Have You Been Playing the Game of Life Wrong? Understanding Power Laws

TL;DR

Power laws change how you should play the game of life because rare, enormous outcomes can dominate the average, making normal-distribution assumptions misleading. Pareto found that income data from several European countries formed straight lines on log-log plots, with England’s gradient around negative 1.5. Recognizing whether outcomes are additive or multiplicative helps you judge long-run payoffs. Read on to compare the distributions and casino games.

Transcript

• Some things are not normal. By that I mean if you go out in the world and start measuring things like human height, IQ or the size of apples on a tree, you will find that for each of these things, most of the data clusters around some average value. This is so common that we call it the normal distribution, but some things in life are not like th... Read More

Key Insights

• Power laws describe distributions where a few large events dominate the average, unlike normal distributions.
• Vilfredo Pareto discovered income distributions follow a power law, not a normal distribution.
• In power laws, the probability of large events is higher than in normal distributions.
• Systems governed by power laws often exist in a critical state, making them unpredictable.
• Self-organized criticality explains how systems naturally reach a critical state without external tuning.
• Power laws and fractals are linked, as both describe scale-free systems.
• Some industries, like venture capital, thrive on power law distributions, relying on rare successes.
• Understanding whether a system follows a power law or normal distribution is crucial for strategic decision-making.

Questions & Answers

Q: What is a power law?

A power law is a relationship in which the frequency of an outcome decreases according to a power of its size. In Pareto’s English income data, the number of people earning at least x was proportional to one divided by x to the power of 1.5.

Q: How are power laws different from normal distributions?

In a normal distribution, most values cluster around a clearly defined average, and extreme outliers are exceptionally rare. Power laws have a much greater likelihood of very large events, so those events can skew the average and make the system harder to predict.

Q: How did Vilfredo Pareto identify a power law in income?

In the late 1800s, Pareto gathered income-tax records from Italy, England, France, and other European countries. When he plotted the logarithms of income and population values on log-log graphs, the broad curves became straight lines with remarkably similar gradients.

Q: What does the negative 1.5 gradient in Pareto’s English income data mean?

It means that each time income doubles, the number of people earning at least that amount falls by two to the power of 1.5, or around 2.8. Pareto expressed this as the number earning at least x being proportional to one over x to the power of 1.5.

Q: Why can averages be misleading in a power-law system?

Rare but extremely large events can dominate and continually increase the measured average. The transcript notes that the average may grow as more observations are collected, unlike a normal-distribution setting where small variations tend to cancel out.

Q: What kinds of outcomes does the transcript say follow power laws?

Nature displays power laws in many places, and the measured sizes of world wars, based on how many people they kill, also follow a power law. Income distributions provide another example, with some people earning five, 10, or even 100 times more than others.

Q: What is the expected payout of the first casino game?

The game provides 100 coin tosses and pays $1 for every head. Multiplying the one-half probability of heads by $1 and 100 tosses gives an expected payout of $50, so the transcript says you should pay less than $50 to play.

Q: What is the expected payout of the multiplicative casino game?

You start with $1 and multiply the total by 1.1 after each head or 0.9 after each tail for 100 tosses. Because both outcomes are equally likely, the expected factor per toss is one, producing an expected payout of $1.

Summary & Key Takeaways

• Power laws describe situations where a few large events dominate the average, unlike normal distributions where data clusters around a mean. This concept is seen in nature and human systems, indicating systems in critical states. Understanding power laws is crucial for making informed decisions in risk-prone environments.

• The video explains the discovery of power laws by Vilfredo Pareto, who found income distributions follow a power law. This contrasts with normal distributions where extreme outliers are rare. Power laws indicate systems in critical states, making them unpredictable and more prone to large events.

• Industries like venture capital benefit from power law distributions, where rare successes dominate returns. Recognizing whether a system follows a power law or normal distribution is essential for strategic decision-making, as it influences risk management and investment strategies.