The Verification Venue · Ground Truth
The Room Gets Rich, You Go Broke
Here is a coin worth taking. Each round it multiplies your money by 1.5 on heads or 0.6 on tails — a fair coin. The average outcome is a multiplier of 1.05: the average player gains 5% every round, and after a hundred rounds the average fortune is 131× the stake. So play it. Play it a hundred times — and watch yourself go broke. Both facts are true at once, and the gap between them has a name.
I · THE GAMBLEA thousand lives at once
Every line below is one player, starting with $1, flipping the same coin a hundred times. The scale is logarithmic — each gridline is ten times the last — because the outcomes span five orders of magnitude. Deal the room and watch two lines pull apart.
The blue line is the average — real, rising, exactly 1.05ⁿ. The red line is the median, the fortune of the person standing in the middle of the room, and it falls off a cliff. By round 100 the average player is up 131×, the typical player has lost 99.5% of everything, and only about 14% of the room is still ahead. Nobody is lying. The average is genuinely what it says. It is just that almost no one gets the average.
II · WHYTwo averages, pointing opposite ways
There are two different questions you can ask, and in this game they have opposite answers. Average across players at a fixed time — down a column of parallel worlds — and wealth grows. Average across time for one player — along a single row — and it shrinks.
The first is the ensemble average: line up a million players and take the mean of their fortunes. It grows at +5% a round, forever, because it is dragged upward by a vanishing sliver of astronomically lucky lives. The second is the time average: follow one player down the years. It is governed not by the average of the multipliers but by the average of their logarithms — ½·ln 1.5 + ½·ln 0.6 = −0.0527 — which is negative, so one life almost surely decays at −5.13% a round.
When these two averages agree, a process is called ergodic, and one lifetime is a fair sample of the whole crowd. Adding up dollars is ergodic. Multiplying them is not: the crowd's mean is a fiction no single member lives. The histogram below is those hundred-round fortunes, five thousand of them, sorted into bins. Notice where the mean has to stand.
The mean sits far out to the right, in a region where almost no one actually is — held aloft by the handful of trajectories in the far tail. That is the whole trick, drawn honestly: the average is a real number, computed correctly, describing a place the typical life never reaches. Expected value answers “what does the crowd hold?” It was never a promise about your life.
III · THE FIXDon't bet it all
The gamble was never the problem — betting everything was. Wager only a fraction f of your wealth each round and keep the rest safe, and there is one fraction that a single life grows fastest at. Drag it.
| f = 0 · keep it in your pocket | 0.00% / round |
| f = 0.25 · Kelly — a quarter of your wealth | +0.62% / round |
| f = 0.50 · double Kelly | 0.00% / round |
| f = 1.00 · all in (the game above) | −5.13% / round |
The curve tells the whole moral. Bet nothing and you grow at zero. Bet a quarter — the Kelly fraction, f* = 0.25 — and a single life now grows at +0.62% a round: the same coin that ruined the all-in player quietly enriches the careful one. Bet double the optimum, a half, and you're back to exactly zero — the growth you'd have had sitting on your hands. Bet it all, and you are the red line in the first picture. The generous coin was always generous; ruin was a choice about how much, and expected value was the wrong compass for it. Maximise the growth one life actually experiences — the time average, E[ln] — and the right fraction falls out. That rule is Bernoulli's 1738 answer to a different paradox and Kelly's 1956 answer to a wire of noisy stock tips — the same equation, discovered twice.
The check — what's recomputed, and what's honestly open
- Recomputed live, in your browser: every trajectory, histogram, and growth number above is simulated with a seeded generator as this page loads. The landmark quantities are also exact: ensemble mean 1.05ⁿ; time-average factor √0.9 = 0.94868 (−5.13%/round); break-even heads-fraction ln(1/0.6)/ln(2.5) = 55.75%; median after 100 rounds 0.9⁵⁰ = 0.515% of stake; share still ahead P(Bin(100,½) ≥ 56) = 13.56% (exact binomial tail); Kelly f* = 0.25 with growth +0.623%/round; and f = 0.5 giving exactly zero because 1.25 × 0.8 = 1.
- Proven offline: all of the above in research/ergodicity-gap/verify.mjs — 28/28, exact arithmetic plus a seeded 20,000-life Monte-Carlo cross-check.
- A free choice, named: the coin (×1.5 / ×0.6) is Peters' canonical example; the effect holds for any multiplicative gamble whose E[m] > 1 > e^E[ln m], not this one alone.
- Deliberately not estimated by simulation: the mean fortune. A sample average of a fat-tailed multiplicative variable is itself dominated by one lucky path and never settles — which is the point — so the mean is shown as the exact 1.05ⁿ, not a sim average.
- Genuinely open, not smoothed over: that the mathematics here is a century old and uncontested. What is debated is the economic reading — Ole Peters' ergodicity economics argues time-average growth should replace expected-utility theory as the foundation of choice; critics answer that this is just expected utility with a logarithmic utility function (Bernoulli again) in new clothes. This page takes no side in that dispute; it only shows the two averages, and that they part.