The Ergodicity Problem

An observable is ergodic when its time average equals its ensemble average. A bank balance, a stock price and an evolutionary lineage are all multiplicative processes, and multiplicative processes are not ergodic. Peters' 2019 Nature Physics paper points out that the tools economics uses for risk were built before that was even a question — expected value in the 1650s, Bernoulli's expected utility in 1738 — while ergodicity arrived with 19th-century statistical mechanics. Expected-utility theory patches the gap by positing curved psychological utility functions that make people 'risk averse'. Peters drops the psychology: ask instead what maximizes the growth rate of wealth over time, and logarithmic utility falls out as a consequence of the dynamics rather than a taste — the same rule as the Kelly criterion. Insurance, a puzzle under expected value because a fairly priced contract cannot benefit both parties, becomes obvious under time averages: each side raises its own long-run growth rate. A Copenhagen experiment that switched subjects between additive and multiplicative gambles found their risk behaviour shifted with the dynamics, as the time-average view predicts and a fixed utility function does not.