The D’Alembert System

french_rouletteAmong casino players this system is also known as the ”Pyramid system”. It carries the name of a French mathematician, as it is based on an equilibrium theory he proposed over two centuries ago. Similar to the Martingales, this system focuses primarily on even-money outside bets.

The D’Alembert System fits the preferences of those players, who wish to maintain the size of their bets and their losses to a minimum, because it is not very likely to earn or to lose huge amounts by using this approach.

The Betting Sequence

The system features a quite comprehensible 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

When using the D’Alembert System, a player does not need to worry about reaching the table limit. However, that is not the case with the table minimum. The sequence we just saw features a greater series of wins and a low initial bet, which may lead to the betting process hitting the table minimum. Such an event would ruin the sequence. Fortunately, there is a way to avoid such a situation – to start with a larger initial bet. Let us see what the sequence will look like, if the initial bet is 10 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

What undoubtedly catches the eye when looking at the three sequences is that in each one of them there have been six wins and six losses, which led to a net result (net gain) of 6 units. This is hardly coincidental! We have come to the very circumstance, which makes the D’Alembert System so appealing, the very circumstance due to which the system is regarded as being a classic. Whenever the number of wins is equal to the number of losses, the net gain will be equal to the number of wins.

A more aggressive system play

The D’Alembert System can be used in a more expansive manner. In case a player increases the bet change to 2 units instead of 1, his/her net result (net gain) will be doubled. Let us take a look at the third sequence, while taking into account a bet variation of 2 units and an initial bet of 10 units:

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

It is important to note that in reality, when playing roulette for a longer period of time, a player is likely to score a lesser number of wins than losses, placing even-money outside bets. If taking into consideration the double-zero wheel, on average there would be 18 wins and 20 losses for every 38 ball spins. If taking into consideration the single-zero wheel, with the application of the ”en prison” or ”le partage” rule, the D’Alembert System would perform reasonably well. It is the reason why this system is widely used at casinos throughout Europe.

Reverse D’Alembert System

This variation of the D’ Alembert system is also known as “contra-Alembert”. It was invented by those players, who failed to achieve consistent gains using the regular system. According to their logic, by reversing the rules of the system, they would find themselves in the position of the house and not in that of the player.

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.

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