Category: Mental Models

Power Laws: How Nonlinear Relationships Amplify Results

“The greatest shortcoming of the human race is our inability to understand the exponential function.”

— Albert Allen Bartlett

Defining A Power Law

Consider a person who begins weightlifting for the first time.

During their initial sessions, they can lift only a small amount of weight. But as they invest more time, they find that for each training session, their strength increases a surprising amount.

For a while, they make huge improvements. Eventually, however, their progress slows down. At first, they could increase their strength by as much as 10% per session; now it takes months to improve by even 1%. Perhaps they resort to taking performance-enhancing drugs or training more often. Their motivation is sapped and they find themselves getting injured, without any real change in the amount of weight they can lift.

Now, let’s imagine that our frustrated weightlifter decides to take up running instead. Something similar happens. While the first few runs are incredibly difficult, the person’s endurance increases rapidly with the passing of each week, until it levels off and diminishing returns set in again.

Both of these situations are examples of power laws — a relationship between two things in which a change in one thing can lead to a large change in the other, regardless of the initial quantities. In both of our examples, a small investment of time in the beginning of the endeavor leads to a large increase in performance.

Power laws are interesting because they reveal surprising correlations between disparate factors. As a mental model, power laws are versatile, with numerous applications in different fields of knowledge.

If parts of this post look intimidating to non-mathematicians, bear with us. Understanding the math behind power laws is worthwhile in order to grasp their many applications. Invest a little time in reading this and reap the value — which is in itself an example of a power law!

A power law is often represented by an equation with an exponent:

Y=MX^B

Each letter represents a number. Y is a function (the result); X is the variable (the thing you can change); B is the order of scaling (the exponent); and M is a constant (unchanging).

If M is equal to 1, the equation is then Y=X^B. If B=2, the equation becomes Y=X^2 (Y=X squared). If X is 1, Y is also 1. But if X=2, then Y=4; if X=3, then Y=9, and so on. A small change in the value of X leads to a proportionally large change in the value of Y.

B=1 is known as the linear scaling law.

To double a cake recipe, you need twice as much flour. To drive twice as far will take twice as long. (Unless you have kids, in which case you need to factor in bathroom breaks that seemingly have little to do with distance.) Linear relationships, in which twice-as-big requires twice-as-much, are simple and intuitive.

Nonlinear relationships are more complicated. In these cases, you don’t need twice as much of the original value to get twice the increase in some measurable characteristic. For example, an animal that’s twice our size requires only about 75% more food than we do. This means that on a per-unit-of-size basis, larger animals are more energy efficient than smaller ones. As animals get bigger, the energy required to support each unit decreases.

One of the characteristics of a complex system is that the behavior of the system differs from the simple addition of its parts. This characteristic is called emergent behavior. “In many instances,” write Geoffrey West in Scale: The Universal Laws of Growth, Innovation, Sustainability, and the Pace of Life in Organisms, Cities, Economies, and Companies, “the whole seems to take on a life of its own, almost dissociated from the specific characteristics of its individual building blocks.”

This collective outcome, in which a system manifests significantly different characteristics from those resulting from simply adding up all of the contributions of its individual constituent parts, is called an emergent behavior.

When we set out to understand a complex system, our intuition tells us to break it down into its component pieces. But that’s linear thinking, and it explains why so much of our thinking about complexity falls short. Small changes in a complex system can cause sudden and large changes. Small changes cause cascades among the connected parts, like knocking over the first domino in a long row.

Let’s return to the example of our hypothetical weightlifter-turned-runner. As they put in more time on the road, constraints will naturally arise on their progress.

Recall our exponential equation: Y=MX^B. Try applying it to the runner. (We’re going to simplify running, but stick with it.)

Y is the distance the runner can run before becoming exhausted. That’s what we’re trying to calculate. M, the constant, represents their running ability: some combination of their natural endowment and their training history. (Think of it this way: Olympic champion Usain Bolt has a high M; film director Woody Allen has a low M.)

That leaves us with the final term: X^B. The variable X represents the thing we have control over: in this case, our training mileage. If B, the exponent, is between 0 and 1, then the relationship between X and Y— between training mileage and endurance — becomes progressively less proportional. All it takes is plugging in a few numbers to see the effect.

Let’s set M to 1 for the sake of simplicity. If B=0.5 and X=4, then Y=2. Four miles on the road gives the athlete the ability to run two miles at a clip.

Increase X to 16, and Y increases only to 4. The runner has to put in four times the road mileage to merely double their running endurance.

Here’s the kicker: With both running and weightlifting, as we increase X, we’re likely to see the exponent, B, decline! Quadrupling our training mileage from 16 to 64 miles is unlikely to double our endurance again. It might take a 10x increase in mileage to do that. Eventually, the ratio of training mileage to endurance will become nearly infinite.

We know this state, of course, as diminishing returns: the point where more input yields progressively less output. Not only is the relationship between training mileage and endurance not linear to begin with, but it also gets less linear as we increase our training.

And what about negative exponents?

It gets even more interesting. If B=−0.5 and X=4, then Y=0.5. Four miles on the road gets us a half-mile of endurance. If X is increased to 16, Y declines to 0.25. More training, less endurance! This is akin to someone putting in way too much mileage, way too soon: the training is less than useful as injuries pile up.

With negative numbers, the more X increases, the more Y shrinks. This relationship is known as an inverse power law. B=−2, for example, is known as the inverse square law and is an important equation in physics.

The relationship between gravity and distance follows an inverse power law. G is the gravitational constant; it’s the constant in Newton's law of gravitation, relating gravity to the masses and separation of particles, equal to:

6.67 × 10−11 N m2 kg−2

Any force radiating from a single point — including heat, light intensity, and magnetic and electrical forces — follows the inverse square law. At 1m away from a fire, 4 times as much heat is felt as at 2m, and so on.

Higher Order Power Laws

When B is a positive integer (a whole number larger than zero), there are names for the power laws.

When B is equal to 1, we have a linear relationship, as we discussed above. This is also known as a first-order power law.

Things really get interesting after that.

When B is 2, we have a second-order power law. A great example of this is kinetic energy. Kinetic energy = 1/2 mv^2

When B is 3, we have a third-order power law. An example of this is the power converted from wind into rotational energy.

Power Available = ½ (Air Density)( πr^2)(Windspeed^3)(Power Coefficient)

(There is a natural limit here. Albert Betz concluded in 1919 that wind turbines cannot convert more than 59.3% of the kinetic energy of the wind into mechanical energy. This number is called the Betz Limit and represents the power coefficient above.)[1]

The law of heat radiation is a fourth-order power law. Derived first by the Austrian physicist Josef Stefan in 1879 and separately by Austrian physicist Ludwig Boltzmann, the law works like this: the radiant heat energy emitted from a unit area in one second is equal to the constant of proportionality (the Stefan-Boltzmann constant) times the absolute temperature to the fourth power.[2]

There is only one power law with a variable exponent, and it’s considered to be one of the most powerful forces in the universe. It’s also the most misunderstood. We call it compounding. The formula looks like this:

Future Value = (Present Value)(1+i)^n

where i is the interest rate and n is the number of years.

Unlike in the other equations, the relationship between X and Y is potentially limitless. As long as B is positive, Y will increase as X does.

Non-integer power laws (where B is a fraction, as with our running example above) are also of great use to physicists. Formulas in which B=0.5 are common.

Imagine a car driving at a certain speed. A non-integer power law applies. V is the speed of the car, P is the petrol burnt per second to reach that speed, and A is the air resistance. For the car to go twice as fast, it must use 4 times as much petrol, and to go 3 times as fast, it must use 9 times as much petrol. Air resistance increases as speed increases, and that is why faster cars use such ridiculous amounts of petrol. It might seem logical to think that a car going from 40 miles an hour to 50 miles an hour would use a quarter more fuel. That is incorrect, though, because the relationship between air resistance and speed is itself a power law.

Another instance of a power law is the area of a square. Double the length of two parallel sides and the area quadruples. Do the same for a 3D cube and the area increases by a factor of eight. It doesn’t matter if the length of the square went from 1cm to 2cm, or from 100m to 200m; the area still quadruples. We are all familiar with second-order (or square) power laws. This name comes from squares, since the relationship between length and area reflect the way second-order power laws change a number. Third-order (or cubic) power laws are likewise named due to their relationship to cubes.

Using Power Laws in Our Lives

Now that we’ve gotten through the complicated part, let’s take a look at how power laws crop up in many fields of knowledge. Most careers involve an understanding of them, even if it might not be so obvious.

“What's the most powerful force in the universe? Compound interest. It builds on itself. Over time, a small amount of money becomes a large amount of money. Persistence is similar. A little bit improves performance, which encourages greater persistence which improves persistence even more. And on and on it goes.”

— Daniel H. Pink, The Adventures of Johnny Bunko

The Power Behind Compounding

Compounding is one of our most important mental models and is absolutely vital to understand for investing, personal development, learning, and other crucial areas of life.

In economics, we calculate compound interest by using an equation with these variables: P is the original sum of money. P’ is the resulting sum of money, r is the annual interest rate, n is the compounding frequency, and t is the length of time. Using an equation, we can illustrate the power of compounding.

If a person deposits $1000 in a bank for five years, at a quarterly interest rate of 4%, the equation becomes this:

Future Value = Present Value * ((1 + Quarterly Interest Rate) ^ Number of Quarters)

This formula can be used to calculate how much money will be in the account after five years. The answer is $2,220.20.

Compound interest is a power law because the relationship between the amount of time a sum of money is left in an account and the amount accumulated at the end is non-linear.

In A Random Walk Down Wall Street, Burton Malkiel gives the example of two brothers, William and James. Beginning at age 20 and stopping at age 40, William invests $4,000 per year. Meanwhile, James invests the same amount per year between the ages of 40 and 65. By the time William is 65, he has invested less money than his brother, but has allowed it to compound for 25 years. As a result, when both brothers retire, William has 600% more money than James — a gap of $2 million. One of the smartest financial choices we can make is to start saving as early as possible: by harnessing power laws, we increase the exponent as much as possible.

Compound interest can help us achieve financial freedom and wealth, without the need for a large annual income. Members of the financial independence movement (such as the blogger Mr. Money Mustache) are living examples of how we can apply power laws to our lives.

As far back as the 1800s, Robert G. Ingersoll emphasized the importance of compound interest:

One dollar at compound interest, at twenty-four per cent., for one hundred years, would produce a sum equal to our national debt. Interest eats night and day, and the more it eats the hungrier it grows. The farmer in debt, lying awake at night, can, if he listens, hear it gnaw. If he owes nothing, he can hear his corn grow. Get out of debt as soon as possible. You have supported idle avarice and lazy economy long enough.

Compounding can apply to areas beyond finance — personal development, health, learning, relationships and more. For each area, a small input can lead to a large output, and the results build upon themselves.

Nonlinear Language Learning

When we learn a new language, it’s always a good idea to start by learning the 100 or so most used words.

In all known languages, a small percentage of words make up the majority of usage. This is known as Zipf’s law, after George Kingsley Zipf, who first identified the phenomenon. The most used word in a language may make up as much as 7% of all words used, while the second-most-used word is used half as much, and so on. As few as 135 words can together form half of a language (as used by native speakers).

Why Zipf’s law holds true is unknown, although the concept is logical. Many languages include a large number of specialist terms that are rarely needed (including legal or anatomy terms). A small change in the frequency ranking of a word means a huge change in its usefulness.

