Latticework of Mental Models: The Power Law

Do you know how much property damage and loss of lives is caused by earthquakes every year?

No idea? Consider this – An earthquake in Japan in 2011 caused an estimated property loss of US$ 235 billion. The deadliest earthquake in recorded history (in terms of loss of lives) occurred in Shaanxi (China) killing more than 800,000 people.

According to one study, annual average losses from earthquakes range from US$ 1.3 to 5.7 billion. That’s loss per year because of all the earthquakes. This is quite huge, considering that it’s caused by an event which we have no control over.

These statistics show that when mother nature’s fury is unleashed, even the modern society with all its technology and resources, finds itself helpless.

Now another question. What do you think is the average loss caused by an earthquake, in terms of dollars per earthquake? Let’s do a quick back of the envelope calculation. So yearly loss of US$ 5.7 billion divided by number of earthquakes per year. How much would that be? Take a guess.

Probably that number would be couple of million dollars per earthquake. Right?

It’s actually little over US$ 10,000. That’s it! How come?

That’s because the total number of earthquakes (excluding the ones that can’t be detected by instruments) occurring every year are more than 500,000 (source: Wikipedia). Yes you read it right.

But the catch is that majority (more than 80 percent) of these tremors are so minor that they can’t even be felt by humans. Animals, they say, are more sensitive to such minute shocks. So next time when you see a dog with raised ears or a kitten with tilted head, they’re probably experiencing an earthquake.

Of these 20 percent (100,000) earthquakes which are felt by humans, about 1 percent actually cause any damage. So we see that more than 99 percent of the damage is caused by less than 1 percent of the earthquakes. That, my friends, is the Power Law, a very important mental model from Algebra.

It’s not just earthquakes but other natural calamities, like floods, also adhere to Power Law. 40 percent of a decade’s damage due to flood usually comes from just one such event.

You might argue that a severe earthquake or a monstrous flood, or any other natural calamity won’t cause any damage if it were to occur at a place where there aren’t any property or human presence. True, but it’s not about loss in terms of money.

Damage is a proxy for the intensity of the earthquake. And intensity of an earthquake is a function of energy released during the earthquake. Mark Buchanan, in his book Ubiquity – Why Catastrophes Happen, describes how some systems, including earthquakes, can be represented by the Power Law. He writes …

In terms of energy, it turns out that the Gutenberg-Richter law boils down to one very simple rule: If earthquakes of type A release twice the energy of type B, then type A quakes happen four times less frequently. Double the energy, that is, and an earthquake becomes four times as rare.

[For wildfires] Once again we find the same geometric pattern: double the area covered by a fire, and it becomes about 2.48 times as rare…In other words, despite the immensely complex picture of how fires spread, a startlingly simple pattern emerges when you look at how often you find fires of different sizes – a kind of Gutenberg-Richter law for ecological conflagration.

In algebra, a Power Law is any curve in which the height changes in proportion to the horizontal distance raised to some power.

Power law is an extension of the Pareto principle, named for Italian economist Vilfredo Pareto. Pareto principle describes the relationship between how common a factor is and how much influence it exerts. It says that the most unusual events will have the greatest impact. Pareto’s study determined that 80 percent of privately held land in Italy was owned by 20 percent of the population.

In his book, The Black Swan, Nassim Taleb writes …

It [Pareto principle] is the common signature of a power law – actually it is how it all started, when Vilfredo Pareto made the observation that 80 percent of the land in Italy was owned by 20 percent of the people. Some use the rule to imply that 80 percent of the work is done by 20 percent of the people. Or that 80 percent worth of effort contributes to only to only 20 percent of results, and vice versa.

…The 80/20 rule is only metaphorical; it is not a rule, even less a rigid law. In the U.S. book business, the proportions are more like 97/20 (i.e., 97 percent of book sales are made by 20 percent of the authors)

Power law is an acute form of Pareto principle where the proportion of 80:20 becomes as much skewed as 99:1. This law can be seen all around us. For example, in most businesses, 80% of the revenue is brought in by top 20% of the activities.

This relationship between low frequency and high impact is found again and again, in various fields of science, business, and elsewhere including Twitter (20 percent of the Twitter users are responsible for 80% of all the tweets, roughly).

In movie business, for instance, 90 percent of the profits come from 10 percent of the movies. Which means many films lose money, and a few are huge hits. But if you compute the average box office receipts, you may be fooled into thinking that all films earn money — which is far from the case. And for directors, 40 percent of their lifetime revenues come from a single film. It’s also known as ‘winner takes all’ effect.

