One coin game, two honest averages — the crowd gets rich while the typical player goes broke.
The two averages, named plainly. The ensemble average asks: pool a crowd of players — what does the mean of all their wealth do? At f = 1 it grows ×1.05 per round, exactly, because 0.5·1.5 + 0.5·0.6 = 1.05. The time average asks: follow one player down their own timeline — what does a round do to them? It multiplies their wealth by √0.9 ≈ 0.9487. Both numbers are correct; they answer different questions, and this game refuses to make them agree — that refusal is what "non-ergodic" means here. After 100 rounds the mean sits near 1.05100 ≈ 131.5 while the median sits near 0.950 ≈ 0.005154. The mean is carried by a sliver of lucky streaks — so thin that at 1000 rounds the simulated mean falls far below the gold line, because 1000 players almost never contain one. That gap is not a bug; it is the exhibit. One honesty note on the blue line: eg(f)·t is the exact median at every even round; at odd rounds the median is a short interval and the line passes through its geometric midpoint.
The fix is a fraction. A player who wagers only f of their wealth each round has time growth g(f) = ½ln(1+0.5f) + ½ln(1−0.4f). It peaks at f* = 1/4, exactly — the Kelly fraction, where the typical player compounds at ×1.00623 per round — and crosses zero at f = 0.5, since 1.25 × 0.8 = 1. Past that, the ensemble mean keeps climbing at ×(1+0.05f) while the typical player shrinks. The right panel is that whole story in one curve.
Lineage. J.L. Kelly Jr., “A New Interpretation of Information Rate”, Bell System Technical Journal 35(4), 917–926 (1956), derived the growth-optimal fraction. Ole Peters & Murray Gell-Mann, “Evaluating gambles using dynamics”, Chaos 26, 023103 (2016), is the earlier published appearance of this exact gamble — start at $1, heads ×1.5, tails ×0.6, fair coin (their Fig. 2). Ole Peters, “The ergodicity problem in economics”, Nature Physics 15, 1216–1221 (2019), presents it as its Eq. 2 and makes it the centrepiece. And Nassim Nicholas Taleb, Skin in the Game (Random House, 2018), ch. 19, “The Logic of Risk Taking”, presses the same point for a general audience — paraphrased: a risk taken once is not the same risk taken again and again, because ruin anywhere along the sequence removes you from every round after it, and the time view, not the ensemble view, is the one your single life actually gets.
Contested, in the open. The arithmetic above is not in dispute — nobody on either side questions these multipliers. What is disputed is what follows from them. Jason N. Doctor, Peter P. Wakker & Tong V. Wang, “Economists’ views on the ergodicity problem”, Nature Physics 16, 1168 (2020) — a Matters Arising reply — argue that economics was never confused about these sums and contest the claim that ergodicity economics rewrites decision theory. Peters replied in turn (Nature Physics 16, 1169 (2020)), standing by the reframing. This page takes no side: the engine wears its EXACT pill, the interpretation wears its CONTESTED one, and both papers are a click away. the wing →