Understanding Zipf’s law is a central component of accelerated language learning. Each new word we learn from the most common 100 words will have a huge impact on our ability to communicate. As we learn less-common words, diminishing returns set in. If each word in a language were listed in order of frequency of usage, the further we moved down the list, the less useful a word would be.

Power Laws in Business, Explained by Peter Thiel

Peter Thiel, the founder of PayPal (as well as an early investor in Facebook and Palantir), considers power laws to be a crucial concept for all businesspeople to understand. In his fantastic book, Zero to One, Thiel writes:

Indeed, the single most powerful pattern I have noticed is that successful people find value in unexpected places, and they do this by thinking about business from first principles instead of formulas.

And:

In 1906, economist Vilfredo Pareto discovered what became the “Pareto Principle,” or the 80-20 rule, when he noticed that 20% of the people owned 80% of the land in Italy—a phenomenon that he found just as natural as the fact that 20% of the peapods in his garden produced 80% of the peas. This extraordinarily stark pattern, when a small few radically outstrip all rivals, surrounds us everywhere in the natural and social world. The most destructive earthquakes are many times more powerful than all smaller earthquakes combined. The biggest cities dwarf all mere towns put together. And monopoly businesses capture more value than millions of undifferentiated competitors. Whatever Einstein did or didn’t say, the power law—so named because exponential equations describe severely unequal distributions—is the law of the universe. It defines our surroundings so completely that we usually don’t even see it.

… [I]n venture capital, where investors try to profit from exponential growth in early-stage companies, a few companies attain exponentially greater value than all others. … [W]e don’t live in a normal world; we live under a power law.

The biggest secret in venture capital is that the best investment in a successful fund equals or outperforms the entire rest of the fund combined.

This implies two very strange rules for VCs. First, only invest in companies that have the potential to return the value of the entire fund. … This leads to rule number two: because rule number one is so restrictive, there can’t be any other rules.

…[L]ife is not a portfolio: not for a startup founder, and not for any individual. An entrepreneur cannot “diversify” herself; you cannot run dozens of companies at the same time and then hope that one of them works out well. Less obvious but just as important, an individual cannot diversify his own life by keeping dozens of equally possible careers in ready reserve.

Thiel teaches a class called Startup at Stanford, where he hammers home the value of understanding power laws. In his class, he imparts copious wisdom. From Blake Masters’ notes on Class 7:

Consider a prototypical successful venture fund. A number of investments go to zero over a period of time. Those tend to happen earlier rather than later. The investments that succeed do so on some sort of exponential curve. Sum it over the life of a portfolio and you get a J curve. Early investments fail. You have to pay management fees. But then the exponential growth takes place, at least in theory. Since you start out underwater, the big question is when you make it above the water line. A lot of funds never get there.

To answer that big question you have to ask another: what does the distribution of returns in [a] venture fund look like? The naïve response is just to rank companies from best to worst according to their return in multiple of dollars invested. People tend to group investments into three buckets. The bad companies go to zero. The mediocre ones do maybe 1x, so you don’t lose much or gain much. And then the great companies do maybe 3-10x.

But that model misses the key insight that actual returns are incredibly skewed. The more a VC understands this skew pattern, the better the VC. Bad VCs tend to think the dashed line is flat, i.e. that all companies are created equal, and some just fail, spin wheels, or grow. In reality you get a power law distribution.

Thiel explains how investors can apply the mental model of power laws (more from Masters’ notes on Class 7):

…Given a big power law distribution, you want to be fairly concentrated. … There just aren’t that many businesses that you can have the requisite high degree of conviction about. A better model is to invest in maybe 7 or 8 promising companies from which you think you can get a 10x return. …

Despite being rooted in middle school math, exponential thinking is hard. We live in a world where we normally don’t experience anything exponentially. Our general life experience is pretty linear. We vastly underestimate exponential things.

He also cautions against over-relying on power laws as a strategy (an assertion that should be kept in mind for all mental models). From Masters’ notes:

One shouldn’t be mechanical about this heuristic, or treat it as some immutable investment strategy. But it actually checks out pretty well, so at the very least it compels you to think about power law distribution.

Understanding exponents and power law distributions isn’t just about understanding VC. There are important personal applications too. Many things, such as key life decisions or starting businesses, also result in similar distributions.

Thiel then explains why founders should focus on one key revenue stream, rather than trying to build multiple equal ones:

Even within an individual business, there is probably a sort of power law as to what’s going to drive it. It’s troubling if a startup insists that it’s going to make money in many different ways. The power law distribution on revenues says that one source of revenue will dominate everything else.

For example, if you’re an entrepreneur who opens a coffee shop, you’ll have a lot of ways you can make money. You can sell coffee, cakes, paintings, merchandise, and more. But each of those things will not contribute to your success in an equal way. While there is value in the discovery process, once you’ve found the variable that matters most, you should place more time on that one and less on the others. The importance of finding this variable cannot be overstated.

He also acknowledges that power laws are one of the great secrets of investing success. From Masters’ notes on Class 11:

On one level, the anti-competition, power law, and distribution secrets are all secrets about nature. But they’re also secrets hidden by people. That is crucial to remember. Suppose you’re doing an experiment in a lab. You’re trying to figure out a natural secret. But every night another person comes into the lab and messes with your results. You won’t understand what’s going on if you confine your thinking to the nature side of things. It’s not enough to find an interesting experiment and try to do it. You have to understand the human piece too.

… We know that, per the power law secret, companies are not evenly distributed. The distribution tends to be bimodal; there are some great ones, and then there are a lot of ones that don’t really work at all. But understanding this isn’t enough. There is a big difference between understanding the power law secret in theory and being able to apply it in practice.

The key to all mental models is knowing the facts and being able to use the concept. As George Box said, “all models are false but some are useful.” Once we grasp the basics, the best next step is to start figuring out how to apply it.

The metaphor of an unseen person sabotaging laboratory results is an excellent metaphor for how cognitive biases and shortcuts cloud our judgement.

Natural Power Laws

Anyone who has kept a lot of pets will have noticed the link between an animal’s size and its lifespan. Small animals, like mice and hamsters, tend to live for a year or two. Larger ones, like dogs and cats, can live to 10-20 years, or even older in rare cases. Scaling up even more, some whales can live for 200 years. This comes down to power laws.

Biologists have found clear links between an animal’s size and its metabolism. Kleiber’s law (identified by Max Kleiber) states that an animal’s metabolic rate increases at three-fourths of the power of the animal’s weight (mass). If an average rabbit (2 kg) weighs one hundred times as much as an average mouse (20g), the rabbit’s metabolic rate will be 32 times the mouse’s. In other words, the rabbit’s structure is more efficient. It all comes down to the geometry behind their mass.

Which leads us to another biological power law: Smaller animals require more energy per gram of body weight, meaning that mice eat around half their body weight in dense foods each day. The reason is that, in terms of percentage of mass, larger animals have more structure (bones, etc.) and fewer reserves (fat stores).

Research has illustrated how power laws apply to blood circulation in animals. The end units through which oxygen, water, and nutrients enter cells from the bloodstream are the same size in all animals. Only the number per animal varies. The relationship between the total area of these units and the size of the animal is a third-order power law. The distance blood travels to enter cells and the actual volume of blood are also subject to power laws.

The Law of Diminishing Returns

As we have seen, a small change in one area can lead to a huge change in another. However, past a certain point, diminishing returns set in and more is worse. Working an hour extra per day might mean more gets done, whereas working three extra hours is likely to lead to less getting done due to exhaustion. Going from a sedentary lifestyle to running two days a week may result in greatly improved health, but stepping up to seven days a week will cause injuries. Overzealousness can turn a positive exponent into a negative exponent. For a busy restaurant, hiring an extra chef will mean that more people can be served, but hiring two new chefs might spoil the proverbial broth.

Perhaps the most underappreciated diminishing return, the one we never want to end up on the wrong side of, is the one between money and happiness.

In David and Goliath, Malcolm Gladwell discusses how diminishing returns relate to family incomes. Most people assume that the more money they make, the happier they and their families will be. This is true — up to a point. An income that’s too low to meet basic needs makes people miserable, leading to far more physical and mental health problems. A person who goes from earning $30,000 a year to earning $40,000 is likely to experience a dramatic boost in happiness. However, going from $100,000 to $110,000 leads to a negligible change in well-being.

Gladwell writes:

The scholars who research happiness suggest that more money stops making people happier at a family income of around seventy-five thousand dollars a year. After that, what economists call “diminishing marginal returns” sets in. If your family makes seventy-five thousand and your neighbor makes a hundred thousand, that extra twenty-five thousand a year means that your neighbor can drive a nicer car and go out to eat slightly more often. But it doesn’t make your neighbor happier than you, or better equipped to do the thousands of small and large things that make for being a good parent.

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Footnotes
  • 1

    http://www.raeng.org.uk/publications/other/23-wind-turbine

  • 2

    https://www.britannica.com/science/Stefan-Boltzmann-law

The Fairness Principle: How the Veil of Ignorance Helps Test Fairness

“But the nature of man is sufficiently revealed for him to know something of himself and sufficiently veiled to leave much impenetrable darkness, a darkness in which he ever gropes, forever in vain, trying to understand himself.”

— Alexis de Tocqueville, Democracy in America

The Basics

If you could redesign society from scratch, what would it look like?

How would you distribute wealth and power?

Would you make everyone equal or not? How would you define fairness and equality?

And — here’s the kicker — what if you had to make those decisions without knowing who you would be in this new society?

Philosopher John Rawls asked just that in a thought experiment known as “the Veil of Ignorance” in his 1971 book, Theory of Justice.

Like many thought experiments, the Veil of Ignorance could never be carried out in the literal sense, nor should it be. Its purpose is to explore ideas about justice, morality, equality, and social status in a structured manner.

The Veil of Ignorance, a component of social contract theory, allows us to test ideas for fairness.

Behind the Veil of Ignorance, no one knows who they are. They lack clues as to their class, their privileges, their disadvantages, or even their personality. They exist as an impartial group, tasked with designing a new society with its own conception of justice.

As a thought experiment, the Veil of Ignorance is powerful because our usual opinions regarding what is just and unjust are informed by our own experiences. We are shaped by our race, gender, class, education, appearance, sexuality, career, family, and so on. On the other side of the Veil of Ignorance, none of that exists. Technically, the resulting society should be a fair one.

In Ethical School Leadership, Spencer J. Maxcy writes:

Imagine that you have set for yourself the task of developing a totally new social contract for today's society. How could you do so fairly? Although you could never actually eliminate all of your personal biases and prejudices, you would need to take steps at least to minimize them. Rawls suggests that you imagine yourself in an original position behind a veil of ignorance. Behind this veil, you know nothing of yourself and your natural abilities, or your position in society. You know nothing of your sex, race, nationality, or individual tastes. Behind such a veil of ignorance all individuals are simply specified as rational, free, and morally equal beings. You do know that in the “real world,” however, there will be a wide variety in the natural distribution of natural assets and abilities, and that there will be differences of sex, race, and culture that will distinguish groups of people from each other.

“The Fairness Principle: When contemplating a moral action, imagine that you do not know if you will be the moral doer or receiver, and when in doubt err on the side of the other person.”

— Michael Shermer, The Moral Arc: How Science and Reason Lead Humanity Toward Truth, Justice, and Freedom

The Purpose of the Veil of Ignorance

Because people behind the Veil of Ignorance do not know who they will be in this new society, any choice they make in structuring that society could either harm them or benefit them.

