Ergodicity
A process is ergodic when the time average for one participant equals the ensemble average across many participants. Roll a fair die repeatedly and your long-run average approaches 3.5 — the same as the average across all die-rollers in the world at any moment. Time and ensemble give the same answer.
Many important processes aren’t ergodic. The expected value of a gamble across a population differs from the expected value for an individual playing repeatedly through time. And this difference changes everything about rational decision-making.
Ole Peters demonstrates with a coin flip: heads, your wealth increases by 50%; tails, it decreases by 40%. Expected value is positive — 0.5 × 1.5 + 0.5 × 0.6 = 1.05, a 5% expected gain. Across a thousand players, total wealth grows. But any individual playing repeatedly goes broke.
Follow one player through ten rounds. Win some, lose some. The math: 1.5^5 × 0.6^5 = 7.6 × 0.078 = 0.59. Despite positive expected value at each round, the time-average outcome is loss. The sequence matters. Multiplicative dynamics create this divergence — what happens to you over time isn’t what happens across everyone at once.
This resolves the St. Petersburg paradox. A game with infinite expected value (flip until tails, win 2^n dollars) should command infinite willingness to pay. But no one would pay much. Traditional explanations invoke diminishing marginal utility. Ergodicity economics says: the expected value calculation is simply wrong for an individual playing through time. Your exposure is sequential — you can’t average across parallel selves.
The same logic applies to insurance, investment, and any multiplicative risk. Positive expected value isn’t enough. You need to survive to the long run where expectations manifest. Losing 100% even once ends the game. Ruin absorbs.
Finance largely ignores this. Modern portfolio theory optimizes expected return for given variance. But expected return is an ensemble average. What matters to an investor is their portfolio’s trajectory through time — and that trajectory can diverge sharply from ensemble expectations.
The Kelly criterion addresses this by maximizing the expected geometric growth rate rather than arithmetic expected value. It yields smaller bets than expected value maximization but ensures long-run survival. The approach acknowledges that you’re one person living through one sequence of outcomes, not the average across parallel universes where you took every possible path.
Go Deeper
Books
- Ergodicity by Luca Dellanna — Accessible introduction to the concept and its life implications. Short and practical.
- The Black Swan by Nassim Nicholas Taleb — Covers related ground on why ensemble statistics mislead about individual fates.
- Fortune’s Formula by William Poundstone — The history of the Kelly criterion. Narrative-driven, covers the math that solves the non-ergodicity problem.
Essays
- Ole Peters’ papers on ergodicity economics (available at London Mathematical Laboratory) — The technical source for the modern formulation.
Related: [[risk]], [[kelly-criterion]], [[fat-tails]], [[antifragility]]