{"id":4509,"date":"2019-11-26T15:35:03","date_gmt":"2019-11-26T19:35:03","guid":{"rendered":"https:\/\/www.shortform.com\/blog\/?p=4509"},"modified":"2022-03-11T16:23:36","modified_gmt":"2022-03-11T20:23:36","slug":"power-law-distribution","status":"publish","type":"post","link":"https:\/\/www.shortform.com\/blog\/power-law-distribution\/","title":{"rendered":"Power-Law Distribution: How It Better Models the World"},"content":{"rendered":"\n

What is a power-law distribution? How is it useful in describing and predicting events in the real world, which is full of uncertainty?<\/p>\n\n\n\n

Power-law distribution is a functional relationship between two quantities in which the relationship remains constant no matter the initial size of the quantities. This relationship is useful in describing events and phenomena that can’t be graphed on a bell curve<\/a>.<\/p>\n\n\n\n

We’ll cover the basics of power-law distribution and look at how it works to accurately describe the world and account for otherwise unpredictable events.<\/p>\n\n\n\n\n\n\n\n

Representing Randomness<\/h2>\n\n\n\n

Before we look at power-law distribution, let’s look at the problem that it can solve: using bell curves for phenomena that don’t increase or decrease exponentially.<\/p>\n\n\n\n

A stunning example of economists\u2019 misinterpretation of the market came in 1998, with the collapse of Long-Term Capital Management (LCTM), a hedge fund that included among its founding partners two Nobel Prize\u2013winners, Myron Scholes and Robert C. Merton (Robert K. Merton\u2019s son). Scholes and Merton won their Nobel Prize for a theory of stock option pricing; the theory, however, was based on a Gaussian model of the market that excluded the possibility of Black Swans<\/a>.
<\/p>\n\n\n\n

When markets were disrupted by the Asian financial crisis of 1997 and the Russian financial crisis of 1998\u2014a one-two punch of Black Swans\u2014Scholes and Merton\u2019s theory was revealed for the fantasy it was. The twin crises resulted in a crushing $4.6 billion loss for LCTM in fewer than four months. The firm had to be bailed out by a consortium of private banks, at the behest of the Federal Reserve. 
<\/p>\n\n\n\n

The problem is that economists\u2019 models are thoroughly Platonified: self-consistent, but bearing almost no resemblance to reality. Economists are like philosopher John Locke\u2019s madmen: they reason \u201ccorrectly from erroneous premises.\u201d<\/strong>
<\/p>\n\n\n\n

Fractals and <\/strong>Power-Law Distribution<\/h2>\n\n\n\n

So can the phenomena of Extremistan<\/a> be modeled\u2014and thereby predicted\u2014at all?  Not precisely; but they can <\/em>be approximated\u2014by the use of fractals<\/em> and power-law distribution.
<\/p>\n\n\n\n

Developed by mathematician Beno\u00eet Mandelbrot in the 1970s, \u201cfractals\u201d are Mandelbrot\u2019s coinage for geometric patterns that repeat at different scales. Fractals, unlike pure geometric shapes like triangles or circles, are seen quite frequently in nature. For example, a leaf\u2019s veins look like little branches, and a tree\u2019s branches look like little trees: the basic shape of the tree is echoed at the smaller scales.
<\/p>\n\n\n\n

The reason fractals are helpful in representing Black Swans is that their internal ratios stay constant across scales. Unlike bell curves, in which ratios decline at accelerating rates the further one gets from the average, fractals exhibit no (or mild) acceleration. They obey power laws<\/em>, which describe a functional relationship between two quantities in which the relationship remains constant no matter what the initial size of the quantities. This is power-law distribution.
<\/p>\n\n\n\n

Take the European wealth example noted just above. As the wealth number doubled, the incidence decreased by 4x (wealth greater than 1 million: 1 in 62.5; wealth greater than 2 million: 1 in 250; wealth greater than 4 million: 1 in 1,000; etc.). The \u201cpower\u201d in this relationship is 2, because 2^2 (2 squared) is 4. 
<\/p>\n\n\n\n

Now, imagine the power was 1: Each time the wealth doubled, the incidence would only decrease by 2x (2^1=2). The probability of extraordinary wealth would increase<\/strong>. This is a power-law distribution.
<\/p>\n\n\n\n

For Extremistan phenomena, the power laws aren\u2019t known with any certainty, but they can be approximated. Say, for example, you want to assess the risk of a stock portfolio, and you know that, based on past data, the worst-case scenario is a -5% move once every 2 years. With a power of 2\u2014remember that the power is estimated\u2014you can assume that a -10% move will happen once every 8 years (2^2=4) and a -20 percent move will happen once every 32 years. (For scale, in the 1987 crash, the U.S. stock market lost almost 23% of its value, an impossibility<\/em> according to Gaussian economic models.) With fractal\/power-law distributions, a 1000-year flood can become a 100-year one, and a 100-year one can become a 10-year one<\/strong>.
<\/p>\n\n\n\n

When we begin to analyze Extremistan in terms of the fractal<\/a> paradigm and the power-law distribution, some Black Swans suddenly become \u201cgray\u201d\u2014they\u2019re not predictable with any sort of precision, but they are, at least, imaginable<\/em>. <\/p>\n","protected":false},"excerpt":{"rendered":"

What is a power-law distribution? How is it useful in describing and predicting events in the real world, which is full of uncertainty? Power-law distribution is a functional relationship between two quantities in which the relationship remains constant no matter the initial size of the quantities. This relationship is useful in describing events and phenomena that can’t be graphed on a bell curve. We’ll cover the basics of power-law distribution and look at how it works to accurately describe the world and account for otherwise unpredictable events.<\/p>\n","protected":false},"author":4,"featured_media":4528,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[25],"tags":[60],"class_list":["post-4509","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-statistics","tag-black-swan","","tg-column-two"],"yoast_head":"\nPower-Law Distribution: How It Better Models the World - Shortform Books<\/title>\n<meta name=\"description\" content=\"Power-law distribution is a functional relationship between two quantities where the relationship remains constant. 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