If they decide men will be superior, for example, they must face the risk that they will be women. If they decide that 10% of the population will be slaves to the others, they cannot be surprised if they find themselves to be slaves. No one wants to be part of a disadvantaged group, so the logical belief is that the Veil of Ignorance would produce a fair, egalitarian society.

Behind the Veil of Ignorance, cognitive biases melt away. The hypothetical people are rational thinkers. They use probabilistic thinking to assess the likelihood of their being affected by any chosen measure. They possess no opinions for which to seek confirmation. Nor do they have any recently learned information to pay undue attention to. The sole incentive they are biased towards is their own self-preservation, which is equivalent to the preservation of the entire group. They cannot stereotype any particular group as they could be members of it. They lack commitment to their prior selves as they do not know who they are.

So, what would these people decide on? According to Rawls, in a fair society all individuals must possess the following:

  • Rights and liberties (including the right to vote, the right to hold public office, free speech, free thought, and fair legal treatment)
  • Power and opportunities
  • Income and wealth sufficient for a good quality of life (Not everyone needs to be rich, but everyone must have enough money to live a comfortable life.)
  • The conditions necessary for self-respect

For these conditions to occur, the people behind the Veil of Ignorance must figure out how to achieve what Rawls regards as the two key components of justice:

  • Everyone must have the best possible life which does not cause harm to others.
  • Everyone must be able to improve their position, and any inequalities must be present solely if they benefit everyone.

However, the people behind the Veil of Ignorance cannot be completely blank slates or it would be impossible for them to make rational decisions. They understand general principles of science, psychology, politics, and economics. Human behavior is no mystery to them. Neither are key economic concepts, such as comparative advantage and supply and demand. Likewise, they comprehend the deleterious impact of social entropy, and they have a desire to create a stable, ordered society. Knowledge of human psychology leads them to be cognizant of the universal desire for happiness and fulfillment. Rawls considered all of this to be the minimum viable knowledge for rational decision-making.

Ways of Understanding the Veil of Ignorance

One way to understand the Veil of Ignorance is to imagine that you are tasked with cutting up a pizza to share with friends. You will be the last person to take a slice. Being of sound mind, you want to get the largest possible share, and the only way to ensure this is to make all the slices the same size. You could cut one huge slice for yourself and a few tiny ones for your friends, but one of them might take the large slice and leave you with a meager share. (Not to mention, your friends won’t think very highly of you.)

Another means of appreciating the implications of the Veil of Ignorance is by considering the social structures of certain species of ants. Even though queen ants are able to form colonies alone, they will band together to form stronger, more productive colonies. Once the first group of worker ants reaches maturity, the queens fight to the death until one remains. When they first form a colony, the queen ants are behind a Veil of Ignorance. They do not know if they will be the sole survivor or not. All they know, on an instinctual level, is that cooperation is beneficial for their species. Like the people behind the Veil of Ignorance, the ants make a decision which, by necessity, is selfless.

The Veil of Ignorance, as a thought experiment, shows us that ignorance is not always detrimental to a society. In some situations, it can create robust social structures. In the animal kingdom, we see many examples of creatures that cooperate even though they do not know if they will suffer or benefit as a result. In a paper entitled “The Many Selves of Social Insects,” Queller and Strassmann write of bees:

…social insect colonies are so tightly integrated that they seem to function as single organisms, as a new level of self. The honeybees' celebrated dance about food location is just one instance of how their colonies integrate and act on information that no single individual possesses. Their unity of purpose is underscored by the heroism of workers, whose suicidal stinging attacks protect the single reproducing queen.

We can also consider the Tragedy of the Commons. Introduced by ecologist Garrett Hardin, this mental model states that shared resources will be exploited if no system for fair distribution is implemented. Individuals have no incentive to leave a share of free resources for others. Hardin’s classic example is an area of land which everyone in a village is free to use for their cattle. Each person wants to maximize the usefulness of the land, so they put more and more cattle out to graze. Yet the land is finite and at some point will become too depleted to support livestock. If the people behind the Veil of Ignorance had to choose how the common land should be shared, the logical decision would be to give each person an equal part and forbid them from introducing too many cattle.

As N. Gregory Mankiw writes in Principles of Microeconomics:

The Tragedy of the Commons is a story with a general lesson: when one person uses a common resource, he diminishes other people's enjoyment of it. Because of this negative externality, common resources tend to be used excessively. The government can solve the problem by reducing use of the common resource through regulation or taxes. Alternatively, the government can sometimes turn the common resource into a private good.

This lesson has been known for thousands of years. The ancient Greek philosopher Aristotle pointed out the problem with common resources: “What is common to many is taken least care of, for all men have greater regard for what is their own than for what they possess in common with others.”

In The Case for Meritocracy, Michael Faust uses other thought experiments to support the Veil of Ignorance:

Let’s imagine another version of the thought experiment. If inheritance is so inherently wonderful — such an intrinsic good — then let’s collect together all of the inheritable money in the world. We shall now distribute this money in exactly the same way it would be distributed in today’s world… but with one radical difference. We are going to distribute it by lottery rather than by family inheritance, i.e, anyone in the world can receive it. So, in these circumstances, how many people who support inheritance would go on supporting it? Note that the government wouldn’t be getting the money… just lucky strangers. Would the advocates of inheritance remain as fiercely committed to their cherished principle? Or would the entire concept instantly be exposed for the nonsense it is?

If inheritance were treated as the lottery it is, no one would stand by it.

[…]

In the world of the 1% versus the 99%, no one in the 1% would ever accept a lottery to decide inheritance because there would be a 99% chance they would end up as schmucks, exactly like the rest of us.

And a further surrealistic thought experiment:

Imagine that on a certain day of the year, each person in the world randomly swaps bodies with another person, living anywhere on earth. Well, for the 1%, there’s a 99% chance that they will be swapped from heaven to hell. For the 99%, 1% might be swapped from hell to heaven, while the other 98% will stay the same as before. What kind of constitution would the human race adopt if annual body swapping were a compulsory event?! They would of course choose a fair one.

“In the immutability of their surroundings the foreign shores, the foreign faces, the changing immensity of life, glide past, veiled not by a sense of mystery but by a slightly disdainful ignorance.”

— Joseph Conrad, Heart of Darkness

The History of Social Contract Theory

Although the Veil of Ignorance was first described by Rawls in 1971, many other philosophers and writers have discussed similar concepts in the past. Philosophers discussed social contract theory as far back as ancient Greece.

In Crito, Plato describes a conversation in which Socrates discusses the laws of Athens and how they are responsible for his existence. Finding himself in prison and facing the death penalty, Socrates rejects Crito’s suggestion that he should escape. He states that further injustice is not an appropriate response to prior injustice. Crito believes that by refusing to escape, Socrates is aiding his enemies, as well as failing to fulfil his role as a father. But Socrates views the laws of Athens as a single entity that has always protected him. He describes breaking any of the laws as being like injuring a parent. Having lived a long, fulfilling life as a result of the social contract he entered at birth, he has no interest in now turning away from Athenian law. Accepting death is essentially a symbolic act that Socrates intends to use to illustrate rationality and reason to his followers. If he were to escape, he would be acting out of accord with the rest of his life, during which he was always concerned with justice.

Social contract theory is concerned with the laws and norms a society decides on and the obligation individuals have to follow them. Socrates’ dialogue with Plato has similarities with the final scene of Arthur Miller’s The Crucible. At the end of the play, John Proctor is hung for witchcraft despite having the option to confess and avoid death. In continuing to follow the social contract of Salem and not confessing to a crime he obviously did not commit, Proctor believes that his death will redeem his earlier mistakes. We see this in the final dialogue between Reverend Hale and Elizabeth (Proctor's wife):

HALE: Woman, plead with him! […] Woman! It is pride, it is vanity. […] Be his helper! What profit him to bleed? Shall the dust praise him? Shall the worms declare his truth? Go to him, take his shame away!

 

ELIZABETH: […] He have his goodness now. God forbid I take it from him!

In these two situations, individuals allow themselves to be put to death in the interest of following the social contract they agreed upon by living in their respective societies. Earlier in their lives, neither person knew what their ultimate fate would be. They were essentially behind the Veil of Ignorance when they chose (consciously or unconsciously) to follow the laws enforced by the people around them. Just as the people behind the Veil of Ignorance must accept whatever roles they receive in the new society, Socrates and Proctor followed social contracts. To modern eyes, the decision both men make to abandon their children in the interest of proving a point is not easily defensible.

Immanuel Kant wrote about justice and freedom in the late 1700s. Kant believed that fair laws should not be based on making people happy or reflecting the desire of individual policymakers, but should be based on universal moral principles:

Is it not of the utmost necessity to construct a pure moral philosophy which is completely freed from everything that may be only empirical and thus belong to anthropology? That there must be such a philosophy is self-evident from the common idea of duty and moral laws. Everyone must admit that a law, if it is to hold morally, i.e., as a ground of obligation, must imply absolute necessity; he must admit that the command, “Then shalt not lie,” does not apply to men only, as if other rational beings had no need to observe it. The same is true for all other moral laws properly so called. He must concede that the ground of obligation here must not be sought in the nature of man or in the circumstances in which he is placed, but sought a priori solely in the concepts of pure reason, and that every other precept which is in certain respects universal, so far as it leans in the least on empirical grounds (perhaps only in regard to the motive involved), may be called a practical rule but never a moral law.

How We Can Apply This Concept

We can use the Veil of Ignorance to test whether a certain issue is fair.

When my kids are fighting over the last cookie, which happens more often than you'd imagine, I ask them to determine who will spilt the cookie. The other person picks. This is the old playground rule, “you split, I pick.” Without this rule, one of them would surely give the other a smaller portion. With it, the halves are as equal as they would be with sensible adults.

When considering whether we should endorse a proposed law or policy, we can ask: if I did not know if this would affect me or not, would I still support it? Those who make big decisions that shape the lives of large numbers of people are almost always those in positions of power. And those in positions of power are almost always members of privileged groups. As Benjamin Franklin once wrote: “Justice will not be served until those who are unaffected are as outraged as those who are.”

Laws allowing or prohibiting abortion have typically been made by men, for example. As the issue lacks real significance in their personal lives, they are free to base decisions on their own ideological views, rather than consider what is fair and sane. However, behind the Veil of Ignorance, no one knows their sex. Anyone deciding on abortion laws would have to face the possibility that they themselves will end up as a woman with an unwanted pregnancy.

In Justice as Fairness: A Restatement, Rawls writes:

So what better alternative is there than an agreement between citizens themselves reached under conditions that are fair for all?

[…]

[T]hreats of force and coercion, deception and fraud, and so on must be ruled out.

And:

Deep religious and moral conflicts characterize the subjective circumstances of justice. Those engaged in these conflicts are surely not in general self-interested, but rather, see themselves as defending their basic rights and liberties which secure their legitimate and fundamental interests. Moreover, these conflicts can be the most intractable and deeply divisive, often more so than social and economic ones.

 

In Ethics: Studying the Art of Moral Appraisal, Ronnie Littlejohn explains:

We must have a mechanism by which we can eliminate the arbitrariness and bias of our “situation in life” and insure that our moral standards are justified by the one thing all people share in common: reason. It is the function of the veil of ignorance to remove such bias.

When we have to make decisions that will affect other people, especially disadvantaged groups (such as when a politician decides to cut benefits or a CEO decides to outsource manufacturing to a low-income country), we can use the Veil of Ignorance as a tool for making fair choices.

