The Martingale Betting System and Other Progressions: D’Alembert, Fibonacci, Labouchère

The Martingale betting system raises the stake after every loss so that the first win recovers all the losses and brings a small profit. At odds of 2.00 that means doubling: 10, 20, 40, 80… The system has relatives — D’Alembert, Fibonacci, Labouchère, Oscar’s Grind and Paroli — that change the stake by other rules. None of them changes the main thing: if a bet has a negative expected value, a sequence of such bets has one too.

One thing up front: betting carries the risk of losing money, and progressions increase that risk. You can test any system on your own numbers in the calculator below; it is also available on its own page — the Martingale calculator.

How the chasing-losses strategy works

Chasing losses is the everyday name for Martingale. The idea seems flawless: sooner or later a bet will win, and since each stake covers all previous losses, the cycle always ends in profit.

Take a bank of 1,000, a base stake of 10 and odds of 2.00. A losing streak looks like this:

After six losses in a row the next stake has to be 640, and only 370 is left. A cycle meant to bring 10 ends with a loss of 630. A bank of a hundred base stakes survives just six losses — the stake grows exponentially.

How likely is such a streak? At odds of 2.00 the real chance of winning is usually below 50% because of the bookmaker margin; take 48%. Then at least one streak of six losses in a row within 200 bets happens in about 86% of cases. More on how often long streaks happen is in the article on losing streaks.

There is a second wall too — limits. The bookmaker caps the maximum stake, and a progression can hit the limit before it hits the bank.

Why a progression creates no edge

Every bet is independent of the previous ones: losing yesterday does not make winning today more likely. The expected value of a bet is the stake multiplied by (probability × odds − 1). At a probability of 48% and odds of 2.00 that is −4% of any amount. Whatever you stake and in whatever order, on average you lose 4% of your total turnover.

What a progression really changes is the shape of the result. Martingale turns the distribution of outcomes into "many small wins and a rare huge loss". Most of the time the system seems to work, and that is exactly why it is so convincing.

Here is what the calculator's simulation shows — 2,000 runs of 200 bets, a bank of 1,000, a base stake of 10, odds of 2.00 and a win probability of 48%:

"Bank too small" is the share of runs in which the bank could not cover the system's next stake; "average result" is the average final bank out of the starting 1,000. The median says even more: with flat staking half of the runs kept at least 920, with Martingale at least 520, and with D’Alembert just 130.

The table shows the trap well. Fibonacci and Oscar’s Grind finish in profit more often than flat staking — more than half of the runs against a quarter. But their average final bank is lower than flat: the rare collapses take more than the frequent small wins bring. Every system averages below the starting 1,000, because the expected value of each single bet is below zero.

Other progressions: D’Alembert, Fibonacci, Labouchère

The D’Alembert system. After a loss the stake grows by one base stake, after a win it falls by one, but never below the base. The growth is linear, so the bank survives long streaks: 13 losses in a row in our example. But after a run of losses the stake stays high for a long time, and the losses build up slowly but surely.

The Fibonacci betting system. Stakes follow the sequence 1, 1, 2, 3, 5, 8, 13… base stakes. After a loss, one step forward; after a win, two steps back. The growth is faster than linear but slower than doubling: the bank in the example survives nine losses in a row.

The Labouchère system. The player writes down a line, for example 1-2-3, and stakes the sum of the outer numbers. A win crosses out the outer numbers, a loss adds the stake to the end of the line. When the line is empty the cycle is over; at odds of 2.00 its profit equals the sum of the original line. The very first stake is four base stakes, and the line grows after losses, so in the simulation Labouchère turned out to be the most destructive.

Positive progressions: Oscar’s Grind and Paroli

Oscar’s Grind. After a loss the stake stays the same; after a win it grows by one base stake — but never more than needed to finish the cycle one base stake up. It is a gentle progression: a single long streak does not wipe out the bank, but the negative expected value does not go away.

The Paroli strategy. A reverse Martingale: the stake grows after a win — you bet the previous stake together with the winnings — and after three wins in a row or after a loss you go back to the base stake. A loss costs one base stake, and a run of wins brings a large plus. In the simulation the bank never ran out, but the average result is still below the start: Paroli does not change the expected value either.

What to use instead of a progression

If the goal is to survive losing streaks rather than win them back, you need a small stake, not a progression: a fixed share of the bank of 1–2% or a percentage of the current bank. These approaches are described in the article on bankroll management. And if you are looking for bets with positive expected value, start with the probability estimate — for example, by comparing odds with the estimated probabilities in football predictions.

The short version

The Martingale betting system and other progressions create no edge: they only change how wins and losses are distributed. Chasing losses brings frequent small wins at the cost of rare but devastating losses; with a bank of a hundred base stakes, six losses in a row are enough to break the system. Gentle progressions make the bank last longer, but with a negative expected value the average result of any of them is below the starting bank.