The D’Alembert is a staking progression for even-money bets: increase the stake by one unit after a loss and decrease it by one unit after a win. This changes exposure, not the probabilities of a fair spin.
The D’Alembert System raises stakes more slowly than doubling, but extended losses can still consume a bankroll or exceed a table limit.
The Betting Sequence
The system features a reasonably comprehensive betting sequence. A player needs to add one unit to his/her next bet after every loss and to subtract one unit from his/her next bet after every win. Let us see what a common sequence of bets looks like if we take into account an initial bet of 4 units.
| Bet Number | Bet Size (Units) | Outcome | Net Gain |
|---|---|---|---|
| 1 | 4 | loss | -4 |
| 2 | 5 | win | 1 |
| 3 | 4 | loss | -3 |
| 4 | 5 | win | 2 |
| 5 | 4 | win | 6 |
| 6 | 3 | loss | 3 |
| 7 | 4 | loss | -1 |
| 8 | 5 | win | 4 |
| 9 | 4 | loss | 0 |
| 10 | 5 | loss | -5 |
| 11 | 6 | win | 1 |
| 12 | 5 | win | 6 |
It appears that the first 4-unit bet lost, thus, the net gain would be (0 – 4 = -4). According to what the system postulates, the next bet needs to be 1 unit larger than the first, or 5 units. It won and the net gain improved to +1 (-4 + 5 = +1). In that case, the next bet needs to be 1 unit smaller than the second bet, or 4 units, and so on.
Let us see another example, which features longer strings of gains and losses. Again, the initial bet is 4 units.
| Bet Number | Bet Size (Units) | Outcome | Net Gain |
|---|---|---|---|
| 1 | 4 | loss | -4 |
| 2 | 5 | loss | -9 |
| 3 | 6 | win | -3 |
| 4 | 5 | win | 2 |
| 5 | 4 | win | 6 |
| 6 | 3 | win | 9 |
| 7 | 2 | win | 11 |
| 8 | 1 | loss | 10 |
| 9 | 2 | loss | 8 |
| 10 | 3 | loss | 5 |
| 11 | 4 | loss | 1 |
| 12 | 5 | win | 6 |
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Both the table maximum and minimum matter. If subtracting a unit produces an invalid stake, the player must stop or apply a predefined floor, changing the progression’s arithmetic. Starting with a larger stake increases exposure; it does not eliminate limits. The next illustration starts at ten units.
| Bet Number | Bet Size (Units) | Outcome | Net Gain |
|---|---|---|---|
| 1 | 10 | win | 10 |
| 2 | 9 | win | 19 |
| 3 | 8 | lose | 11 |
| 4 | 9 | lose | 2 |
| 5 | 10 | lose | -8 |
| 6 | 11 | win | 3 |
| 7 | 10 | lose | -7 |
| 8 | 11 | win | 4 |
| 9 | 10 | win | 14 |
| 10 | 9 | lose | 5 |
| 11 | 10 | lose | -5 |
| 12 | 11 | win | 6 |
Each selected example has six wins and six losses and ends six units ahead. This illustrates the arithmetic of an uninterrupted progression, not the likelihood of obtaining those results.
With equal win and loss counts, and every prescribed stake remaining positive and permitted, the one-unit progression produces a profit equal to the number of wins. There is no guarantee of reaching that balance before funds or limits stop play.
A more aggressive system play
A two-unit step increases the potential gains and losses. In this selected balanced sequence, a ten-unit starting stake produces a twelve-unit profit rather than six:
| Bet Number | Bet Size (Units) | Outcome | Net Gain |
|---|---|---|---|
| 1 | 10 | win | 10 |
| 2 | 8 | win | 18 |
| 3 | 6 | lose | 12 |
| 4 | 8 | lose | 4 |
| 5 | 10 | lose | -6 |
| 6 | 12 | win | 6 |
| 7 | 10 | lose | -4 |
| 8 | 12 | win | 8 |
| 9 | 10 | win | 18 |
| 10 | 8 | lose | 10 |
| 11 | 10 | lose | 0 |
| 12 | 12 | win | 12 |
Ordinary double-zero even-money bets win with probability 18/38 and lose with probability 20/38. These are averages, not a promised mix in 38 spins. Special zero rules can reduce expected losses on eligible single-zero bets, but do not make this staking system profitable in expectation.
Reverse D’Alembert System
Reverse D’Alembert increases stakes after wins and decreases them after losses. Reversing a progression does not put the player in the position of the house: the same unfavourable probabilities and payouts apply.
What the reverse variation postulates is quite obvious. A player needs to increase his/her bet by 1 unit after every win and, respectively, decrease the bet by 1 unit after every loss. Similar to the reverse Martingale system, a player does not need to chase losses, but to rely on winning streaks. And again, as in the Reverse Martingale, the Reverse D’ Alembert system encounters the problem of how to determine when a winning streak has to be abandoned. A possible way to deal with winning streaks is to revert to the original bet after a specified number of successive wins (typically, four wins).
Regardless of its imperfections, this system has gained a certain popularity among casino players.
Further reading: Gambling Commission explanation of house edge.