As Robert F. Kennedy (the younger brother of John F. Kennedy) said in the 1960s:

Few will have the greatness to bend history itself, but each of us can work to change a small portion of events. It is from numberless diverse acts of courage and belief that human history is shaped. Each time a man stands up for an ideal, or acts to improve the lot of others, or strikes out against injustice, he sends forth a tiny ripple of hope, and crossing each other from a million different centers of energy and daring, those ripples build a current which can sweep down the mightiest walls of oppression and resistance.

When we choose to position ourselves behind the Veil of Ignorance, we have a better chance of creating one of those all-important ripples.

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The Power of Incentives: Inside The Hidden Forces that Shape Behavior

“Never, ever, think about something else when you should be thinking about the power of incentives.”

— Charlie Munger

According to Charlie Munger, there are only a few forces more powerful than incentives. In his speech “The Psychology of Human Misjudgment,” he reflects on how the power of incentives never disappoints him:

Well, I think I’ve been in the top 5% of my age cohort all my life in understanding the power of incentives, and all my life I’ve underestimated it. And never a year passes but I get some surprise that pushes my limit a little farther.

Sometimes the solution to a behavior problem is simply to revisit incentives and make sure they align with the desired goal. Munger talks about Federal Express, which is one of his favorite examples of the power of incentives:

The heart and soul of the integrity of the system is that all the packages have to be shifted rapidly in one central location each night. And the system has no integrity if the whole shift can’t be done fast. And Federal Express had one hell of a time getting the thing to work.
And they tried moral suasion, they tried everything in the world, and finally somebody got the happy thought that they were paying the night shift by the hour, and that maybe if they paid them by the shift, the system would work better. And lo and behold, that solution worked.

If you’re trying to change a behavior, reason will take you only so far. Reflecting on another example where misaligned incentives hampered the sales of a superior product, Munger said:

Early in the history of Xerox, Joe Wilson, who was then in the government, had to go back to Xerox because he couldn’t understand how their better, new machine was selling so poorly in relation to their older and inferior machine. Of course when he got there, he found out that the commission arrangement with the salesmen gave a tremendous incentive to the inferior machine.

Ignoring incentives almost never works out well. Thinking about the incentives of others is necessary to create win-win relationships.

We can turn to psychology to obtain a more structured and thorough understanding of how incentives shape our actions.

The Science of Reinforcement

The science of reinforcement was furthered by Burrhus Frederic Skinner (usually called B.F. Skinner), a professor of psychology at Harvard from 1958 to 1974.

Skinner, unlike his contemporaries, refused to hypothesize about what happened on the inside (what people or animals thought and felt) and preferred to focus on what we can observe. To him, focusing on how much people ate meant more than focusing on subjective measures, like how hungry people were or how much pleasure they got from eating. He wanted to find out how environmental variables affected behavior, and he believed that behavior is shaped by its consequences.

If we don’t like the consequences of an action we’ve taken, we’re less likely to do it again; if we do like the consequences, we’re more likely to do it again. That assumption is the basis of operant conditioning, “a type of learning in which the strength of a behavior is modified by [its] consequences, such as reward or punishment.” 1

One of Skinner’s most important inventions was the operant conditioning chamber, also known as a “Skinner box,” which was used to study the effects of reinforcers on lab animals. The rats in the box had to figure out how to do a task (such as pushing a lever) that would reward them with food. Such an automated system allowed Skinner and thousands of successors to study conditioned behavior in a controlled setting.

What years of studies on reinforcement have revealed is that consistency and timing play important roles in shaping new behaviors. Psychologists argue that the best way for us to learn complex behaviors is via continuous reinforcement, in which the desired behavior is reinforced every time it’s performed.

If you want to teach your dog a new trick, for example, it is smart to reward him for every correct response. At the very beginning of the learning curve, your failure to immediately respond to a positive behavior might be misinterpreted as a sign of incorrect behavior from the dog’s perspective.

Intermittent reinforcement is reinforcement that is given only some of the times that the desired behavior occurs, and it can be done according to various schedules, some predictable and some not (see “Scheduling Reinforcement,” below). Intermittent reinforcement is argued to be the most efficient way to maintain an already learnt behavior. This is due to three reasons.

First, rewarding the behavior takes time away from the behavior’s continuation. Paying a worker after each piece is assembled on the assembly line simply does not make sense.

Second, intermittent reinforcement is better from an economic perspective. Not only is it cheaper not to reward every instance of a desired behavior, but by making the rewards unpredictable, you trigger excitement and thus get an increase in response without increasing the amount of reinforcement. Intermittent reinforcement is how casinos work; they want people to gamble, but they can’t afford to have people win large amounts very often.

Finally, intermittent reinforcement can induce resistance to extinction (stopping the behavior when reinforcement is removed). Consider the example of resistance outlined in the textbook Psychology: Core Concepts:

Imagine two gamblers and two slot machines. One machine inexplicably pays off on every trial and another, a more usual machine, pays on an unpredictable, intermittent schedule. Now, suppose that both devices suddenly stop paying. Which gambler will catch on first?

Most of us would probably guess it right:

The one who has been rewarded for each pull of the lever (continuous reinforcement) will quickly notice the change, while the gambler who has won only occasionally (on partial reinforcement) may continue playing unrewarded for a long time.

Scheduling Reinforcement

Intermittent reinforcement can be used on various schedules, each with its own degree of effectiveness and situations to which it can be appropriately applied. Ratio schedules are based on the number of responses (the amount of work done), whereas interval schedules are based on the amount of time spent.

  • Fixed-ratio schedules are used when you pay your employees based on the amount of work they do. Fixed-ratio schedules are common in freelancing, where contractors are paid on a piecework basis. Managers like fixed-ratio schedules because the response to reinforcement is usually very high (if you want to get paid, you do the work).
  • Variable-ratio schedules are unpredictable because the number of responses between reinforcers varies. Telemarketers, salespeople, and slot machine players are on this schedule because they never know when the next sale or the next big win will occur. Skinner himself demonstrated the power of this schedule by showing that a hungry pigeon would peck a disk 12,000 times an hour while being rewarded on average for only every 110 pecks. Unsurprisingly, this is the type of reinforcement that normally produces more responses than any other schedule. (Varying the intervals between reinforcers is another way of making reinforcement unpredictable, but if you want people to feel appreciated, this kind of schedule is probably not the one to use.)
  • Fixed-interval schedules are the most common type of payment — they reward people for the time spent on a specific task. You might have already guessed that the response rate on this schedule is very low. Even a rat in a Skinner box programmed for a fixed-interval schedule learns that lever presses beyond the required minimum are just a waste of energy. Ironically, the “9-5 job” is a preferred way to reward employees in business.

While the design of scheduling can be a powerful technique for continuing or amplifying a specific behavior, we may still fail to recognize an important aspect of reinforcement — individual preferences for specific rewards.

Experience suggests that survival is propelled by our need for food and water. However, most of us don’t live in conditions of extreme scarcity and thus the types of reinforcement appealing to us will differ.

Culture plays an important role in determining effective reinforcers. And what’s reinforced shapes culture. Offering tickets to a cricket match might serve as a powerful reward for someone in a country where cricket is a big deal, but would be meaningless to most Americans. Similarly, an air-conditioned office might be a powerful incentive for employees in Indonesia, but won’t matter as much to employees in a more temperate area.

What About Punishment?

So far we’ve talked about positive reinforcement — the carrot, if you will. However, there is also a stick.

There is no doubt that our society relies heavily on threat and punishment as a way to keep ourselves in line. Still, we keep arriving late, forgetting birthdays, and receiving parking fines, even though we know there is the potential to be punished.

There are several reasons that punishment might not be the best way to alter someone’s behavior.

First of all, Skinner observed that the power of punishment to suppress behavior usually disappears when the threat of punishment is removed. Indeed, we all refrain from using social networks during work hours, when we know our boss is around, and we similarly adhere to the speed limit when we know we are being watched by a police patrol.

Second, punishment often triggers a fight-or-flight response and renders us aggressive. When punished, we seek to flee from further punishment, and when the escape is blocked, we may become aggressive. This punishment-aggression link may also explain why abusing parents come from abusing families themselves.

Third, punishment inhibits the ability to learn new and better responses. Punishment leads to a variety of responses — such as escape, aggression, and learned helplessness — none of which aid in the subject’s learning process. Punishment also fails to show subjects what exactly they must do and instead focuses on what not to do. This is why environments that forgive failure are so important in the learning process.

Finally, punishment is often applied unequally. We are ruled by bias in our assessment of who deserves to be punished. We scold boys more often than girls, physically punish grade-schoolers more often than adults, and control members of racial minorities more often (and more harshly) than whites.

What Should I Do Instead?

There are three alternatives that you can try the next time you feel tempted to punish someone.

The first we already touched upon — extinction. A response will usually diminish or disappear if it ceases to produce the rewards it once did. However, it is important that all possible reinforcements are withheld. This is far more difficult to do in real life than in a lab setting.

What makes it especially difficult is that during the extinction process, organisms tend to look for novel techniques to obtain reinforcement. This means that a whining child will either redouble her efforts or change tactics to regain the parent’s attention before ceasing the behavior. In this case, a better extinction strategy is to combine methods by withholding attention after whining occurs and rewarding more desirable behaviors with attention before the whining occurs.

The second alternative is positively reinforcing preferred activities. For example, people who exercise regularly (and enjoy it) might use a daily run as a reward for getting other tasks done. Similarly, young children learn to sit still by being rewarded with occasional permission to run around and make noise. The main principle of this idea is that a preferred activity, such as running around, can be used to reinforce a less preferred activity. This idea is also called the Premack principle.

Finally, prompting and shaping are two actions we can use together to change behavior in an iterative manner. A prompt is a cue or stimulus that encourages the desired behavior. When shaping begins, any approximation of the target response is reinforced. Once you see the approximation occurring regularly, you can make the criterion for the target more strict (the actual behavior has to match the desired behavior more closely), and you continue narrowing the criteria until the specific target behavior is performed. This tactic is often the preferred method of developing a habit gradually and of training animals to perform a specific behavior.

***

I hope that you are now better equipped to recognize incentives as powerful forces shaping the way we and others behave. The next time you wish someone would change the way they behave, think about changing their incentives.

Like any parent, I experiment with my kids all the time. One of the most effective things I do when one of them has misbehaved is to acknowledge my child’s feelings and ask him what he was trying to achieve.

When one kid hits the other, for example, I ask him what he was trying to accomplish. Usually, the response is “He hit me. (So I hit him back.)” I know this touches on an automatic human response that many adults can’t control. Which makes me wonder how I can change my kids’ behavior to be more effective.

“So, you were angry and you wanted him to know?”

“Yes.”

“People are not for hitting. If you want, I’ll help you go tell him why you’re angry.”

Tensions dissipate. And I’m (hopefully) starting to get my kids thinking about effective and ineffective ways to achieve their goals.

Punishment works best to prevent actions whereas incentives work best to encourage them.

Let’s end with an excellent piece of advice that has been given regarding incentives. Here is Charlie Munger, speaking at the University South California commencement:

You do not want to be in a perverse incentive system that’s causing you to behave more and more foolishly or worse and worse — incentives are too powerful a control over human cognition or human behavior. If you’re in one [of these systems], I don’t have a solution for you. You’ll have to figure it out for yourself, but it’s a significant problem.

