In 2022, Jaime Vich, founder of AniSoft, wrote a small Delphi program to settle a question that had been on his mind for years. He called it Apuestas («Bets»). It tossed a coin a thousand times and compared five ways of betting: always the same amount, doubling after a loss (the famous martingale), doubling after a win… The question was simple: can you win through money management alone, in a game where there is nothing to win?
Now we have rebuilt it inside AniQuant, with twelve betting systems, millions of tosses in seconds and the maths done properly. The answer is no. The interesting part is how it fools you.
The experiment
A fair coin: 50% heads, 50% tails, and every dollar bet is either won or lost. We start with $1,000, the minimum bet is $1 and each series has 1,000 tosses. We repeat the series 100,000 times.
Every betting system plays on the same sequence of tosses in each series. If one of them gets a streak of ten tails, so do all the others. That keeps the comparison fair.
- Flat: always $1.
- Martingale: doubles after a loss (1, 2, 4, 8…) up to ten times and goes back to $1 after a win. If it loses eleven in a row, the cycle is lost and it starts again with $1.
- Paroli: the opposite; doubles after a win, up to three wins in a row.
- D'Alembert, Fibonacci, Labouchère and Oscar's Grind: the classic casino progressions.
What happened
| Betting system | Ends up ahead | Average final capital | Median | Ruined |
|---|---|---|---|---|
| Flat | 48.7% | $1,000 | $1,000 | 0% |
| Martingale | 61.2% | $997 | $1,485 | 27.5% |
| Paroli | 49.3% | $1,000 | $999 | 0% |
| D'Alembert | 65.1% | $1,001 | $1,261 | 19.6% |
| Fibonacci | 82.3% | $999 | $1,185 | 14.5% |
| Labouchère | 47.1% | $1,003 | $0 | 52.5% |
| Oscar's Grind | 83.4% | $1,001 | $1,150 | 11.5% |
Look at the average capital column: on average they all end around $1,000 (between $997 and $1,003), the same as they started with. Those small differences fit within the simulation's own margin of error: with more series, they get even closer to $1,000. Now look at «ends up ahead»: Oscar's Grind wins 83% of the time, Fibonacci 82%, the martingale 61%.
How can something that wins nothing on average win almost every time?
Shape is everything
This is how capital ends up across the 100,000 series. Flat betting is a narrow bell around $1,000. The martingale is something else: six out of ten series pile up near $1,485, with a small, steady gain, and almost three out of ten end at zero.
The martingale wins one dollar per cycle, almost always. But in almost four out of ten series, at some point a streak of ten tails in a row appears, and then it has to bet $1,024 after having already lost $1,023. Often it does not have it.
There is a theorem that guarantees this. What matters about each bet is its expected gain: what you win on average per dollar wagered. With the fair coin you win 50% of the time and lose the other 50%, so per dollar the expected gain is 0.5 × $1 − 0.5 × $1 = $0. If you decide how much to bet before seeing the result, every bet keeps that zero expected gain, whether you bet $1 or $1,000, and a sum of zeros is still zero, however you combine them. Money management does not change the average. It changes the shape.
And the shape is deceptive. A system that almost always wins a little and sometimes loses a lot looks like it works: you try it a few times, it almost always wins, and you never see the losing day coming.
The race: the one that wins most is the one that loses most
In each series we look at which betting system finishes first and which finishes last. Labouchère wins the race most often: 46% of the series. It also finishes last most often: 34%. The martingale finishes first 29% of the time and last 19% of the time.
It is not that they are better. They are the ones that swing the most. In a race of pure chance, whoever moves the most is the one who most often arrives first… and most often arrives last. Flat betting, which barely moves, finishes first only 2% of the time.
The longer you play, the more certain the ruin
The martingale does not need bad luck: only time. We repeated the experiment with shorter and longer series, always 100,000 times:
| Tosses per series | 10 | 100 | 1,000 | 5,000 | 10,000 |
|---|---|---|---|---|---|
| Ends up ahead | 84% | 94% | 61% | 35% | 25% |
| Ruined | 0% | 3.7% | 27.5% | 63% | 74.5% |
With 100 tosses, the martingale ends up ahead 94% of the time: over an afternoon at the casino it looks foolproof. With 10,000, three out of four are ruined. And the average stays around $1,000 in every case: what is won on the good afternoons is lost, all at once, on the bad ones. Each cycle is a small bet with a tiny probability of disaster; play long enough and the disaster arrives.
At the casino it is worse
At European roulette, betting on red, you win 18 times out of 37: the house has a 2.7% edge. With the same experiment and a $500 table limit:
| Betting system | Average wagered | Average loss | Loss per dollar wagered |
|---|---|---|---|
| Flat | $1,000 | $27 | 2.70% |
| Martingale | $4,485 | $123 | 2.74% |
| D'Alembert | $15,173 | $408 | 2.69% |
All three lose the same per dollar wagered: the house edge. Systems that raise the bet after a loss only manage to bet more… and lose more. D'Alembert loses fifteen times more than flat betting because it bets fifteen times more.
