D’Alembert system

The D’Alembert system, also called montant et demontant (from French, meaning upwards and downwards), is often called a pyramid system. It is based on a mathematical equilibrium theory devised by a French mathematician of the same name. Like the martingale, this system is mainly applied to the even-money outside bets, and is favored by players who want to keep the amount of their bets and losses to a minimum. The betting progression is very simple: After each loss, you add one unit to the next bet, and after each win, one unit is deducted from the next bet. Starting with an initial bet of, say, 1 unit, a loss would raise the next bet to 2 units. If this is followed by a win, the next bet would be 1 units.
This betting system relies on the gambler’s fallacy—that the player is more likely to lose following a win, and more likely to win following a loss.

How it works

The basic system is to add 1 unit on a loss, and to deduct 1 unit on a win, with the minimum bet being 1 unit.
An example of betting with D’Alembert:

Bet Amount Bet Result Total Profit
1 Loss -1
2 Loss -3
3 Win -1
2 Loss -3
3 Loss -6
4 Win -2
3 Win 1
2 Win 3
1 Win 4
1 Loss 3
2 Loss 1
3 Win 4

While D’Alembert works well for a short period of time, it is fundamentally flawed for any casino with a house edge, because the size of the bet is dependent on the total number of wins and losses. Assuming a house edge of 1%, you need to bet at 49.5% chance to win to make 2x profit per bet. This means that statistically, you will lose 1% more bets than you will win, causing your bets to slowly and continually increase in size until it becomes larger than your balance, in spite of possibly being in profit over all.

While all betting systems are essentially a gamblers fallacy, d’Alembert is the most obviously flawed for lengthy use.

Deviations of the d’Alembert strategy include “stretching” your wins or losses. This means that you need to hit more than one win or loss in a row to decrement or increment your bet. Below is the example bets from above, while stretching losses to 2 (increment only on 2 losses in a row)

Bet Amount Bet Result Total Profit
 1 Loss -1
 1 Loss -2
 2 Win 0
 1 Loss -1
 1 Loss -2
 2 Win 0
 1 Win 1
 1 Win 2
 1 Win 3
 1 Loss 2
 1 Loss 1
 2 Win  3

D’Alembert can be reversed as a positive progression system, meaning you increase the size of your bet on a win and decrease the size of the bet on a loss.

Bet Amount Bet Result Total Profit
1  Loss  -1
1  Loss -2
1  Win -1
 2  Loss -3
1  Loss -4
1  Win -3
2  Win -1
3  Win 2
4  Win 6
5  Loss 1
4  Loss -3
3  Win 0