6 min read

Nonlinearity: The Mathematical Concept Everyone Is Trying To Explain

Nonlinearity is counter-intuitive. Viewing relationships as linear when they aren’t is a frequently unconsidered mental flaw that humans have. From traffic, to being more politically savvy, to evaluating businesses, to being more aware of how you value your time.

Nassim Taleb

Let’s start with the first example of non-linearity that made me say “so THATS why that happens.” It comes from the “Traffic in New York” chapter of Antifragile:

“Traffic is highly nonlinear. When I take the day flight from New York to London, and I leave my residence around five in the morning (yes, I know), it takes me around 26 minutes to reach the British Air terminal at JFK airport. At that time, New York is empty, eerily non– New York. When I leave my place at six o’clock for the later flight, there is almost no difference in travel time, although traffic is a bit denser. One can add more and more cars on the highway, with no or minimal impact on time spent in traffic. Then, a mystery — increase the number of cars by 10 percent and watch the travel time jump by 50 percent (I am using approximate numbers). Look at the convexity effect at work: the average number of cars on the road does not matter at all for traffic speed. If you have 90,000 cars for one hour, then 110,000 cars for another hour, traffic would be much slower than if you had 100,000 cars for two hours. … Every additional car increases travel time more than the previous one.”

A core thesis of Antifragile is that non-linearity is extremely hard for people to wrap their heads around which leads to poor decision-making.

From here, let’s move to the world of politics.

Jordan Ellenberg

Jordan Ellenberg, a math professor I had an college, wrote the insanely good book How Not To Be Wrong. The concept of linearity and nonlinearity are so crucial that he dedicates the entire first section of the book to them.

The first example of linearity he discusses is about a paper written by a libertarian think tank titled “Why Is Obama Trying to Make America More Like Sweden when Swedes Are Trying to Be Less Like Sweden?” (stick with me here). This paper argued that since Sweden was working to reduce the amount of welfare it provided for its citizens, why was the US attempting to increase the welfare it provided?

Jordan provides the following chart for this point of view:

As you might have already guessed, this argument hinges on the assumption that the relationship between welfare and prosperity is a linear one. Jordan provides a second chart to show what the relationship looks like if we don’t assume it’s linear:

He explains:

“Usually, when someone announces they’re a “nonlinear thinker” they’re about to apologize for losing something you lent them. But nonlinearity is a real thing! And in this context, thinking nonlinearly is crucial, because not all curves are lines. A moment of reflection will tell you that the real curves of economics look like the second picture, not the first. They’re nonlinear. Mitchell’s reasoning is an example of false linearity — he’s assuming, without coming right out and saying so, that the course of prosperity is described by the line segment in the first picture, in which case Sweden stripping down its social infrastructure means we should do the same. … Nonlinear thinking means which way you should go depends on where you already are.”

I highly recommend picking up the book as it dives into several other political examples.

And speaking of business and the welfare state, let’s get a businessman’s perspective on nonlinearity.

Charlie Munger

Investor Charlie Munger’s take on non-linearity is what he calls “lollapalooza effects”. He evaluates businesses by passing them through his mental models, hoping to find businesses where their business model has a positive non-linearity. From Poor Charlie’s Almanack:

“The most important thing to keep in mind is the idea that especially big forces often come out of these one hundred models. When several models combine, you get lollapalooza effects; this is when two, three or four forces are all operating in the same direction. And, frequently, you don’t get simple addition. It’s often like a critical mass in physics where you get a nuclear explosion if you get to a certain point of mass — and you don’t get anything much worth seeing if you don’t reach the mass. Sometimes the forces just add like ordinary quantities and sometimes they combine on a breakpoint or critical-mass basis.”

Later in the Almanack Charlie poses a business problem as an example of evaluating businesses with potential lollapalooza results — how would one build Coca Cola into a company worth $2 trillion in 150 years, starting in the year 1884? First he goes into further detail of what a $2 trillion business even looks like:

“… We will next use numerical fluency to ascertain what our target implies. We can guess reasonably that by 2034 there will be about eight billion beverage consumers in the world. On average, each of these consumers will be much more prosperous in real terms than the average consumer of 1884. Each consumer is composed mostly of water and must ingest about sixty-four ounces of water per day. This is eight, eight-ounce servings. Thus, if our new beverage, and other imitative beverages in our new market, can flavor and otherwise improve only twenty-five percent of ingested water worldwide, and we can occupy half of the new world market, we can sell 2.92 trillion eight-ounce servings in 2034. And if we can then net four cents per serving, we will make $117 billion. This will be enough, if our business is still growing at a good rate, to make it easily worth $2 trillion.
… These are lollapalooza results. Accordingly, we must attack our problem by causing every favorable factor we can think of to work for us. Plainly, only a powerful combination of many factors is likely to cause the lollapalooza consequences we desire.”

First off, wow, what a simultaneously enlightening, frightening, and astounding way to look at business. Building a business out of the fact that humans need to drink water to stay alive with a target market of all humans is kinda crazy. It is a scale of thinking that blew my head open the first time I read it. Anyways, the factors he lists that would combine non-linearly to create these results include:

  • Strong trademark to act as stimuli for the desired responses of purchasing and ingesting Coca-Cola
  • Operant conditioning via the rewards of taste (flavor, texture, aroma), stimulus (sugar and caffeine), cooling effect (drinking a coke on a hot day)
  • Pavlovian conditioning so that “as long as we are in business, our beverage and its promotion must be associated in consumer minds with all other things consumers like or admire.” This means artificially coloring the beverage so it looks more like wine than water, carbonating it to make it seem like champagne or other expensive beverages, and giving it an expensive sounding name, Coca-Cola, rather than “Glotz’s Sugared, Caffeinated Water”.
  • Using social proof to spread the product by having the product be so ubiquitous just seeing other people drink it will trigger a response. Or as Munger puts it — “… increased selling power will come from each increase in sales”.

He ties these all together:

“We can now see … that by combining (1) much Pavlovian conditioning, (2) powerful social-proof effects, and (3) a wonderful-tasting, energy-giving, stimulating, and desirably cold beverage that causes much operant conditioning, we are going to get sales that speed up for a long time by reason of the huge mixture of factors we have chosen. Therefore, we are going to start something like an autocatalytic reaction in chemistry, precisely the sort of multifactor-triggered lollapalooza effect we need.”

Boom. Trillion dollar business via non-linearities.

Neal Stephenson

Neal Stephenson explains nonlinearity via a slightly-harder-to-measure metric: his productivity as a writer.

“Writing novels is hard, and requires vast, unbroken slabs of time. Four quiet hours is a resource that I can put to good use. Two slabs of time, each two hours long, might add up to the same four hours, but are not nearly as productive as an unbroken four. If I know that I am going to be interrupted, I can’t concentrate, and if I suspect that I might be interrupted, I can’t do anything at all. Likewise, several consecutive days with four-hour time-slabs in them give me a stretch of time in which I can write a decent book chapter, but the same number of hours spread out across a few weeks, with interruptions in between them, are nearly useless.
The productivity equation is a non-linear one, in other words.”

What are aspects of your life that you had presumed to be linear? Once you start passing the world through the lense of “is this actually linear?” you’ll notice a shift. It’s a mental model that maps more cleanly to the real world than a simple “as x goes up, so in equal proportion must y”. And the closer your models can map to the world, hopefully the better your understanding and decision-making will become.