Footnotes

Complex Adaptive Cities

Complex adaptive systems are hard to understand. Messy and complicated, they cannot be broken down into smaller bits. It would be easier to ignore them, or simply leave them as mysteries. But given that we are living in one such system, it might be more useful to buckle down and sort it out. That way, we can make choices that are aligned with how the world actually operates.

In his book Diversity and Complexity, Scott E. Page explains, “Complexity can be loosely thought of as interesting structures and patterns that are not easily described or predicted. Systems that produce complexity consist of diverse rule-following entities whose behaviors are interdependent. Those entities interact over a contact structure or network. In addition, the entities often adapt.”

Understanding complexity is important, because sometimes things are not further reducible. While the premise of Occam’s Razor is that things should be made as simple as possible but not simpler, sometimes there are things that cannot be reduced. There is, in fact, an irreducible minimum. Certain things can be properly contemplated only in all their complicated, interconnected glory.

Take, for example, cities.

Cities cannot be created for success from the top down by the imposition of simple rules.

For those of us who live in cities, we all know what makes a particular neighborhood great. We can get what we need and have the interactions we want, and that’s ultimately because we feel safe there.

But how is this achieved? What magic combination of people and locations, uses and destinations, makes a vibrant, safe neighborhood? Is there a formula for, say, the ratio of houses to businesses, or of children to workers?

No. Cities are complex adaptive systems. They cannot be created for success from the top down by the imposition of simple rules.

In her seminal book The Death and Life of Great American Cities, Jane Jacobs approached the city as a complex adaptive system, turned city planning on its head, and likely saved many North American cities by taking them apart and showing that they cannot be reduced to a series of simple behavioral interactions.

Cities fall exactly into the definition of complexity given above by Page. They are full of rule-following humans, cars, and wildlife, the behaviors of which are interdependent on the other entities and respond to feedback.

These components of a city interact over multiple interfaces in a city network and will adapt easily, changing their behavior based on food availability, road closures, or perceived safety. But the city itself cannot be understood by looking at just one of these behaviors.

Jacobs starts with “the kind of problem which cities pose — a problem in handling organized complexity” — and a series of observations about that common, almost innocuous, part of all cities: the sidewalk.

What makes a particular neighborhood safe?

Jacobs argues that there is no one factor but rather a series of them. In order to understand how a city street can be safe, you must examine the full scope of interactions that occur on its sidewalk. “The trust of a city street is formed over time from many, many little public sidewalk contacts.” Nodding to people you know, noticing people you don’t. Recognizing which parent goes with which kid, or whose business seems to be thriving. People create safety.

Given that most of them are strangers to each other, how do they do this? How come these strangers are not all perceived as threats?

Safe streets are streets that are used by many different types of people throughout the 24-hour day. Children, workers, caregivers, tourists, diners — the more people who use the sidewalk, the more eyes that participate in the safety of the street.

Safety on city streets is “kept primarily by an intricate, almost unconscious, network of voluntary controls and standards among the people themselves, and enforced by the people themselves.” Essentially, we all contribute to safety because we all want safety. It increases our chances of survival.

Jacobs brings an amazing eye for observational detail in describing neighborhoods that work and those that don’t. In describing sidewalks, she explains that successful, safe neighborhoods are orderly. “But there is nothing simple about that order itself, or the bewildering number of components that go into it. Most of those components are specialized in one way or another. They unite in their joint effect upon the sidewalk, which is not specialized in the least. That is its strength.” For example, restaurant patrons, shopkeepers, loitering teenagers, etc. — some of whom belong to the area and some of whom are transient — all use the sidewalk and in doing so contribute to the interconnected and interdependent relationships that produce the perception of safety on that street. And real safety will follow perceived safety.

To get people participating in this unorganized street safety, you have to have streets that are desirable. “You can’t make people use streets they have no reason to use. You can’t make people watch streets they do not want to watch.” But Jacobs points out time and again that there is no predictable prescription for how to achieve this mixed use where people are unconsciously invested in the maintenance of safety.

This is where considering the city as a complex adaptive system is most useful.

Each individual component has a part to play, so a top-down imposition of theory that doesn’t allow for the unpredictable behavior of each individual is doomed to fail. “Orthodox planning is much imbued with puritanical and Utopian conceptions of how people should spend their free time, and in planning, these moralisms on people’s private lives are deeply confused with concepts about the workings of cities.” A large, diverse group of people is not going to conform to only one way of living. And it’s the diversity that offers the protection.

For example, a city planner might decide to not have bars in residential neighborhoods. The noise might keep people up, or there will be a negative moral impact on the children who are exposed to the behavior of loud, obnoxious drunks. But as Jacobs reveals, safe city areas can’t be built on the basis of this type of simplistic assumption.

By stretching the use of a street through as many hours of the day as possible, you might create a safer neighborhood. I say “might” because in this complex system, other factors might connect to manifest a different reality.

Planning that doesn’t respect the spectrum of diverse behavior and instead aims to insist on an ideal based on a few simple concepts will hinder the natural ability of a system to adapt.

As Scott Page explains, “Creating a complex system from scratch takes skill (or evolution). Therefore, when we see diverse complex systems in the real world, we should not assume that they’ve been assembled from whole cloth. Far more likely, they’ve been constructed bit by bit.”

Urban planning that doesn’t respect the spectrum of diverse behavior and instead aims to insist on an ideal based on a few simple concepts (fresh air, more public space, large private space) will hinder the natural ability of a city system to adapt in a way that suits the residents. And it is this ability to adapt that is the cornerstone requirement of this type of complex system. Inhibit the adaptive property and you all but ensure the collapse of the system.

As Jacobs articulates:

Under the seeming disorder of the old city, wherever the old city is working successfully, is a marvelous order for maintaining the safety of the streets and the freedom of the city. It is a complex order. Its essence is intricacy of sidewalk use, bringing with it a constant succession of eyes. This order is all composed of movement and change, and although it is life, not art, we may fancifully call it the art form of the city and liken it to the dance — … to an intricate ballet in which the individual dancers and ensembles all have distinctive parts which miraculously reinforce each other and compose an orderly whole. The ballet of the good city sidewalk never repeats itself from place to place, and in any one place is always replete with new improvisations.

This is the essence of complexity. As Scott Page argues, “Adaptation occurs at the level of individuals or of types. The system itself doesn’t adapt. The parts do; they alter their behaviors leading to system level adaptation.”

Jacobs maintains that “the sight of people attracts still other people.” We feel more secure when we know there are multiple eyes on us, eyes that are concerned only with the immediate function that might affect them and are not therefore invasive.

Our complex behavior as individuals in cities, interacting with various components in any given day, is multiplied by everyone, so a city that produces a safe environment seems to be almost miraculous. But ultimately our behavior is governed by certain rules — not rules that are imposed by theory or external forces, but rules that we all feel are critical to our well-being and success in our city.

Thus, the workings of a desirable city are produced by a multitude of small interactions that have evolved and adapted as they have promoted the existence of the things that most support the desires of individuals.

“The look of things and the way they work are inextricably bound together, and in no place more so than cities,” claims Jacobs. Use is not independent of form. That is why we must understand the system as a whole. No matter how many components and unpredictable potential interactions there are, they are all part of what makes the city function.

As Jacobs concludes, “There is no use wishing it were a simpler problem, because in real life it is not a simpler problem. No matter what you try to do to it, a city park behaves like a problem in organized complexity, and that is what it is. The same is true of all other parts or features of cities. Although the inter-relations of their many factors are complex, there is nothing accidental or irrational about the ways in which these factors affect each other.”

Reciprocation Bias

“There are slavish souls who carry their appreciation for favors done
them so far that they strangle themselves with the rope of gratitude.”

—Friedrich Nietzsche

***

If you are like me, whenever receiving a favor, you too feel an immense need, almost an obligation, to pay it back in kind.

If a friend invites you over for dinner, you are almost sure to invite them over to your place for dinner as well. It almost seems as if we were meant to do each other favors and, more important, return them.

Have you ever wondered why?

A large part of the reason is that this behavior seems to have strong evolutionary benefits. It’s so pervasive in human culture, it’s believed that there is no society that does not feel reciprocation’s pull. The archaeologist Richard Leakey believes reciprocation is the foundation on which we have evolved: “We are human because our ancestors learned to share their food and their skills in an honored network of obligation.”

The web of indebtedness created by reciprocation allows for the division of tasks, eases the exchange of goods and services, and helps create interdependencies that bind us into units that are more productive than each of us is on our own. Reciprocation allows one person to give something to another with the expectation that the favor will be returned and the giver will not be taken advantage of.

Throughout human history, reciprocation lowered the cost of transactions, as almost everything begins with one person trusting another. Land could be farmed with one person lending seeds to another. Gifts could be given. Currency could be lent. Aid could be given to the weak. Moreover, reciprocation is not a human concept — it exists in the physical world. Newton's third law is that for every action there is an equal and opposite reaction. You might push on a wall, but the wall pushes back on you.

There is such an advantage to be gained from reciprocation that it’s become imprinted onto our subconscious. For example, we teach our kids to invite others they may not like to their birthday parties because our kids were invited to those kids’ parties. Deeper still, we negatively label people who violate the rule: untrustworthy, moocher, welsher. Because social sanctions can be tough on those who fail to cooperate, the rule of reciprocity often evokes guilt.

As with most things, however, reciprocation has a darker side. Just as we tend to reciprocate good behavior, sometimes we also pay back bad deeds. One of the most effective game-theory strategies is tit for tat.

“Pay every debt, as if God wrote the bill.”

— Ralph Waldo Emerson

Hate

The reciprocation of bad behavior is best evidenced in wars. Brutality escalates as each side feels obliged to return the violence it experienced from its counterpart. This spiral can lead to more mindlessly destructive behavior, including torture and mass deaths. There are plenty of examples of this negative reciprocation; consider World War II, the Crusades, and the Mongolian invasions led by Genghis Khan.

It might seem that we humans have exclusively caused much suffering in the world in a relatively short period of time. However, the reciprocation rule is overarching — the human species is not the only one capable of extreme cruelty. Charlie Munger believes that reciprocal aggression appears to be more of a rule rather than an exception among other species, too:

One interesting mental exercise is to compare Genghis Khan, who exercised extreme, lethal hostility toward other men, with ants that display extreme, lethal hostility toward members of their own species that are not part of their breeding colony. Genghis looks sweetly lovable when compared to the ants. The ants are more disposed to fight and fight with more extreme cruelty.

If the reciprocation rule is so overpowering, the natural question here would be, is there a way we can still control our response to it?

Munger advises us to train our patience.

The standard antidote to one’s overactive hostility is to train oneself to defer reaction. As my smart friend Tom Murphy so frequently says, “You can always tell the man off tomorrow if it is such a good idea.”

There’s also another way. Because the reciprocation tendency is so extreme, we can reverse the course of events by doing good rather than harm to the other party.

Particularly in WWI, the fighting sometimes paused after a positive feedback loop of less severe damage occurred. Here is how a British staff officer described his surprise about the degree of trust between the British and German soldiers:

[I was] astonished to observe German soldiers walking about within rifle range behind their own line. Our men appeared to take no notice. I privately made up my mind to do away with that sort of thing when we took over; such things should not be allowed. These people evidently did not know there was a war on. Both sides apparently believed in the policy of “live and let live.” (Dugdale 1932, p. 94)

Such behavior was not restricted to this one case, but was rather common in trench warfare during the later stages of the war.

And this makes me think that if such things could happen even during a war, there is little doubt that we could improve our relationships by doing a little undeserved good for the other person.

