Counting Calculi

Thoughts On The Linearity Fallacy

Life is not linear

by Kevin McLaughlin

During World War I, the presence of battleships prevented local fisherman from practicing their trade in the Mediterranean. As a result, people in 1918 intuitively expected the amount of fish to rise dramatically. Instead, the amount of fish actually declined, while the number of sharks dramatically rose. This is perhaps this first recorded example of a common and important error that we will call, the linearity fallacy.

The reason for the unexpected decline of fish was explained by Vito Volterra in his now famous predator-prey equations. Qualitatively, these equations are easy to understand. Because local fishermen could not catch their usual amount of fish, the number of fish initially increased. But that increase made it possible for more sharks to survive, who ate more fish. The fluctuation between the predator-shark and prey-fish populations is an example of a non-linear system because the amount of prey affects the number of predators and vice versa. The people who expected the amount of fish to rise proportionally to the decrease of fisherman made the mistake of oversimplifying a non-linear system to a linear system, thus committing the linearity fallacy.

The amount of gas in your car’s tank and the distance you can drive form a linear system because the gas and distance are directly proportional to each other.

“Virtuous circle”, “downward spiral”, and “self-fulfilling prophecy”, are linguistic metaphors for non-linear systems like the fish and sharks in Volterra’s Mediterranean.

Unlike linear systems, non-linear ones don’t maintain the direct relationship between inputs and outputs like the gas in your tank and distance you can drive. Frequently, the inputs themselves are actually affected by the outputs. For example, in Austin Powers: The Spy Who Shagged Me, Fat Bastard describes his downward spiral of weight gain thusly:

“I eat because I’m unhappy and I’m unhappy because I eat.”

The joke is funny (and relatable) because Fat Bastard’s weight gain is not determined just by the amount of food he eats, but by how unhappy and fat he already is. Thus, unlike the car and gas tank, his weight gain is not directly proportional to the amount food he eats. In fact, the amount of food he eats is affected by his current weight and unhappiness.

As the car and Fat Bastard examples show, linear systems are much less complicated and we intuitively understand them. Thus we are, like the people of 1918, prone to oversimplify non-linear system to linear ones.

For example, as Gary Taubes, author of Why We Get Fat, describes, most people have the intuition that to lose weight we must simply burn more calories than we consume. Let’s call this the calories in-calories out theory. Since most people are not continuously gaining weight, calories in-calories out would lead us to believe that if they cut their calorie intake by, say, 100 calories a day, they would lose weight. As most of us have experienced, however, this is not the case.

Why? Because our metabolism is much more complex than the gas tanks in our cars. The amount of calories we consume affects the amount of calories we burn. And, as anyone who’s trained for a marathon will tell you, the amount of calories we burn affects the amount of calories we consume. Our metabolism is a non-linear system and the calories in-calories out theory commits the linearity fallacy.1

Because of the linearity fallacy, some demographers can be dangerously overconfident when predicting the future population growth of a city. Population does not simply rise proportionally to factors like the price of housing. Instead, population growth both affects and is affected by the housing prices in the city. Thus housing prices can either be a virtuous circle or a downward spiral for the growth of a city. Either way, housing prices alone can make a simple linear model of population growth not just wrong, but wrong by orders of magnitude.

As another example, some criminologists too easily dismiss how small changes in our environment, like the broken window theory, can cascade into a serious crime decline. When “respectable residents” (ie non-criminals) feel safer in a neighborhood, they spend more time in that neighborhood, making it harder to commit crimes, which, in turn, brings out more respectable people. This virtuous circle, unlike a linear model, makes an exponential decrease in crime possible from a comparatively small change.

This leads me to believe that “data-driven” people, like demographers and criminologists, may be particularly vulnerable to the linearity fallacy. As evidence, most data analyses use linear regression, which, by definition, assumes a linear relationship between dependent and independent variables. As the physicists Stanislaw Ulam put it, assuming most of our world is linear and treating non-linear system as the unusual case is “like referring to the bulk of zoology as the study of non-elephant animals”.

Our world is complex. To make sense of it, we have to strive to make sure that we understand the internal mechanisms of a system, like the chemical reactions involved in our metabolism, or why people choose to live in the city, or what makes people feel safe in a neighborhood. When we refuse to embrace the complexity of our world and ignore the things that can’t easily be quantified we risk being overconfident in our predictions. But if we do embrace complexity we may find that small changes can sometimes make a big difference in many of our most intractable problems.

1. If our metabolism was truly linear, then anybody eating 100 more calories a day than necessary would gain weight ad infinitum (as Fat Bastard perhaps does) instead of just being slightly overweight.) ^