The Coin Flip That Ruins You

Back to the rigged coin. It comes up heads 60% of the time and pays even money: stake a fraction of your bankroll, win that much on heads, lose it on tails. You may play as long as you like. This is a good bet — genuinely, provably in your favour — and I am going to show you that the “obvious” way to exploit it, the way that literally maximises your expected wealth, will bankrupt almost everyone who tries it. Not through bad luck. By design.

That sentence should feel wrong. A favourable bet and a wealth-maximising strategy that ruins you? Surely if a plan grows your average fortune, it grows your fortune? It does not, and the reason is one of the most important and least intuitive ideas in all of risk-taking. The average of many gamblers is not the experience of one gambler over time. This lesson takes that idea apart on our coin, and the repair — maximising log wealth — is what the rest of the course is built on.

The bet that’s brilliant on paper

Let’s make the edge concrete. Stake a fraction $f$ of your bankroll on each flip. Win (probability $p = 0.6$) and your wealth is multiplied by $1 + f$; lose (probability $q = 0.4$) and it’s multiplied by $1 - f$. The expected multiplier for one round is:

$E[\text{factor}] = p(1 + f) + q(1 - f) = 1 + f(p - q) = 1 + 0.2f.$

Read that formula. It is a straight line that rises with $f$. The more you bet, the higher your expected wealth after the round — with no peak, no catch, no diminishing returns in sight. Push $f$ all the way to 1 (bet the farm) and you maximise it: your expected wealth grows a fat 20% per round. By the logic of expected value — the same logic that correctly tells you to take a +EV bet in the first place — you should bet everything, every time.

Everything about this table says bet more. And every cell of it is true. The expected wealth really is maximised at $f = 1$. The catch is not in the arithmetic — it’s in the quiet assumption that “expected wealth” is the thing you should care about. Watch it fall apart.

Why betting everything is a death sentence

Follow a single player who bets everything. Round one, they flip heads: bankroll doubles. Round two, heads again: doubles again. This can go on for a thrilling while. But wealth here is multiplicative — each round multiplies the last — and the losing factor is $1 - f = 1 - 1 = \mathbf{0}$. The very first time this player sees tails, their bankroll is multiplied by zero. Not dented. Zeroed. And here is the cruelty of compounding: zero times anything is zero, forever. No future winning streak, however long, can lift a bankroll of $0 back above $0. There is no road back from ruin.

Now ask how likely that fatal tail is. The chance of never flipping tails in $n$ rounds is $0.6^n$ — which collapses fast:

Play long enough and survival probability races to zero. The strategy that maximises your average wealth ruins you with probability approaching 1. Both statements are true at once, and they are not in contradiction — they are just measuring different things. The average is propped up entirely by the vanishing sliver of players who get impossibly lucky; the typical player — the median one, the one in the middle of the pack — is flat broke.

Two averages that disagree: ensemble vs time

We’ve been sloppily saying “average.” There are in fact two averages, and the whole lesson is that on a multiplicative bet they disagree.

• The ensemble average (arithmetic mean) is what you get by lining up a huge crowd of players who each bet once, and averaging their wealth across the crowd. This is $E[\text{wealth}]$, and for our coin at $f = 1$ it grows +20%/round. It answers: if I could clone myself into a million parallel bettors, what’s my average outcome?
• The time-average (geometric growth) is what a single player experiences down one sequence of flips, over time. Because the rounds multiply, the right way to average them is on a log scale: the long-run growth rate per round is

$g(f) = p\ln(1 + f) + q\ln(1 - f).$

At $f = 1$ that second term is $q\ln(0) = -\infty$: the time-average growth is negative infinity — a mathematician’s way of writing “certain ruin.” The two averages don’t just differ in size; they point in opposite directions. Ensemble says soar; time says sink.

A process where these two averages agree is called ergodic; one where they part ways — like multiplicative wealth — is non-ergodic. Almost every bet you will ever size in real life is non-ergodic, which is exactly why “maximise expected wealth” is such a treacherous default.

The single most useful move you can make with this sandbox: drag the bet-fraction slider downward and watch the typical line lift off the floor. You are not changing the coin — the edge is the same — you are changing how much of it you expose to ruin each round. Below a certain fraction the time-average growth flips from negative to positive, and the typical player finally compounds upward. Finding that fraction is the Kelly problem.

Seeing the gap directly

Put the coin back to exactly 60/40 at even money and watch three staking fractions race: half-Kelly, full Kelly, and a deliberate over-bet. Run it on the Typical player (median) view first, then flip to Ensemble average (mean) and watch the over-bet line’s story completely change.

This is the ergodicity gap made visible. On the median view, the over-bet strategy is a catastrophe. Flip to the mean and it looks like the best idea you’ve ever had — because the mean is a story told by survivors. When someone quotes you the expected return of an aggressive strategy, they are quoting the ensemble average, the number that describes the lucky few and almost no one else. You are not the ensemble. You are one path. Kelly sizing maximises the number that describes your path: the time-average, equivalently the median long-run wealth.

The fix, in one word: logarithms

Here’s the escape, set up now and derived in full next lesson. The trouble with $E[\text{wealth}]$ is that it averages wealth on a linear scale, where a doubling and a halving don’t cancel (up $100 then down $100 from $200 leaves you at $100 — but the multiplicative round-trip ×2 then ×0.5 lands you right back where you started). The honest scale for a compounding bankroll is the logarithm, because logs turn multiplication into addition, so the growth of a single path is the sum of per-round log-factors — a thing the law of large numbers actually governs.

So the winning objective is not “maximise expected wealth” but maximise expected log wealth, $E[\ln W]$ — which is exactly the time-average growth rate $g(f)$ we wrote above. Maximise that, and you maximise the growth of the typical path and the median long-run wealth. Do the calculus on $g(f)$ and out drops a single optimal fraction: the Kelly fraction $f^* = p - \dfrac{q}{b}$, which for our coin is $0.6 - 0.4 = 0.20$. Twenty percent — not one hundred, not fifty. That derivation is Lesson 2.