One Average Says Take the Bet. The Other Says It Will Ruin You. Both Numbers Are Correct.

A COIN FLIP WITH A +5% EXPECTED RETURN TURNS $100 INTO 52 CENTS. THE ERROR IS 300 YEARS OLD AND IT IS STILL IN EVERY FINANCE TEXTBOOK.

In 1713, a Swiss mathematician named Nicolas Bernoulli sent a letter to a French correspondent describing a game whose expected value was infinite and which nobody in their right mind would pay much to play. The problem sat unresolved for twenty-five years. In 1738, his cousin Daniel Bernoulli published a solution in the Commentarii of the Imperial Academy of Sciences in Petersburg - a journal read by perhaps a few hundred people in Europe - and that solution became the foundation of every model of financial decision-making taught since.

Daniel's answer was that people do not value money linearly. A second million matters less than the first. So instead of maximising expected wealth, we maximise expected utility of wealth, and the utility function bends. The whole apparatus of modern risk analysis - risk aversion, utility curves, the assumption that a nervous investor is expressing a psychological preference - descends from that paper.

In December 2019, a physicist at the London Mathematical Laboratory published a Perspective in Nature Physics arguing that Bernoulli got the right answer for the wrong reason, and that the wrong reason has been quietly breaking financial reasoning for three centuries.

Ole Peters' claim is not that people are irrational. It is that the mathematics they were handed answers a question nobody is actually asking. The prevailing formulations of economic theory make an indiscriminate assumption called ergodicity, and the assumption is false for almost every financial decision you will ever make.

His demonstration takes one coin flip and thirty seconds.

Here is the coin flip, the two averages, the code, the trap, and the reason your instinct to refuse a good-looking bet was mathematically correct all along.

Layer 1 - The Averaging Illusion

You do not have a risk tolerance problem. You have an averaging problem.

Here is the game. You start with $100. A fair coin is flipped. Heads, your wealth increases by 50%. Tails, your wealth decreases by 40%. You play 100 rounds.

Run the standard analysis, the one every finance course teaches. Compute the expected value of one round:

$$0.5 × 1.50 + 0.5 × 0.60 = 1.05$$

A 5% expected gain per round. Positive expected value. In classical terms, this bet is equivalent to a guaranteed 5% return, and refusing it means leaving free money on the table.

Compound that expectation over 100 rounds:

$100 × 1.05^100 = $13,150

The textbook says: play. Enthusiastically.

Now run the other calculation. Over 100 flips of a fair coin, the most likely outcome is roughly 50 heads and 50 tails. Multiply the actual sequence:

$100 × 1.50^50 × 0.60^50 = $100 × 0.9^50 = $0.52

Fifty-two cents.

The expected value says $13,150. The typical path says you lose 99.5% of your money.

Both numbers are computed correctly. Neither is a trick. They are answers to two different questions, and almost nobody is told there are two questions.

Layer 2 - The Proof That Two Averages Exist

Peters showed this is not a paradox. It is two distinct operations that were quietly assumed to be one.

The $13,150 figure is an ensemble average. It answers: if a very large number of people each played this game once, what would the average outcome be across all of them at that moment?

The $0.52 figure is a time average. It answers: if one person played this game repeatedly, what happens to that person's wealth over time?

Ensemble average ⟨x⟩ = Σ pᵢ · xᵢ average across parallel worlds Time average g = ⟨ln(x_{t+1} / xₜ)⟩ growth rate along one trajectory

Ergodicity is the property that these two are the same number. Where a system is ergodic, you may compute whichever is convenient and use it for the other. Time drops out of the model entirely, which is enormously convenient and is precisely why economics adopted it.

The term comes from Ludwig Boltzmann's thermodynamics in the 1870s. The economic concepts that assume it were built in the seventeenth and eighteenth centuries - roughly two hundred years before anyone had the word. The assumption was not made carelessly. It was made before there was language to notice it was being made.

For this coin flip, run the time-average growth rate:

$$g = √(1.5 × 0.6) − 1 = √0.9 − 1 = −5.1% per round$$

Positive expected value. Negative growth rate. Simultaneously, from the same game, both exactly true.

Where does the $13,150 come from, if the typical player is left with pocket change? From the tail. A vanishingly small number of paths - the ones that flip heads far more often than tails - end up with astronomical sums, and those few paths carry the entire ensemble average. The mean is real. It just does not live anywhere near where anyone actually ends up.

