Why a five percent edge still goes broke

Here is a game. You stake part of your money. A fair coin decides. On heads the stake comes back multiplied by 1.5. On tails it comes back multiplied by 0.6.

Work out what a dollar staked returns on average:

0.5 × 1.5 + 0.5 × 0.6 = 1.05

Five percent, per round, forever. Every lesson about expected value says the same thing about a game like this: bet the maximum.

Bet the maximum and you go broke. A $100 bankroll that goes all in every round is worth 52 cents after 100 rounds. After 500 rounds it is worth 3.6e-10 dollars. The edge is real. The arithmetic above is right. The player is ruined anyway.

Expected value adds, and wealth multiplies

That is the whole of it, and everything below is a consequence.

The 1.05 is an average over players. It is what you get by adding up every possible outcome, weighted by how likely it is, and dividing. Addition is the right operation for that question, and the answer is correct.

But you are not adding. You are multiplying. Your bankroll after two rounds is your bankroll times one factor times another factor, and multiplication cares about a different average. Take logarithms and the multiplication becomes addition again:

0.5 × ln(1.5) + 0.5 × ln(0.6) = −0.0527

Negative. The typical path loses about 5.1% of its value every round and halves roughly every 13 rounds.

Both numbers are correct, and they disagree because they answer different questions. The mean bankroll really does climb to 131.5 times the stake after 100 rounds. It climbs because a vanishingly rare player wins nearly every flip and carries the entire average on their back. That player is not you.

Set the stake yourself

The figure below runs three strategies against the same coin flips. Betting everything, betting a quarter, and betting whatever you choose. The luck is held still on purpose, because the question is about the strategy.

Three things to try, in order:

1. Drag your stake to 100%. Your line leaves the bottom of the chart and stops when the bankroll can no longer cover the table minimum. On the seed it opens with, that happens on round 95.
2. Drag your stake to 50%. Your line wanders a long way, as high as $1,819, and then finishes on exactly $100, where it started. A game paying a five percent edge, played at half your bankroll, grows at exactly nothing, and the marker on the lower chart sits on zero. This is the most surprising thing on the page, and no amount of expected-value arithmetic prepares you for it.
3. Press Set to Kelly. Your line and the Kelly line become one line, and the marker on the lower chart sits on the top of the hump.

That lower chart is the one that settles the argument. It is not a simulation. It is the closed form for the growth rate at every stake, computed directly, so you cannot make it go away by changing the seed. The shape is the point: growth climbs to a peak, comes back to exactly zero, and falls off a cliff after that.

Now the formula

The stake at the peak has a name. It is the Kelly criterion, and for a bet paying b per dollar on a win and losing a per dollar on a loss:

f* = (p·b − q·a) / (a·b)

For the game above that is (0.5 × 0.5 − 0.5 × 0.4) / (0.4 × 0.5), which is exactly 0.25.

Every introduction to Kelly states that formula and stops there. I put it here, underneath the picture, on purpose. A formula teaches you a constant. The curve teaches you why there is a constant, which is the part you keep.

One thing the curve will save you from believing. For a fair coin, break-even sits at exactly twice the Kelly stake, and it is tempting to take that away as a rule. It is not one. Set the win chance to 60% and Kelly moves to 70%, while break-even moves to about 134%, not 140%. The identity holds for p = 0.5 and nowhere else.

Everyone who plays it

The first figure follows one player. This one follows a thousand, all playing one strategy against their own coins.

The mean line goes up. The median line goes down. Both are computed from the same thousand players, and both are correct.

That gap is the lesson, and it is enormous: after 100 rounds of betting everything, the mean player holds about $1,965 and the median player holds about 90 cents. A factor of roughly 2,181 between the average outcome and the ordinary one.

Now compare that mean against the one I quoted near the top of this page. The expectation is 131.5 times the stake, which is $13,150, and the thousand players in the figure came up with $1,965. They are short by a factor of about seven, and nothing is wrong with either number.

This is the same phenomenon eating its own tail. The expectation is carried by outcomes so rare that a thousand players usually do not contain one. More than half of that $13,150 comes from players who win 71 or more of their 100 flips, and a fair coin does that about once in 62,000 tries. A thousand players contain one less than 2% of the time. Raise the population and the mean climbs and jumps around. Lower it and the mean collapses toward the median. Try it: at forty two players it lands near $29.

So the average here is worse than useless for predicting your own result. It is hard to measure even when you run the experiment a thousand times, because the thing it is mostly made of almost never happens.

The average player in this game does not exist.

Four things worth taking away

Betting everything needs 56 wins out of 100 to break even, and expects 50. Not 51, not 52. The asymmetry between multiplying by 1.5 and multiplying by 0.6 means you have to win noticeably more often than you lose just to stand still. Only about 13.6% of all-in players finish above where they started.

A real table takes some of even that away. That 13.6% assumes you can keep betting fractions of a cent forever. Put a $1 minimum on a $100 bankroll and the share drops to about 12.8%, because ruin absorbs: a player who dips under the table minimum early is out for good, even when the coin later turns and would have carried them home. Nearly 70% of all-in players never reach round 100 at all.

(The figure above reports 12.6% rather than 12.8%, and the difference is the same one again. 12.8% is the exact probability. 12.6% is what a particular thousand players actually did.)

Half Kelly gives up 25% of the growth rate and roughly halves the volatility. That trade is why almost nobody who uses this in earnest bets the full amount.

Full Kelly is not a smooth ride. Watch the drawdown column. On the default run the Kelly player ends up at $186, having been down about 70% from their peak along the way. Over a hundred rounds, about four Kelly players in five watch their bankroll halve from a peak at some point. Over a long run, the chance of falling to half of what you started with is close to a coin flip. If that would make you change strategy partway through, you were never running this strategy.

What this is not

This is not financial advice, and the game is made up.

The most important thing the widget hides is that it knows p exactly. You do not. Every real version of this involves an estimate, and betting Kelly on an overestimated edge overshoots the peak in the direction where the curve falls away fastest. That is the strongest argument there is for betting less than Kelly, and it is one the figure above cannot make for you, because the figure always knows the answer.

Also absent, and deliberately: fees, taxes, table limits, more than one bet at a time, correlated bets, and any objective other than the growth rate.

Where it comes from

The 1.5 and the 0.6 are not invented. They are Ole Peters' coin toss from the ergodicity economics papers, and they are the right default because the numbers come out clean: a five percent edge, a Kelly stake of exactly 25%, break-even at exactly 50%, and ruin above that.

• J. L. Kelly Jr., 1956, A New Interpretation of Information Rate. It came out of Shannon's information theory rather than out of finance, which is why the original paper is about a gambler with a noisy wire rather than about a portfolio.
• Edward Thorp took it to blackjack, then to a hedge fund, and wrote the most readable account of using it in practice.
• Ole Peters on ergodicity economics, which is the modern framing of exactly this coin game and of why the two averages part company.
• Paul Samuelson objected, at length, and made his final objection in a paper written entirely in words of one syllable. It is a better joke than most papers manage.

The one sentence I would keep, if I kept one: the edge tells you whether to play, and the size tells you whether you survive.