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Betting Systems

d'Alembert - Wikipedia

```D’Alembert is negative progression betting system, similar to martingale and Labouchere. This means that when a bet is lost, the next bet will be increased, and when a bet is won, the next bet will be decreased.

The system, also called montant et demontant, 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.  -- Wikipedia

As mentioned in the quote above, 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

1Loss-1
2Loss-3
3Win-1
2Loss-3
3Loss-6
4Win-2
3Win1
2Win3
1Win4
1Loss3
2Loss1
3Win4

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
1Loss-2
2Win0
1Loss-1
1Loss-2
2Win0
1Win1
1Win2
1Win3
1Loss2
1Loss1
2Win3

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

1Loss -1
1Loss -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
3Win 0

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