The mechanism is that wealth compounds multiplicatively. Each outcome is a multiplication, not an addition. Under multiplication, the order does not matter but the arithmetic mean stops describing anything anyone experiences. A 50% gain and a 40% loss do not average to +5% along a path. They compound to −5.1%.

You cannot spend an ensemble average. You only ever get one trajectory.

Layer 3 - The Code

Twenty lines. Run it and watch the two numbers separate.

import numpy as np

def coin_game(n_people=10_000, n_rounds=100, seed=0):
    """
    +50% on heads, -40% on tails. Fair coin. Start at $100.
    Ensemble average vs. what actually happens to a person.
    """
    rng = np.random.default_rng(seed)
    flips = rng.random((n_people, n_rounds)) < 0.5
    multipliers = np.where(flips, 1.5, 0.6)
    paths = 100 * np.cumprod(multipliers, axis=1)
    return paths

paths = coin_game()
final = paths[:, -1]

print(f"Expected value (theory):  ${100 * 1.05**100:,.0f}")
print(f"Ensemble mean (simulated): ${final.mean():,.0f}")
print(f"Median person:             ${np.median(final):,.2f}")
print(f"Share of people below $100: {(final < 100).mean():.1%}")

# Time-average growth rate, per round:
g = np.exp(np.mean(np.log(final / 100)) / 100) - 1
print(f"Time-average growth:       {g:.2%} per round")

Run it and the structure is immediately visible. The mean is dragged into the tens of thousands by a handful of paths. The median person is close to broke. The overwhelming majority finish below where they started. The growth rate along the typical path is roughly −5% per round, exactly as the closed form predicts.

Increase n_people and the simulated mean drifts upward, not toward stability - because the mean is controlled by rarer and rarer outliers, so more samples find bigger ones. That is the signature of a non-ergodic system, and it is the thing to look for. If your sample mean keeps climbing as you add data instead of settling, you are averaging something you should not be averaging.

The comparison, in one table:

Layer 4 - The Trap Nobody Talks About

The dangerous move is not refusing this bet. It is applying the correction everywhere.

Having seen the coin flip, the temptation is to conclude that expected value is broken and geometric growth is the truth. That overcorrection is its own error, and it is worth being precise about when each applies.

The time-average argument requires three conditions, all of them structural:

The dynamics must be multiplicative. Your outcome must be a percentage of your current position. Buy a house at 5x leverage and the dynamics are multiplicative. Bet $50 on a coin flip out of a $2 million net worth and they are effectively additive. Under additive dynamics the two averages converge and the classical analysis is correct.

There must be no reset. The argument assumes losses persist and compound. If you have outside income, an employer, a floor beneath you, part of every drawdown is repaired from outside the process, and the pure multiplicative model overstates the damage.

The sequence must be long enough to matter. For a single one-off bet, the ensemble average is a defensible guide, because there is no trajectory to speak of yet. The divergence grows with repetition.

Here is what makes this genuinely hard. A trader who does not know about time averages will overbet a positive-EV strategy into ruin. A trader who learns about time averages and applies them indiscriminately will refuse every risk with variance, including risks that are structurally additive for them, and underperform for thirty years while feeling mathematically rigorous.

The failure modes are symmetric and the second one is invisible, because nothing dramatic happens to the person committing it.

The correct question is never "is the expected value positive?" nor "is the growth rate positive?" It is:

> Is this outcome a percentage of everything I have, and will I face it repeatedly?

Yes to both: use the growth rate. No to either: the classical analysis is fine.

Layer 5 - The Logarithm Hiding Inside Three Centuries of Finance

Bernoulli got the right formula in 1738. He attributed it to psychology. It was arithmetic.

Look again at the time-average growth rate:

$$g = ⟨ln(x_{t+1} / xₜ)⟩$$

To find how wealth grows along a path, you take the logarithm. Now look at Daniel Bernoulli's 1738 solution to his cousin's paradox: maximise not wealth but the logarithm of wealth, because each additional unit is worth less than the last.

Same function. Radically different justification.

Bernoulli said the logarithm belongs there because of how people feel about money - diminishing marginal utility, a psychological fact about human beings. Peters' argument is that the logarithm belongs there because of how multiplicative wealth actually behaves over time - a mathematical fact about the process, present whether anyone is feeling anything or not.

