Martingale and other roulette systems

Martingale — doubling your bet after every loss — wins small amounts often and loses big amounts rarely. In our simulation of 1,000,000 sessions it reached a +1,000 target 86.7% of the time, and the other 13.3% lost about 8,400 each. Overall it lost 2.69% of everything staked, the same as flat betting. No system changes the 2.70% edge.

Bet one unit on an even chance — red, say. Win, and you start again. Lose, and you double: 100, 200, 400, 800… The first win of a series pays back every loss before it plus one unit, because each bet is bigger than all the previous ones put together. Red has to come up sooner or later, the reasoning goes, and when it does you are always 100 ahead.

For every series that ends in a win, that is true. The reasoning leaves out two things: you can run out of money, and the table will not take a bet above its limit.

At every table in the Rotunda of Shai the limit on an even chance is 50 times the smallest chip: 5,000 at the Scribes’ Table (smallest bet 100), 50,000 at the Priests’, 500,000 at the Pharaoh’s. Starting from the smallest chip you can double six times — 100, 200, 400, 800, 1,600, 3,200 — and the seventh bet, 6,400, is refused.

So a Martingale series at our table wins 100 unless red misses six times running. The chance of six losses in a row is (19/37)⁶ = 1.83%, about once in 55 series. An hour of play at our table is around 53 series, which puts the chance of at least one break at 63% per hour and 95% over three hours.

On average a series puts 642 debens on the table and loses 17.35 of them — exactly 2.70%.

To see the whole picture we simulated sessions with settings a real player might pick at our cheapest table: a bank of 10,000 debens (a tenth of what a new player starts with), a base bet of 100 on red, the 5,000 limit, and a target of +1,000 — the player gets up as soon as the bank reaches 11,000. If the system asks for a bet the table refuses, the series is abandoned and the player starts the system over; if it asks for more than is left in the bank, the session is over. Sessions are capped at 1,000 spins, about nine hours at our table. Each system played 1,000,000 sessions, and every spin’s number was drawn uniformly from 37, as on our server.

Martingale’s results fall into two clusters. Most sessions — 86.7% — end at exactly +1,000, and a Martingale session lasts only 30 spins on average, about 17 minutes. The rest end between −7,000 and −10,000, at −8,423 on average. Flat betting of 100 a spin reaches the target only 53.1% of the time, but its losses are spread out and rarely total.

Averaged over all sessions, Martingale lost 251 per session and flat betting 1,552. That does not make Martingale cheaper: it simply puts less money through the wheel — 9,332 per session against 57,427 — and both lose the same share of it, 2.69% and 2.70%. And one failure costs as much as eight successful sessions win: out of ten such evenings, at least one breaks with a probability of 76%.

Each popular system is just a different rule for the size of the next bet:

• Reverse Martingale (Paroli) doubles after a win instead of a loss and banks the profit after three wins in a row. Losses are small; wins come in rare bursts.
• D’Alembert adds one unit after a loss and takes one away after a win — a gentler Martingale.
• Fibonacci moves one step up the sequence 1, 1, 2, 3, 5, 8, 13… after a loss and two steps back after a win.
• Labouchère writes down a line of numbers — we used 1-2-3-4, which adds up to the 1,000 target — and bets the sum of the two ends. A win crosses them out, a loss adds the lost amount to the end of the line.

The cards tell the same story six times. Aggressive systems hit the target more often and lose harder when they fail; slow ones keep you at the table longer. The line that matters is the one about every 100 staked: each system lost between 2.66 and 2.71. The theory says 2.70; the rest is the noise of a million random sessions.

Every bet at our table, whatever its size, loses 1/37 of its stake on average. A system only decides how big the next bet is, and it decides from what has already happened. The wheel knows none of it: each number is drawn independently with the same 1 in 37 chance, however many reds came before. Add up bets that each lose 2.70% on average and the total loses 2.70% of everything staked, whatever order and sizes you chose.

That is also why “red is due” does not work. Ten reds in a row happen about once in 1,350 tries; the spin after them is still red with a chance of 48.65%.

Systems change the shape of the results, not their average. Martingale turns an evening into many small wins and a rare heavy loss. Paroli does the opposite. D’Alembert and flat betting sit somewhere in between. Knowing that, you can choose the shape you enjoy — and decide in advance how much you are prepared to lose, since that, unlike the outcome, is up to you.

In our game it costs nothing to try: chips are free debens and cannot be cashed out. Play a system for real money and the maths is exactly the same, only the losses are real. If gambling stops being fun, see our responsible gaming page.