Love

Reciprocation is just as important in breeding love as it is in breeding hate.

Andy Warhol said, in The Philosophy of Andy Warhol (From A to B and Back Again):

Love affairs get too involved, and they’re not really worth it. But if, for some reason, you feel that they are, you should put in exactly as much time and energy as the other person. In other words, “I’ll pay you if you pay me.”

This is the reciprocation tendency at its finest. Truth is, love and marriage would lose much of their allure if there were no reciprocation tendency among partners. By loving, we literally may become loved.

As lovers and spouses, we promise loyalty to our partners and we expect it to be returned. We are encouraged to practice the virtues of marriage in front of not only our partners, but also society. These effects reinforcing each other can be thought of as the fabric of many of today’s relationships.

Furthermore, reciprocation not only holds us together, but can also bring us together in the first place. Displaying generosity can be a powerful way to advance a relationship by setting up implicit expectations of compliance from the other person.

Women, in particular, often report on the pressure they feel after receiving expensive gifts or dinners. In Influence, professor of psychology Robert Cialdini quotes the words of one of his (female) students:

After learning the hard way, I no longer let a guy I meet in a club buy me a drink because I don't want either of us to feel that I am obligated sexually.

Perhaps the key to genuine relationships lies at least partially in each party being kind without expectations. Indeed, in communal relationships like marriage, friendship, and the parent-child relationship, the accounting is unnecessary, and if you think about it, you’ll see that it is hardly ever practiced.

What is exchanged reciprocally instead is the near-unconditional willingness to provide what the other side needs, when it is needed. Still, some symmetry seems to be best; even in close friendships, strong inequalities will eventually make themselves noticed.

Abusing Reciprocity

As with any human tendency, reciprocity holds a great potential for abuse. Charlie Munger recalls how the eccentric hedge-fund manager Victor Niederhoffer managed to get good grades with an impressive course load when he was an undergraduate student at Harvard.

Contrary to what one may expect, Niederhoffer was not a very hard-working student. Instead of studying, he liked spending his time playing world-class checkers, gambling in high-stakes card games, and playing amateur-level tennis and professional-level squash. So how did he manage to get those good grades?

Munger explains:

He thought he was up to outsmarting the Harvard Economics Department. And he was. He noticed that the graduate students did most of the boring work that would otherwise go to the professors, and he noticed that because it was so hard to get to be a graduate student at Harvard, they were all very brilliant and organized and hard working, as well as much needed by grateful professors.

And therefore, by custom, and as would be predicted from the psychological force called reciprocity tendency, in a really advanced graduate course, the professors always gave an A. So Victor Niederhoffer signed up for nothing but the most advanced graduate courses in the Harvard Economics Department, and of course, he got A, after A, after A, after A, and was hardly ever near a class. And for a while, some people at Harvard may have thought it had a new prodigy on its hands. That’s a ridiculous story, but the scheme will work still. And Niederhoffer is famous: they call his style “Niederhoffering the curriculum.”

There are cases that are less innocent than Niederhoffer’s gaming the system. For example, when a salesman offers us a cup of coffee with cookies, we are likely to be subconsciously tricked into compliance by even such a minor favor, which combines reciprocity and association. Buying can be just as much about the actual experience as it is about acquiring goods at an optimal price, and salesmen know this.

Your Costs Are My Benefits

In our personal expenses, we are the ones suffering from our follies, but an important problem arises when we buy on someone else’s behalf. Imagine that you are the purchasing agent for an employer. Now the extra costs that are paid in return for the minor favor you receive are incurred not by you but by your employer.

Gifts and favors tend to create perverse incentives on the purchaser’s part and allow the seller to maximize his advantage. Smart employers know this and therefore do not allow their purchasing personnel to accept gifts. Sam Walton is one notable example; he wouldn't let Walmart’s purchasing agents accept even a hot dog from a vendor.

The exchange of favors at another’s expense is not restricted to purchasing on someone’s behalf.

Munger notes that the reciprocation tendency can also be held responsible for some wicked pay dynamics in the boardroom of public companies:

It’s incredible the reciprocity that happens when CEOs keep recommending that directors get paid more, and then the directors raise the CEO’s pay — it’s a big game of pitty pat. And then they hire compensation consultants to make sure no-one else is getting paid more. This is true even if the CEO is a klutz and a little dishonorable. I think the existing system is very bad and my system would work better, but it’s not going to happen.

In order to prevent these dynamics, he suggests that the board of directors does not get paid at all.

I think tons of eminent people would serve on boards of companies like Exxon without being paid. The lower courts in England are run by unpaid magistrates. And Harvard is run by boards of people who don’t get paid — in fact, they have to pay [in the form of donations to the school]. I think boards would be better if they were run like Berkshire Hathaway’s.

For these same reasons, Munger believes that the reciprocity tendency should be part of the compulsory law curriculum; otherwise, students may unknowingly steer away from representing their clients’ best interests. Ignorance of the reciprocation rule may explain why malpractice still occurs even among lawyers with the best intentions. The law schools simply don’t know, or care to teach, what Sam Walton knew so well.

The Concession

Besides the obvious doing of favors, there is a more subtle technique that may lure us into reciprocal and cooperative behavior. Rob Cialdini recalls an incident that made him aware of the technique:

I was walking down the street when I was approached by an 11- or 12-year-old boy. He introduced himself and said he was selling tickets to the annual Boy Scouts Circus to be held on the upcoming Saturday night. He asked if I wished to buy any tickets at $5 apiece. Since one of the last places I wanted to spend Saturday evening was with the Boy Scouts, I declined. “Well,” he said, “if you don't want to buy any tickets, how about buying some of our chocolate bars? They're only $1 each.”

Cialdini automatically bought two chocolates and immediately realized that something was wrong:

I knew that to be the case because (a) I do not like chocolate bars; (b) I do like dollars; (c) I was standing there with two of his chocolate bars; and (d) he was walking away with two of my dollars.

After meeting with his research assistants and conducting experiments with a similar setup on his students, Cialdini arrived at a rule that explains this behavior: The person who acts in a certain way toward us is entitled to a similar return action.

The person who acts in a certain way toward us is entitled to a similar return action.

This rule has two consequences:

  1. We feel obliged to repay favors we have received.
  2. We feel obliged to make a concession to someone who has made a concession to us.

As Cialdini and his research group reflected, they increasingly saw that the Boy Scout had brought him under the rule. The request to purchase the chocolates was introduced as a concession — a retreat from the request that Cialdini buy some $5 tickets.

If Cialdini was to live up to the dictates of the reciprocation rule, there had to be a concession on his part. And there was — after all, Cialdini moved from rejection to compliance after the boy had moved from a larger to a smaller request. The remarkable thing, and this is where bias comes in, was that Cialdini was not at all interested in either of the things the boy had offered.

Why would this rule be so important? Because it can lead to a lot of unnecessary trouble.

Both Cialdini and Munger believe that a subconscious reciprocation tendency was an important lever that allowed Watergate, one of the biggest political scandals in history, to occur.

Breaking into the Watergate offices of the Democratic party was a plan that was conceived by G. Gordon Liddy, an aggressive subordinate with a questionable reputation. Liddy pulled the same trick on his superiors that the twelve-year-old boy did on Cialdini. The $250,000 break-in plan was not the first that Liddy proposed — it was a significant concession from the previous two. The first of these plans, for $1 million, entailed a program that included a specially equipped “chase plane,” break-ins, kidnapping and mugging squads, and a yacht featuring “high-class call girls,” all meant to blackmail the Democratic politicians.

The second plan was a little more modest, at half of the initial price and reductions in the program. After the two initial plans were rejected by his superiors, Liddy submitted the third, “bare bones” plan, which was a little less stupid and cost “a mere” quarter of the initial price.

Do you see what Liddy did there?

Unsurprisingly, his superiors gave in; eventually, the plan was approved and it started the snowball that caused Nixon to resign. As the Watergate example illustrates, an unwatched reciprocation tendency may subtly cause mindless behavior with many extreme or dangerous consequences.

***

One of the reasons reciprocation can be used so effectively as a device for gaining another's compliance is that it combines power and subtlety. Especially in its concessionary form, the reciprocation rule often produces a yes response to a request that otherwise would surely have been refused.

I hope that the next time you come across a situation where you feel the need to return a favor, you will think twice about the possible consequences of accepting it in the first place. You may think, for example, that someone offering you a free pen will not influence you at all, but there is an entire human history arguing otherwise. Perhaps Sam Walton’s policy, of not accepting favors at all in matters where impartiality is preferred, is best.

Yet there is some truth to saying that reciprocal behavior also represents the best part of human nature. There are times when successful trade, good friendships, and even romantic relationships develop out of the need to feel symmetrical in our relationships. Indeed, it could well be that the very best parts of our lives lie in relationships of affection in which both we and the other party want to please each other.

The Butterfly Effect: Everything You Need to Know About This Powerful Mental Model

“You could not remove a single grain of sand from its place without thereby … changing something throughout all parts of the immeasurable whole.”

— Fichte, The Vocation of Man (1800)
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The Basics

In one of Stephen King’s greatest works, 11/22/63, a young man named Jake discovers a portal in a diner’s pantry which leads back to 1958. After a few visits and some experiments, Jake deduces that altering history is possible. However long he stays in the past, only two minutes go by in the present. He decides to live in the past until 1963 so he can prevent the assassination of President John F. Kennedy, believing that this change will greatly benefit humanity. After years of stalking Lee Harvey Oswald, Jake manages to prevent him from shooting Kennedy.

Upon returning to the present, he expects to find the world improved as a result. Instead, the opposite has happened. Earthquakes occur everywhere, his old home is in ruins, and nuclear war has destroyed much of the world. (As King wrote in an article for Marvel Spotlight, “Not good to fool with Father Time.”) Distraught, Jake returns to 1958 once again and resets history.

In addition to being a masterful work of speculative fiction, 11/22/63 is a classic example of how everything in the world is connected together.

The butterfly effect is the idea that small things can have non-linear impacts on a complex system. The concept is imagined with a butterfly flapping its wings and causing a typhoon.

Of course, a single act like the butterfly flapping its wings cannot cause a typhoon. Small events can, however, serve as catalysts that act on starting conditions.

And as John Gribbin writes in his cult-classic work Deep Simplicity, “some systems … are very sensitive to their starting conditions, so that a tiny difference in the initial ‘push’ you give them causes a big difference in where they end up, and there is feedback, so that what a system does affects its own behavior.”

In the foreword to The Butterfly Effect in Competitive Markets by Dr. Rajagopal, Tom Breuer writes:

Simple systems, with few variables, can nonetheless show unpredictable and sometimes chaotic behavior…[Albert] Libchaber conducted a series of seminal experiments. He created a small system in his lab to study convection (chaotic system behavior) in a cubic millimeter of helium. By gradually warming this up from the bottom, he could create a state of controlled turbulence. Even this tightly controlled environment displayed chaotic behavior: complex unpredictable disorder that is paradoxically governed by “orderly” rules.

… [A] seemingly stable system (as in Libchaber’s 1 ccm cell of helium) can be exposed to very small influences (like heating it up a mere 0.001 degree), and can transform from orderly convection into wild chaos. Although [such systems are] governed by deterministic phenomena, we are nonetheless unable to predict how [they] will behave over time.