When there is an edge, size decides everything
Now a coin loaded in our favour: we win 55% of the time. Every dollar bet wins ten cents on average.
- Always betting $1, the median ends at $1,100. Safe and boring.
- Betting 10% of capital on each toss (the Kelly criterion, the fraction that grows the account fastest in the long run), the median reaches almost $150,000.
- Betting 25% of capital, more than is wise, the median ends at 70 cents and 68% of the series are ruined. With an edge. While being right.
A curious detail: in that last case the average final capital is in the billions, because a few lucky series shoot up. It is the best example of why the average alone is useless for deciding: the median is what will happen to most people.
With a real strategy
Coins are good for understanding the idea, but a trader does not toss coins. So we repeated the experiment with a real strategy from our portfolio, which trades the Nasdaq 100 future (NQ) on a daily chart. Its history has 721 trades and it wins 61.4% of them.
To compare like with like, we measure each result in risk units: one unit is what a losing trade loses on average, $2,612 with one contract. Measured that way, the strategy wins on average 0.263 units for every unit it risks. It has an edge, and a good one.
The Lab draws those 721 trades, with replacement, to build 100,000 different histories of the same length: the same trades, in a different order and with different streaks. We start with 100 units (about $261,000) and try several ways of deciding the size:
| Betting system | Typical final capital | Ruined | Loses more than half at some point | Worst drawdown, on average |
|---|---|---|---|---|
| Flat (always 1 unit) | 287 | 0.01% | 1.8% | 20% |
| Paroli (3 wins) | 447 | 2.4% | 20.8% | 37% |
| Martingale (10 doublings) | 501 | 19.9% | 43.9% | 52% |
| Half Kelly (1.95% of capital) | 1,560 | 0.13% | 53.2% | 52% |
| Kelly (3.9% of capital) | 4,484 | 3.6% | 99.2% | 82% |
| Double Kelly (8% of capital) | 0 | 82.9% | 100% | 99.5% |
- With an edge, the martingale is still Russian roulette. Its typical capital beats flat betting (501 versus 287), but one history in five ends in ruin. The strategy is the same; the only change is doubling after a loss.
- Kelly grows the account the most: in the typical history it multiplies capital by 45. But in 99% of the histories it loses, at some point, more than half from its peak, and its worst drawdown averages 82%. It is optimal on paper; in practice, almost nobody sits through that without touching anything.
- Half Kelly gives up two thirds of that growth (1,560 versus 4,484) and, in exchange, ruin drops to 0.13% and the average worst drawdown stays at 52%. That is why many managers bet a fraction of Kelly rather than full Kelly.
- And double Kelly destroys everything. Risking 8% of capital on each trade, the typical history ends at zero and 83% of them are ruined. Its average capital, on the other hand, is 78 million units: a few lucky histories shoot up and drag the average with them. It is the same trap as the loaded coin.
One more detail pushes towards half Kelly: a strategy's edge is never known exactly. If the real one were half the measured one, betting «Kelly» would really be betting double Kelly: the last row of the table.
A warning: drawing the trades at random assumes the future will look like the past and breaks the real streaks. It is useful for comparing ways of deciding size, not for predicting what the strategy will earn.
What it has to do with trading
A lot. Averaging down a losing position, doubling size to «win it back», raising risk after a bad streak: they are martingales by another name. They work almost every time, and that is precisely why they are so dangerous.
And scaling them with the account does not save them. If the unit is 1% of capital, that is, «I risk 1% and, if I lose, I double», the 10-doubling martingale is ruined in 98% of the series. Ten doublings of 1% means betting ten times the account: the bad streak arrives almost for sure. The average, once again, stays around $1,000.
The lesson is the same for a trader as for a casino player: size does not create the edge, but it can multiply or destroy it. First you need a strategy with a real edge, validated out of sample. Then, a size that lets it survive its worst streaks.
What we built in AniQuant
The whole experiment is in AniQuant 1.2.0, under the Capital & Risk menu: the Chance Lab. You choose the game (a coin, roulette or your own) and twelve betting systems, plus one with your own rules, and run up to a million series in seconds. You see the race, the shape of each result, how capital evolves and one series in slow motion. You can repeat the experiment as many times as you like, make it longer or shorter, and choose whether the progressions use a fixed unit or a % of capital.
And the most useful part, what we did above with the NQ one: instead of the coin you can use the trades of one of your strategies. That way you see what each betting system does with your real edge, and how much you can risk without a bad streak knocking you out of the game.
What does each betting system do with your real edge?
AniQuant's Chance Lab compares twelve betting systems on a coin, a roulette wheel or the trades of your own strategies. Try it free for 14 days.
Download AniQuantSoftware company based in Palma de Mallorca since 1995, makers of AniQuant. We write about honest backtesting, strategy validation and the mistakes that cost money.