A betting system, also called a progression, is a rule that sets the size of the next stake according to previous results. The system does not change the game: the same bets, with the same probabilities and the same payouts. It changes only how much is staked and when. This article describes the best-known systems and shows why none of them can change the average loss per unit staked.
Martingale
Martingale is the system of doubling after a loss, usually applied to bets paid 1 to 1, such as red or black in roulette. The player starts with a base stake; after each loss they double the stake, and after the first win they go back to the base stake. The stakes in a series are therefore 1, 2, 4, 8 and so on. When the series ends with a win, that win covers all the earlier losses and leaves net winnings equal to the base stake.
The impression is of a small win obtained almost every time. The problem lies in long runs of losses, rare but unavoidable in the long run, in which the required stake grows exponentially. The table shows what happens with a base stake of one unit on an even-money bet in European roulette:
| Losses in a row | Probability | Lost so far | Next stake |
|---|---|---|---|
| 3 | 13.54% | 7 | 8 |
| 5 | 3.57% | 31 | 32 |
| 7 | 0.94% | 127 | 128 |
| 10 | 0.13% | 1,023 | 1,024 |
"Probability" is the chance that a series starts with that many losses in a row; "Lost so far" is the amount lost at that point, and "Next stake" is the stake that would be needed to continue the series.
If the table has, for example, a limit of 1,000 units per bet, after ten losses in a row the next stake is no longer allowed, and the series ends with the loss shown in the last row of the table. A single series gets there with a probability of 0.13%, but over 100 series the chance that this happens at least once is 11.98%. Such a failed series wipes out the winnings of as many successful series as the units shown in the "Lost so far" column, since each successful series brings in a single unit.
The long-run result is a mix of many small wins and rare but large losses. Their average, relative to everything staked, is still the house edge.
Reverse Martingale
The reverse Martingale (also known as Paroli) does the opposite: it doubles the stake after each win and goes back to the base stake after a loss or after a number of consecutive wins fixed in advance. The profile of results is reversed: many small losses, one base stake each, and larger but rare wins when a run of wins comes along. The house edge applies to every stake, so the average stays the same.
D'Alembert
The D'Alembert system raises the stake by one unit after each loss and lowers it by one unit after each win. It starts from the idea that, in a game of chance, wins and losses tend to balance out, so a loss would make the next win more likely. The idea is false: results are independent, and balancing does not happen by compensation, as the article on the gambler's fallacy explains. Stakes grow more slowly than with Martingale, but they still grow during unfavourable spells, that is, precisely when money has already been lost.
Fibonacci
The Fibonacci system uses the sequence in which each term is the sum of the two before it: 1, 1, 2, 3, 5, 8, 13, 21 and so on. After a loss, the stake moves one term forward in the sequence; after a win, it moves two terms back. Growth is slower than with doubling but still exponential, and a long run of losses still leads to large stakes and a large cumulative loss.
Why no system changes the average result
Each bet has its own house edge, and it does not depend on what happened before. The stake of a bet is set before the spin, based on earlier results, and the spin does not depend on them. Therefore the average result of each bet is its stake multiplied by the average loss per unit, that is, 2.70% of the stake in European roulette, however the stake was chosen.
The average result of a session is the sum of the average results of its bets. However the stakes are chosen, that sum is the house edge multiplied by the total amount staked. A system can change only two things:
- the shape of the distribution of results: many small wins and rare but large losses (Martingale, D'Alembert, Fibonacci), or many small losses and rare but larger wins (reverse Martingale);
- the total amount staked, which with rising-stake systems is usually larger than with flat stakes, so it increases the average loss in money rather than reducing it.
The ratio between the average loss and the amount staked remains the house edge. In probability theory, this result is formalised by the optional stopping theorem: in a sequence of bets with a negative average, no rule for choosing stakes and the moment to stop, with finite resources, can produce a positive average. Incidentally, the system's name was taken over by mathematics, where a "martingale" is the model of a fair game.
Table limits and bankroll
On paper, Martingale would work only with unlimited money, unlimited time and no stake limit. In reality all three are finite:
- The table limit. Casinos set a maximum stake for each table, which stops the progression after a number of losses. The value of the limit differs from one table to another.
- The player's bankroll. The money available has the same effect: after a few losses in a row, the next stake exceeds the money left.
- Time. The more series are played, the more likely the long run of losses becomes, as the figure above for 100 series shows.
The limits do not create the house edge, which exists on every bet; they determine how it shows up when a system is used: as one large, rare loss instead of many small ones. The long-run mechanism is explained in Variance and the law of large numbers.
See also
- House edge and RTP
- Probability basics
- House edge and variance in the glossary