What the Butterfly Effect Is Not

The point of the butterfly effect is not to get leverage. As General Stanley McChrystal writes in Team of Teams:

In popular culture, the term “butterfly effect” is almost always misused. It has become synonymous with “leverage”—the idea of a small thing that has a big impact, with the implication that, like a lever, it can be manipulated to a desired end. This misses the point of Lorenz’s insight. The reality is that small things in a complex system may have no effect or a massive one, and it is virtually impossible to know which will turn out to be the case.

Benjamin Franklin offered a poetic perspective in his variation of a proverb that’s been around since the 14th century in English and the 13th century in German, long before the identification of the butterfly effect:

For want of a nail the shoe was lost,
For want of a shoe the horse was lost,
For want of a horse the rider was lost,
For want of a rider the battle was lost,
For want of a battle the kingdom was lost,
And all for the want of a horseshoe nail.

The lack of one horseshoe nail could be inconsequential, or it could indirectly cause the loss of a war. There is no way to predict which outcome will occur. (If you want an excellent kids book to start teaching this to your children, check out If You Give a Mouse a Cookie.)

In this post, we will seek to unravel the butterfly effect from its many incorrect connotations, and build an understanding of how it affects our individual lives and the world in general.

Edward Lorenz and the Discovery of the Butterfly Effect

“It used to be thought that the events that changed the world were things like big bombs, maniac politicians, huge earthquakes, or vast population movements, but it has now been realized that this is a very old-fashioned view held by people totally out of touch with modern thought. The things that change the world, according to Chaos theory, are the tiny things. A butterfly flaps its wings in the Amazonian jungle, and subsequently a storm ravages half of Europe.”

— from Good Omens, by Terry Pratchett and Neil Gaiman
***

Although the concept of the butterfly effect has long been debated, the identification of it as a distinct effect is credited to Edward Lorenz (1917–2008). Lorenz was a meteorologist and mathematician who successfully combined the two disciplines to create chaos theory. During the 1950s, Lorenz searched for a means of predicting the weather, as he found linear models to be ineffective.

In an experiment to model a weather prediction, he entered the initial condition as 0.506, instead of 0.506127. The result was surprising: a somewhat different prediction. From this, he deduced that the weather must turn on a dime. A tiny change in the initial conditions had enormous long-term implications. By 1963, he had formulated his ideas enough to publish an award-winning paper entitled Deterministic Nonperiodic Flow. In it, Lorenz writes:

Subject to the conditions of uniqueness, continuity, and boundedness … a central trajectory, which in a certain sense is free of transient properties, is unstable if it is nonperiodic. A noncentral trajectory … is not uniformly stable if it is nonperiodic, and if it is stable at all, its very stability is one of its transient properties, which tends to die out as time progresses. In view of the impossibility of measuring initial conditions precisely, and thereby distinguishing between a central trajectory and a nearby noncentral trajectory, all nonperiodic trajectories are effectively unstable from the point of view of practical prediction.

In simpler language, he theorized that weather prediction models are inaccurate because knowing the precise starting conditions is impossible, and a tiny change can throw off the results. In order to make the concept understandable to non-scientific audiences, Lorenz began to use the butterfly analogy.

A small error in the initial data magnifies over time.

In speeches and interviews, he explained that a butterfly has the potential to create tiny changes which, while not creating a typhoon, could alter its trajectory. A flapping wing represents the minuscule changes in atmospheric pressure, and these changes compound as a model progresses. Given that small, nearly imperceptible changes can have massive implications in complex systems, Lorenz concluded that attempts to predict the weather were impossible. Elsewhere in the paper, he writes:

If, then, there is any error whatever in observing the present state—and in any real system such errors seem inevitable—an acceptable prediction of an instantaneous state in the distant future may well be impossible.

… In view of the inevitable inaccuracy and incompleteness of weather observations, precise very-long-range forecasting would seem to be nonexistent.

Lorenz always stressed that there is no way of knowing what exactly tipped a system. The butterfly is a symbolic representation of an unknowable quantity.

Furthermore, he aimed to contest the use of predictive models that assume a linear, deterministic progression and ignore the potential for derailment. Even the smallest error in an initial setup renders the model useless as inaccuracies compound over time. The exponential growth of errors in a predictive model is known as deterministic chaos. It occurs in most systems, regardless of their simplicity or complexity.

The butterfly effect is somewhat humbling—a model that exposes the flaws in other models. It shows science to be less accurate than we assume, as we have no means of making accurate predictions due to the exponential growth of errors.

Prior to the work of Lorenz, people assumed that an approximate idea of initial conditions would lead to an approximate prediction of the outcome. In Chaos: Making a New Science, James Gleick writes:

The models would churn through complicated, somewhat arbitrary webs of equations, meant to turn measurements of initial conditions … into a simulation of future trends. The programmers hoped the results were not too grossly distorted by the many unavoidable simplifying assumptions. If a model did anything too bizarre … the programmers would revise the equations to bring the output back in line with expectation… Models proved dismally blind to what the future would bring, but many people who should have known better acted as though they believed the results.

One theoretician declared, “The basic idea of Western science is that you don’t have to take into account the falling of a leaf on some planet in another galaxy when you’re trying to account for the motion of a billiard ball on a pool table on earth.”

An illustration of two weather conditions with very slightly different initial conditions. The trajectories are similar at first, before deviating further and further.

Lorenz’s findings were revolutionary because they proved this assumption to be entirely false. He found that without a perfect idea of initial conditions, predictions are useless—a shocking revelation at the time.

During the early days of computers, many people believed they would enable us to understand complex systems and make accurate predictions. People had been slaves to weather for millennia, and now they wanted to take control. With one innocent mistake, Lorenz shook the forecasting world, sending ripples which (appropriately) spread far beyond meteorology.

Ray Bradbury, the Butterfly Effect, and the Arrow of Time

Ray Bradbury’s classic science fiction story A Sound of Thunder predates the identification of chaos theory and the butterfly effect. Set in 2055, it tells of a man named Eckels who travels back 65 million years to shoot a dinosaur. Warned not to deviate from the tour guide’s plan, Eckels (along with his guide and the guide’s assistant) heads off to kill a Tyrannosaurus Rex who was going to die soon anyway when a falling tree lands on it. Eckels panics at the sight of the creature and steps off the path, leaving his guide to kill the T Rex. The guide is enraged and orders Eckels to remove the bullets before the trio returns to 2055. Upon arrival, they are confused to find that the world has changed. Language is altered and an evil dictator is now in charge. A confused Eckels notices a crushed butterfly stuck to his boot and realizes that in stepping off the path, he killed the insect and changed the future. Bradbury writes:

Eckels felt himself fall into a chair. He fumbled crazily at the thick slime on his boots. He held up a clod of dirt, trembling, “No, it cannot be. Not a little thing like that. No!”

Embedded in the mud, glistening green and gold and black, was a butterfly, very beautiful and very dead.

“Not a little thing like that! Not a butterfly!” cried Eckels.

It fell to the floor, an exquisite thing, a small thing that could upset balances and knock down a line of small dominoes and then big dominoes and then gigantic dominoes, all down the years across Time. Eckels' mind whirled. It couldn't change things. Killing one butterfly couldn't be that important! Could it?

Bradbury envisioned the passage of time as fragile and liable to be disturbed by minor changes. In the decades since the publication of A Sound of Thunder, physicists have examined its accuracy. Obviously, we cannot time–travel, so there is no way of knowing how plausible the story is, beyond predictive models. Bradbury’s work raises the questions of what time is and whether it is deterministic.

Physicists refer to the Arrow of Time—the non-reversible progression of entropy (disorder.) As time moves forward, matter becomes more and more chaotic and does not spontaneously return to its original state. If you break an egg, it remains broken and cannot spontaneously re-form, for example. The Arrow of Time gives us a sense of past, present, and future. Arthur Eddington (the astronomer and physicist who coined the term) explained:

Let us draw an arrow arbitrarily. If as we follow the arrow we find more and more of the random element in the state of the world, then the arrow is pointing towards the future; if the random element decreases the arrow points towards the past. That is the only distinction known to physics. This follows at once if our fundamental contention is admitted that the introduction of randomness is the only thing which cannot be undone.

In short, the passage of time as we perceive it does exist, conditional to the existence of entropy. As long as entropy is non-reversible, time can be said to exist. The closest thing we have to a true measurement of time is a measurement of entropy. If the progression of time is nothing but a journey towards chaos, it makes sense for small changes to affect the future by amplifying chaos.

We do not yet know if entropy creates time or is a byproduct of it. Subsequently, we cannot know if changing the past would change the future. Would stepping on a butterfly shift the path of entropy? Did Eckels move off the path out of his own free will, or was that event predetermined? Was the dictatorial future he returned to always meant to be?

These interconnected concepts — the butterfly effect, chaos theory, determinism, free will, time travel — have captured many imaginations since their discoveries. Films ranging from It’s a Wonderful Life to Donnie Darko and the eponymous Butterfly Effect have explored the complexities of cause and effect. Once again, it is important to note that works of fiction tend to view the symbolic butterfly as the cause of an effect. According to Lorenz’s original writing, though, the point is that small details can tip the balance without being identifiable.

The Butterfly Effect in Business

Marketplaces are, in essence, chaotic systems that are influenced by tiny changes. This makes it difficult to predict the future, as the successes and failures of businesses can appear random. Periods of economic growth and decline sprout from nowhere. This is the result of the exponential impact of subtle stimuli—the economic equivalent of the butterfly effect. Breuer explains:

We live in an interconnected, or rather a hyper-connected society. Organizations and markets “behave” like networks. This triggers chaotic (complex) rather than linear behavior.

Preparing for the future and seeing logic in the chaos of consumer behaviour is not easy. Once-powerful giants collapse as they fall behind the times. Tiny start-ups rise from the ashes and take over industries. Small alterations in existing technology transform how people live their lives. Fads capture everyone’s imagination, then disappear.

Businesses have two options in this situation: build a timeless product or service, or race to keep up with change. Many businesses opt for a combination of the two. For example, Doc Martens continues selling the classic 1460 boot, while bringing out new designs each season. This approach requires extreme vigilance and attention to consumer desires, in an attempt to both remain relevant and appear timeless. Businesses leverage the compounding impact of small tweaks that aim to generate interest in all they have to offer.

In The Butterfly Effect in Competitive Markets, Dr. Rajagopal writes that

most global firms are penetrating bottom-of-the-pyramid market segments by introducing small changes in technology, value perceptions, [and] marketing-mix strategies, and driving production on an unimagined scale of magnitude to derive a major effect on markets. …Procter & Gamble, Kellogg’s, Unilever, Nestlé, Apple, and Samsung, have experienced this effect in their business growth…. Well-managed companies drive small changes in their business strategies by nipping the pulse of consumers….

Most firms use such effect by making a small change in their strategy in reference to produce, price, place, promotion, … posture (developing corporate image), and proliferation…to gain higher market share and profit in a short span.

For most businesses, incessant small changes are the most effective way to produce the metaphorical typhoon. These iterations keep consumers engaged while preserving brand identity. If these small tweaks fail, the impact is hopefully not too great. But if they succeed and compound, the rewards can be monumental.

By nature, all markets are chaotic, and what seem like inconsequential alterations can propel a business up or down. Rajagopal explains how the butterfly effect connects to business:

Globalization and frequent shifts in consumer preferences toward products and services have accelerated chaos in the market due to the rush of firms, products, and business strategies. Chaos theory in markets addresses the behavior of strategic and dynamic moves of competing firms that are highly sensitive to existing market conditions triggering the butterfly effect.

