Power laws and criticality: the world without a scale

 

Introduction to power laws and the failure of normal intuition

Power laws describe systems where extreme events are far more probable than in normal distributions. The video traces how data from incomes to wars follow 1/x^p patterns, creating heavy tails that skew averages and defy conventional risk assessment.

"Power laws reveal something fundamental about a system's structure." - Veritasium

Pareto, income, and the log-log reveal

Vilfredo Pareto observed that country income data collapses onto a simple rule of 1/x^alpha when plotted on log-log axes, with alpha about 1.5. This pattern recurs across nations and time, signaling a universal mechanism behind wealth distribution.

"This phenomenon is called self-organized criticality." - Veritasium

The three classic casino thought experiments

Three coin-toss scenarios illustrate different distributions: a normal (additive) payout, a multiplicative log-normal variant with a fat tail, and the St. Petersburg paradox which yields an infinite expected value and a pure power-law payout distribution.

"The payout of the St. Petersburg Paradox follows a power law." - Veritasium

From math to nature and society

Power laws appear in earthquakes, forest fires, and even internet and venture-capital networks. They arise from interacting components and often signal universality, where details fade and broad patterns emerge. The lesson for risk and strategy is clear: in power-law worlds, most outcomes are small but a few colossal events dominate results.

"if you're playing some sort of game that is dominated by power law, then you better do the work as much of it early as possible so you get the benefit from the snowball effect, essentially." - Veritasium