develop a “second curve”…

Those are all very important, and we’ll talk about them specifically. But most importantly, this world loves to produce an extreme minority.

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Whether you like it or not, this is not an average world where kindergarten children line up to eat fruit. Averaging is an artificially created environment in small worlds; it requires someone to deliberately allocate it. Averaging does not follow natural laws.

As an ordinary person in a heavy-tailed world, I will try to join the multiplicative world. I hope to be associated with capital, network effects, and prestige. I will try to avoid jobs that are “do one task, get one payment, unrelated to current assets.” I hope my past achievements can influence my future opportunities. I hope to build personal credibility and a brand, accumulate compound interest, and form positive feedback.

We will provide many tools to help you do this. For now, don’t you think the heavy-tailed world is a lot of fun?

Closing Poem

A few stand on the peaks, surrounded by flourishing flowers; The many dwell in the valleys, where starlight is scarce. Multiplication gathers mountains, compound interest builds towers; Black swans cast shadows, toppling them in an instant. Inequality is the norm, opportunity follows the long tail; Build your battlements first, let resilience be your fort. Embrace positive feedback, use your strength to pry the summit; If the lamp of the heart stays bright, all things will be clear.

Notes

[1] Elliott, Larry. “World’s Eight Richest People Have Same Wealth as Poorest 50%.” The Guardian, January 16, 2017. [2] Lotka, Alfred J. “The Frequency Distribution of Scientific Productivity.” Journal of the Washington Academy of Sciences 16 (1926): 317–323. [3] Social Justice Fallacies 6: We Cannot Expect the Universe to Uphold Justice. [4] Social Justice Fallacies 3: The Emotional Value of Intellectuals. [5] Wiltbank, Robert E., and Wade T. Brooks. Angel Returns 2016: Tracking Angel Returns. Angel Resource Institute, 2016. [6] Gompers, Paul, Will Gornall, Steven N. Kaplan, and Ilya A. Strebulaev. “How Do Venture Capitalists Make Decisions?” NBER Working Paper 22587, 2016. [7] Nassim Taleb and the Philosopher’s Stone. [8] Compound Interest Chicken Soup and Real World Growth.