The initial conditions (economic, social, cultural, political) in which a business sets up are vital influences on its success or failure. Lorenz found that the smallest change in the preliminary conditions created a different outcome in weather predictions, and we can consider the same to be true for businesses. The first few months and years are a crucial time, when rates of failure are highest and the basic brand identity forms. Any of the early decisions, achievements, or mistakes have the potential to be the wing flap that creates a storm.

Benoit Mandelbrot on the Butterfly Effect in Economics

International economies can be thought of as a single system, wherein each part influences the others. Much like the atmosphere, the economy is a complex system in which we see only the visible outcomes—rain or shine, boom or bust. With the advent of globalization and improved communication technology, the economy is even more interconnected than in the past. One episode of market volatility can cause problems for the entire system. The butterfly effect in economics refers to the compounding impact of small changes. As a consequence, it is nearly impossible to make accurate predictions for the future or to identify the precise cause of an inexplicable change. Long periods of stability are followed by sudden declines, and vice versa.

Benoit Mandelbrot (the “father of fractals”) began applying the butterfly effect to economics several decades ago. In a 1999 article for Scientific American, he explained his findings. Mandelbrot saw how unstable markets could be, and he cited an example of a company which saw its stock drop 40% in one day, followed by another 6%, before rising by 10%—the typhoon created by an unseen butterfly. When Benoit looked at traditional economic models, he found that they did not even allow for the occurrence of such events. Standard models denied the existence of dramatic market shifts. Benoit writes in Scientific American:

According to portfolio theory, the probability of these large fluctuations would be a few millionths of a millionth of a millionth of a millionth. (The fluctuations are greater than 10 standard deviations.) But in fact, one observes spikes on a regular basis—as often as every month—and their probability amounts to a few hundredths.

If these changes are unpredictable, what causes them? Mandelbrot’s answer lay in his work on fractals. To explain fractals would require a whole separate post, so we will go with Mandelbrot’s own simplified description: “A fractal is a geometric shape that can be separated into parts, each of which is a reduced-scale version of the whole.” He goes on to explain the connection:

In finance, this concept is not a rootless abstraction but a theoretical reformulation of a down-to-earth bit of market folklore—namely that movements of a stock or currency all look alike when a market chart is enlarged or reduced so that it fits the same time and price scale. An observer then cannot tell which of the data concern prices that change from week to week, day to day or hour to hour. This quality defines the charts as fractal curves and makes available many powerful tools of mathematical and computer analysis.”

In a talk, Mandelbrot held up his coffee and declared that predicting its temperature in a minute is impossible, but in an hour is perfectly possible. He applied the same concept to markets that change in dramatic ways in the short term. Even if a long-term pattern can be deduced, it has little use for those who trade on a shorter timescale.

Mandelbrot explains how his fractals can be used to create a more useful model of the chaotic nature of the economy:

Instead, multifractals can be put to work to “stress-test” a portfolio. In this technique, the rules underlying multifractals attempt to create the same patterns of variability as do the unknown rules that govern actual markets. Multifractals describe accurately the relation between the shape of the generator and the patterns of up-and-down swings of prices to be found on charts of real market data… They provide estimates of the probability of what the market might do and allow one to prepare for inevitable sea changes. The new modeling techniques are designed to cast a light of order into the seemingly impenetrable thicket of the financial markets. They also recognize the mariner’s warning that, as recent events demonstrate, deserves to be heeded: On even the calmest sea, a gale may be just over the horizon.

In The Misbehaviour of Markets, Mandelbrot and Richard Hudson expand upon the topic of financial chaos. They begin with a discussion of the infamous 2008 crash and its implications:

The worldwide market crash of autumn 2008 had many causes: greedy bankers, lax regulators and gullible investors, to name a few. But there is also a less-obvious cause: our all-too-limited understanding of how markets work, how prices move and how risks evolve. …

Markets are complex, and treacherous. The bone-chilling fall of September 29, 2008—a 7 percent, 777 point plunge in the Dow Jones Industrial Average—was, in historical terms, just a particularly dramatic demonstration of that fact. In just a few hours, more than $1.6 trillion was wiped off the value of American industry—$5 trillion worldwide.

Mandelbrot and Hudson believe that the 2008 credit crisis can be attributed in part to the increasing confidence in financial predictions. People who created computer models designed to guess the future failed to take into account the butterfly effect. No matter how complex the models became, they could not create a perfect picture of initial conditions or account for the compounding impact of small changes. Just as people believed they could predict and therefore control the weather before Lorenz published his work, people thought they could do the same for markets until the 2008 crash proved otherwise. Wall Street banks trusted their models of the future so much that they felt safe borrowing growing sums of money for what was, in essence, gambling. After all, their predictions said such a crash was impossible. Impossible or not, it happened.

According to Mandelbrot and Hudson, predictive models view markets as “a risky but ultimately … manageable world.” As with meteorology, economic predictions are based on approximate ideas of initial conditions—ideas that, as we know, are close to useless. As Mandelbrot and Hudson write:

[C]auses are usually obscure. … The precise market mechanism that links news to price, cause to effect, is mysterious and seems inconsistent. Threat of war: Dollar falls. Threat of war: Dollar rises. Which of the two will actually happen? After the fact, it seems obvious; in hindsight, fundamental analysis can be reconstituted and is always brilliant. But before the fact, both outcomes may seem equally likely.

In the same way that apparently similar weather conditions can create drastically different outcomes, apparently similar market conditions can create drastically different outcomes. We cannot see the extent to which the economy is interconnected and we cannot identify where the butterfly lies. Mandelbrot and Hudson disagree with the view of the economy as separate from other parts of our world. Everything connects:

No one is alone in this world. No act is without consequences for others. It is a tenet of chaos theory that, in dynamical systems, the outcome of any process is sensitive to its starting point—or in the famous cliché, the flap of a butterfly’s wings in the Amazon can cause a tornado in Texas. I do not assert that markets are chaotic…. But clearly, the global economy is an unfathomably complicated machine. To all the complexity of the physical world… you add the psychological complexity of men acting on their fleeting expectations….

Why do people prefer to blame crashes (such as the 2008 credit crisis) on the folly of those in the financial industry? Jonathan Cainer provides a succinct explanation:

Why do we love the idea that people might be secretly working together to control and organise the world? Because we do not like to face the fact that our world runs on a combination of chaos, incompetence, and confusion.

Historic Examples of the Butterfly Effect

“A very small cause which escapes our notice determines a considerable effect that we cannot fail to see, and then we say the effect is due to chance. If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment. But even if it were the case that the natural laws had no longer any secret for us, we could still only know the initial situation *approximately*. If that enabled us to predict the succeeding situation with *the same approximation*, that is all we require, and we should say that the phenomenon had been predicted, that it is governed by laws. But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon.”

— Jules Henri Poincaré (1854–1912)

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Many examples exist of instances where a tiny detail led to a dramatic change. In each case, the world we live in could be different if the situation had been reversed. Here are some examples of how the butterfly effect has shaped our lives.

  • The bombing of Nagasaki. The US initially intended to bomb the Japanese city of Kuroko, with the munitions factory as a target. On the day the US planned to attack, cloudy weather conditions prevented the factory from being seen by military personnel as they flew overhead. The airplane passed over the city three times before the pilots gave up. Locals huddled in shelters heard the hum of the airplane preparing to drop the nuclear bomb and prepared for their destruction. Except Kuroko was never bombed. Military personnel decided on Nagasaki as the target due to improved visibility. The implications of that split-second decision were monumental. We cannot even begin to comprehend how different history might have been if that day had not been cloudy. Kuroko is sometimes referred to as the luckiest city in Japan, and those who lived there during the war are still shaken by the near miss.
  • The Academy of Fine Arts in Vienna rejecting Adolf Hitler’s application, twice. In the early 1900s, a young Hitler applied for art school and was rejected, possibly by a Jewish professor. By his own estimation and that of scholars, this rejection went on to shape his metamorphosis from a bohemian aspiring artist into the human manifestation of evil. We can only speculate as to how history would have been different. But it is safe to assume that a great deal of tragedy could have been avoided if Hitler had applied himself to watercolors, not to genocide.
  • The assassination of Archduke Franz Ferdinand. A little-known fact about the event considered to be the catalyst for both world wars is that it almost didn’t happen. On the 28th of June, 1914, a teenage Bosnian-Serb named Gavrilo Princip went to Sarajevo with two other nationalists in order to assassinate the Archduke. The initial assassination attempt failed; a bomb or grenade exploded beneath the car behind the Archduke’s and wounded its occupants. The route was supposed to have been changed after that, but the Archduke’s driver didn’t get the message. Had he actually taken the alternate route, Princip would not have been on the same street as the car and would not have had the chance to shoot the Archduke and his wife that day. Were it not for a failure of communication, both world wars might never have happened.
  • The Chernobyl disaster. In 1986, a test at the Chernobyl nuclear plant went awry and released 400 times the radiation produced by the bombing of Hiroshima. One hundred fifteen thousand people were evacuated from the area, with many deaths and birth defects resulting from the radiation. Even today, some areas remain too dangerous to visit. However, it could have been much worse. After the initial explosion, three plant workers volunteered to turn off the underwater valves to prevent a second explosion. It has long been believed that the trio died as a result, although there is now some evidence this may not have been the case. Regardless, diving into a dark basement flooded with radioactive water was a heroic act. Had they failed to turn off the valve, half of Europe would have been destroyed and rendered uninhabitable for half a million years. Russia, Ukraine, and Kiev also would have become unfit for human habitation. Whether they lived or not, the three men—Alexei Ananenko, Valeri Bezpalov and Boris Baranov—stilled the wings of a deadly butterfly. Indeed, the entire Chernobyl disaster was the result of poor design and the ineptitude of staff. The long-term result (in addition to the impact on residents of the area) was a widespread anxiety towards nuclear plants and bias against nuclear power, leading to a preference for fossil fuels. Some people have speculated that Chernobyl is responsible for the acceleration of global warming, as countries became unduly slow to adopt nuclear power.
  • The Cuban Missile Crisis. We all may owe our lives to a single Russian Navy officer named Vasili Arkhipov, who has been called “the man who saved the world.” During the Cuban Missile Crisis, Arkhipov was stationed on a nuclear-armed submarine near Cuba. American aircraft and ships began using depth charges to signal the submarine that it should surface so it could be identified. With the submarine submerged too deep to monitor radio signals, the crew had no idea what was going on in the world above. The captain, Savitsky, decided the signal meant that war had broken out and he prepared to launch a nuclear torpedo. Everyone agreed with him—except Arkhipov. Had the torpedo launched, nuclear clouds would have hit Moscow, London, East Anglia and Germany, before wiping out half of the British population. The result could have been a worldwide nuclear holocaust, as countries retaliated and the conflict spread. Yet within an overheated underwater room, Arkhipov exercised his veto power and prevented a launch. Without the courage of one man, our world could be unimaginably different.

From these handful of examples, it is clear how fragile the world is, and how dire the effects of tiny events can be on starting conditions.

We like to think we can predict the future and exercise a degree of control over powerful systems such as the weather and the economy. Yet the butterfly effect shows that we cannot. The systems around us are chaotic and entropic, prone to sudden change. For some kinds of systems, we can try to create favorable starting conditions and be mindful of the kinds of catalysts that might act on those conditions – but that’s as far as our power extends. If we think that we can identify every catalyst and control or predict outcomes, we are only setting ourselves up for a fall.

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