The D'Alembert Strategy in Roulette: principle and example

The D'Alembert strategy is a bet-management system in roulette in which the bet size changes by one unit: after a loss it grows, after a win it falls. The D'Alembert system in roulette belongs to the gentle progressions and is named after the 18th-century French mathematician Jean le Rond d'Alembert. Unlike the sharp Martingale, it does not double the bet but moves it smoothly, so the bankroll is spent more slowly and a losing streak does not hit the account as hard. Below we break down the principle, a step-by-step example with a table, the pros and cons, the reverse version and why no progression cancels the house edge.

The principle of the system

The player chooses a base unit - for example, 1 dollar, and starts with one unit. Then a pyramid logic applies:

• after a loss the bet increases by one unit (+1);
• after a win the bet decreases by one unit (-1);
• the bet does not drop below one unit.

The bet rises and falls in steps, forming a gentle pyramid rather than a steep ladder of doublings. The idea is that on bets with a chance of about 50-50 the wins and losses roughly balance out over time, and then the sum of the bet's decreases covers the sum of its increases.

The balance logic and the house edge

The author of the system started from the law of equilibrium: if you toss a coin for a long time, the number of heads and tails tends to even out. Hence the conclusion that after a run of losses the outcomes will supposedly start to be compensated by wins.

The problem is that the wheel has no memory. Every spin is independent, past outcomes do not change the probability of the next one. The European wheel has 37 pockets (0-36), the house edge is 2.7%, the return (RTP) is 97.3%. The zero lowers the chance of an even-money bet to 18 out of 37, about 48.6% to win, and this share in the house's favour holds under any progression. How the zero affects the odds - in the piece on the role of the zero. The D'Alembert redistributes bet sizes over time but does not touch the maths of the outcome: the distance becomes smoother, while an edge over the casino does not appear.

Step-by-step example

Let's take a base unit of 10 dollars and a bet on red. The bet changes by the rule +1 after a loss and -1 after a win.

After six rounds the account is up by 30 dollars. Wins and losses are equal (three each), but thanks to the structure of the progression the result is positive - this is how the balance idea works on a lucky stretch. If, however, the losses come in a row, several wins at already raised bets are needed to return to zero.

Pros and cons

The strengths are tied to the gentleness of the progression:

• smooth bet growth - one unit is added rather than a doubling, so bets grow more slowly than in the Martingale;
• a gentle regime for the bankroll - short swings of outcomes are borne without sharp drops;
• simplicity - the rule is memorised in a minute.

The weaknesses stem from the same mechanics:

• a drawn-out losing streak is not recovered - the bet rises slowly, and a long run of bad luck takes the account into the minus;
• recovery needs wins at already high bets, and no one promises them;
• the zero remains, and over distance the account tends toward the minus.

The reverse D'Alembert

The counter-D'Alembert flips the rule: the bet grows by one unit after a win and falls by one unit after a loss. The logic is different - to swing the bet up on the wave of a lucky streak and lower it when the outcomes go against the player. On a run of bad luck the approach behaves more gently, but a noticeable plus needs a drawn-out winning streak. Like the direct version, the reverse D'Alembert does not change the probabilities and does not eliminate the house edge.

Which bets to use it on

The system is designed for bets that pay 1 to 1, that is, for even-money chances:

• red or black;
• odd or even;
• low (1-18) or high (19-36) numbers.

On these bets the outcomes are close to 50-50, and a step of one unit makes sense. On high-paying bets - straight (35 to 1), split (17 to 1), dozen or column (2 to 1) - the balance logic does not work: the hit frequency there is completely different.

What bankroll you need

There are no sharp swings of the bank, but a drawn-out streak still takes the account into the minus. A reasonable guide is to keep the bankroll at the level of 50-100 base units. At a unit of 10 dollars that is 500-1000 dollars that you are ready to spend on entertainment, not as a source of income.

It is useful to set two limits in advance: a loss limit at which you stop, and a profit target after which you lock in the result. Without such boundaries a gentle progression draws you in: it seems that another couple of rounds and the balance will recover, but the wheel does not promise that. More on discipline - in the article on the roulette betting system.

Comparison with the Martingale and Fibonacci

Briefly on the differences from two other popular progressions:

The Martingale is the most aggressive of all: bets grow like an avalanche and quickly run into the table limit. The Fibonacci is a compromise, its progression is steeper than the D'Alembert's but gentler than doublings. The D'Alembert is the flattest of the three and suits those who want to stretch out the game and smooth the swings of the account. A review of other approaches is gathered in the breakdown of roulette strategies. None of the three systems gives a mathematical edge - they all merely distribute the bets differently.

How to track streaks

Any progression on even-money chances is tied to streaks: how many times in a row one colour has come up, how long there has been no even number. These data are more convenient to view in live statistics. The service collects the results of live-dealer tables online and shows how long colours, odd and even, dozens and sectors have not appeared. You can open them in the roulette statistics section.

Statistics do not predict the next spin - the wheel remains random - but they help you see the picture soberly. Use the D'Alembert as a way to structure your bets and protect your bankroll, not as a method to beat the casino, and play within the planned amount.

Open roulette statistics →