The distinction sounds academic. It is not, and here is where it bites.

If the logarithm is psychological, then "risk aversion" is a preference. Preferences vary between people, can be measured with surveys, and can in principle be overcome by education or discipline. Whole literatures exist on correcting for it. The nervous investor is making an emotional error to be managed.

If the logarithm is dynamic, then declining a negative-growth bet is not a preference at all. It is arithmetic. The investor refusing the coin flip is not being timid. They are being correct, and the model calling them irrational is the thing that is broken.

That reframe pulls the rug out from under a large amount of standard financial advice, most of which is built on the premise that the client's caution is a bias requiring correction rather than a signal requiring respect.

It also explains something odd about the accepted formula for compound returns. The finance industry's own honest performance metric - CAGR, the geometric mean, the number in every prospectus footnote - is the time average. It has been sitting in the disclosures the entire time, computed correctly, while the arithmetic mean sits in the marketing. Nobody hid it. It was simply never explained why the two differ, or which one you actually receive.

You receive the geometric one. You are shown the arithmetic one. The gap between them is not fraud. It is the ergodicity assumption, quietly doing its work.

Layer 6 - The Reality Layer

The framework is contested, and the honest version says so.

This is not settled science and it would be dishonest to present it as such.

The 2019 Nature Physics piece drew published pushback, including a critique by Doctor and colleagues in Nature Physics itself in 2020, and a longer 2023 critique by Ford and Kay in Econ Journal Watch. The substance of the objection is roughly this: expected utility theory does not actually assume ergodicity in the way the argument claims; economists have understood multiplicative dynamics and log wealth since at least Kelly and Latané in the 1950s; and framing a well-known result as a discovery overstates the case.

There is force in that. The mathematics of the coin flip is not in dispute by anyone - it is a straightforward calculation that has been reproducible for centuries. What is disputed is how much of standard economics it actually invalidates.

Three further limits worth carrying:

Pooling changes the dynamics. If a group agrees to share outcomes, each participant's exposure becomes closer to the ensemble average, and a gamble that destroys an individual can be positive for the group. Proponents of the framework argue this is the real mathematical basis for insurance and cooperation that a contract can raise the growth rate of both parties even where it looks like a transfer. That is an appealing account and it is contested; treat it as a live argument, not a settled result.

Real returns are not independent coin flips. They are autocorrelated, fat-tailed, and regime-dependent. The coin flip is a teaching device with the clarity of a laboratory, not a model of any actual market.

Sizing is a separate problem with a separate literature. The question "how much of my capital should I risk?" was answered in 1956 and has its own seventy-year body of work. Knowing that growth rates and expected values diverge tells you the direction; it does not tell you the fraction.

None of this dissolves the coin flip. A bet with a +5% expected return and a −5.1% growth rate is exactly what it appears to be, and any framework that calls refusing it irrational has a problem to answer for.

The Stack

Six layers. Two averages. Public since 1738 and re-argued since 2019.

1. A positive expected value can be a negative growth rate. $13,150 and $0.52 from the same game. - the coin flip
2. There are two averages, not one. Ensemble across people, time along a path. - Peters, Nature Physics, 2019
3. You can simulate the divergence in twenty lines. Watch the mean climb as the median falls. - implementation
4. The correction has boundaries. Multiplicative, non-resetting, repeated. Otherwise classical holds. - the symmetric failure mode
5. The logarithm was always there. Bernoulli called it psychology; the dynamics say arithmetic. - Bernoulli, 1738
6. The framework is contested. The math is not. - Doctor et al. 2020; Ford & Kay 2023

The Close

The reason this matters is not that expected value is wrong. It is that expected value answers a question about a population, and you are not a population.

Every financial decision you will ever make happens on exactly one trajectory. You do not get to average across the versions of yourself who called it right. You get the sequence you actually lived, in the order it actually happened, and multiplication does not forgive.

Three centuries ago a mathematician noticed that people refuse bets the arithmetic says they should take, and concluded that something must be wrong with the people. The alternative explanation - that something was wrong with the arithmetic being applied - took until 2019 to be argued properly in a physics journal, and is still being fought over.

Meanwhile the coin flip has been sitting there the whole time, taking $100 to fifty-two cents, at a positive expected return, in full view.

You do not need a higher expected value. You need to know which average you are going to be paid in.