In his talk A Lesson on Elementary Worldly Wisdom As It Relates to Investment Management and Business, Charlie Munger, while talking about advantages of scale, explains this effect. He says …

There’s another kind of advantage to scale. In some businesses, the very nature of things is to sort of cascade toward the overwhelming dominance of one firm. The most obvious one is daily newspapers. There’s practically no city left in the United States, aside from a few very big ones, where there’s more than one daily newspaper.

And, again, that’s a scale thing. Once I got most of the circulation, I got most of the advertising. And once I got most of the advertising and circulation, why would anyone want the thinner paper with less information in it? So it tends to cascade to a winner-take-all situation.

My friend Jana Vembunarayanan has written a wonderful post on this mental model. Please read it for additional insights and interesting examples.

Here is an fascinating example of how Power Law can be used in the context of health and fitness. Arthur De Vany in his book, The New Evolution Diet, writes…

There is a power law of exercise, too: Your least frequent most extreme exertions will have the greatest influence on your fitness. The peak moments of a workout count far more than the amount of time you spend working out. This is why a series of 40-yard sprints at full speed benefits you more than a half an hour of jogging. It’s also the reason why lifting a weight heavy enough to make your heart pound your muscles burn counts more than spending hours at the gym always in your comfort zone, never truly challenging your body. When a workout becomes an unvarying, monotonous routine, it loses its effectiveness.

In modern world, the increased complexity around us makes our world more prone to Power Law. There are few other mental models which are relevant here – Complex Adaptive Systems, Network Effect and Feedback Loop. These, to some extent, explain the occurrence of Power Law.

As systems become more complex, with individual parts interacting and evolving (like stock market), the behaviour of the systems become highly unpredictable. And because of the mutual interaction and network effect (e.g. more people join Facebook because they find more people there instead of any other social network) the outcome becomes highly skewed.

Nassim Taleb has named these complex systems as ‘Extremistan’. And Black Swan is the term he has coined to represent those rare and unexpected outliers which have high impact (like those rare but severe earthquakes).

In Investing

When it comes to investing, power of compounding is one of the most important fundamental principle to learn. Rs 100,000 compounded at the rate of 15 percent for 40 years becomes Rs 26,800,000 or Rs 2.6 crore. What’s the most important part of this compounding process is that more than 50 percent of this Rs 2.6 crore comes in the last 5 years!

For that matter, the world’s greatest investor, Warren Buffett, made 80 percent of his current total wealth after his 60th birthday. Call it compounding, call it Power law – it’s very counter intuitive to human brain. And that’s why most people just can’t comprehend the huge element of nonlinearity present in the process of long term value investing. Patience has non-linear payoff.

Peter Theil writes in his book Zero To One …

In venture capital, where investors try to profit from exponential growth in early-stage companies, a few companies attain exponentially greater value than all others. This is because venture returns don’t follow a normal distribution overall. Rather, they follow a power law: a small handful of companies radically outperform all others. If you focus on diversification instead of single-minded pursuit of the very few companies that can become overwhelmingly valuable, you’ll miss those rare companies in the first place.

How To Benefit From The Power Law

Rolf Dobelli, in his book The Art of Thinking Clearly, writes …

Back in the Stone Age, we hardly ever encountered anything truly extraordinary. The deer we chased was sometimes a bit faster or slower, sometimes a little bit fatter or thinner. Everything revolved around a stable mean.

Today is different. With one breakthrough, you can increase your income by a factor of 10,000. Just ask Larry Page [founder of Google], George Soros, J.K. Rowling [author of Harry Potter series] or Bono. Such fortunes didn’t exist previously…

So what can be done? Put yourself in situations where you can catch a ride on a positive Black Swan…become an artist, inventor or entrepreneur with a scaleable product. If you sell you time (e.g. as an employee, dentist or journalist), you’re waiting in vain for such a break. But even if you feel compelled to continue as such, avoid surroundings where negative Black Swans thrive. This means: stay out of debt, invest your savings as conservatively as possible and get used to modest standard of living – no matter whether your big breakthrough comes or not.

Conclusion

So far we have discussed more than 20 different mental models in this series. If you have been a regular reader and gone through most of the posts in Latticework series, you would have found that sometimes the ideas are contradictory.

I totally agree with you on that. But that’s in the nature of truth, it feels contradictory and sometimes outright paradoxical.

But don’t forget, just like a carpenter’s toolbox contains tools to break things and also instruments to join them back, a multidisciplinary thinker’s mental toolbox contains tool of thinking which may sometimes point in different directions. The key to learn is how and when to use a particular tool, which can only come by practicing and implementing the mental models.

In that sense multidisciplinary thinking is also an art which becomes more and more useful as you start using it.

I would love to know about your personal experience as to where have you found these ideas useful and how it has helped you in thinking through a real world problem.

Take care and keep learning.