# D'Alembert Betting System

## Introduction

The d'Alembert is a popular and classic betting system. Like most betting systems, it usually results in a small win, at the cost of huge losses sometimes. Like all betting systems, not only can't it overcome the house edge, it can't even dent it.

Like the Martingale, Labouchere and Fibonacci betting, the d'Alembert has the player chasing losses with larger bets. However, it is not as aggressive as those other progressive betting systems, resulting in more time at the table and less volatility. It is also at a cost of a lower probability of overall goal success.

## Rules

The only thing that everybody seems to agree on about the d'Alembert is that the player increases his bet by a unit after a loss and decreases by a unit after a win. Other sources generally don't address the initial bet and winning or losing benchmarks. For purposes of my analysis, I start the player with a bet of one unit with a goal of winning one unit. Here is putting the whole system more formally.

1. The player shall define his winning goal and bankroll size.
2. The player's "unit size" shall be equal to his winning goal.
3. The player starts with a one-unit bet.
4. If the player ties, then he repeats the same bet.
5. Otherwise, if the last bet results is a win and the player has achieved his winning goal, then he walks away happy.
6. Otherwise, after a win, if the bet size was one unit, he keeps it the same. Otherwise, he decreases his bet size by one unit.*
7. Otherwise, after a loss, the player increases his bet size by one unit.**
8. The player bets.
9. Go back to rule 4, until the player either achieves his winning goal or loses his entire bankroll.

*: If such a bet would cause the player to overshoot his winning goal if he wins, then drop the bet size to whatever would result in achieving exactly the winning goal the next bet.
**: If the player does not have enough money to make the next bet, then drop the bet size to whatever money the player has left.

An interesting facet of the d'Alembert is it doesn't matter in what order wins and losses are, like flat betting. The only thing that matters in terms of session results is the number of wins and losses. The d'Alembert can show a profit even when losses outnumber wins, as long as the disparity isn't too much. The following table shows the net win according to various totals of wins and losses. For example, the player can have 22 wins and 28 losses and still show a profit of one unit.

### Net Win by Total Wins and Losses

Wins Losses Net Win
2 3 1
3 4 2
4 6 1
5 7 2
6 8 3
7 10 1
8 11 2
9 12 3
10 13 4
11 15 1
12 16 2
13 17 3
14 18 4
15 19 5
16 21 1
17 22 2
18 23 3
19 24 4
20 25 5
21 26 6
22 28 1
23 29 2
24 30 3
25 31 4
26 32 5
27 33 6
28 34 7
29 36 1
30 37 2
31 38 3
32 39 4
33 40 5
34 41 6
35 42 7
36 43 8
37 45 1
38 46 2
39 47 3
40 48 4
41 49 5
42 50 6
43 51 7
44 52 8
45 53 9
46 55 1
47 56 2
48 57 3
49 58 4
50 59 5

In situations where the number of losses is equal or greater to the number of wins, the general formula for the net win is W - D*(D+1)/2, where:

W = Number of wins
D = Difference between wins and losses. In other words, losses minus wins.

In the example above of 22 wins and 28 losses, the net win is 22 - 6*7/2 = 21.

Despite being able to win in moderately losing sessions, it is at a cost of small wins most of the time and huge wins in extremely cold sessions that wipe out those small wins over the long run.

## Simulation Results

To show what to expect from using the d'Alembert, I wrote a simulation that followed the rules above, based on various bets and games. The simulation used a Mersenne Twister random number generator. For each simulation, the initial bet and winning goal were both one unit. I tested the simulation on the following bankrolls: 10, 25, 50, 100, and 250 units.

The first simulation is based on betting the Player bet in baccarat. The simulation size is over 73 billion sessions. As a reminder, the theoretical house edge on the Player bet is 1.235%.

### Baccarat Simulation — Player Bet

Statistic 10 Units 25 Units 50 Units 100 Units 250 Units
Probability winning goal reached 90.36% 95.74% 97.73% 98.78% 99.45%
Average number of bets 2.422 3.297 3.719 4.169 4.837
Average units bet 4.857 8.727 12.670 18.456 30.939
Expected win per session -0.060 -0.108 -0.157 -0.228 -0.382
Ratio money lost to Money bet 1.234% 1.236% 1.235% 1.235% 1.235%

The first simulation is based on betting the pass bet in craps. The simulation size is over 65 billion sessions. As a reminder, the theoretical house edge on the pass bet is 1.41%.

### Craps Simulation — Pass Bet

Statistic 10 Units 25 Units 50 Units 100 Units 250 Units
Probality winning goal reached 90.34% 95.72% 97.72% 98.78% 99.44%
Average number bet 2.423 3.300 3.724 4.176 4.850
Average total bet 4.399 7.908 11.489 16.752 28.134
Expected win per session -0.062 -0.112 -0.162 -0.237 -0.398
Ratio money lost to Money bet 1.414% 1.414% 1.414% 1.414% 1.414%

The next simulation is based on the don't pass bet in craps. The simulation size was over 76 billion sessions. As a reminder, the house edge on the don't pass bet is 1.364%.

### Craps Simulation — Don't Pass Bet

Statistic 10 Units 25 Units 50 Units 100 Units 250 Units
Probality winning goal reached 90.35% 95.73% 97.72% 98.78% 99.44%
Average number bet 2.423 3.299 3.723 4.175 4.847
Average total bet 4.523 8.131 11.811 17.218 28.903
Expected win per session -0.062 -0.111 -0.161 -0.235 -0.394
Ratio money lost to Money bet 1.364% 1.364% 1.364% 1.365% 1.363%

The next simulation is based on any even money bet in single-zero roulette. The simulation size was over 25 billion sessions. As a reminder, the theoretical house edge is 1/37 = 2.7027%.

### Roulette Simulation — Single Zero

Statistic 10 Units 25 Units 50 Units 100 Units 250 Units
Probality winning goal reached 89.81% 95.30% 97.40% 98.52% 99.26%
Average number bet 2.456 3.381 3.851 4.371 5.190
Average total bet 4.485 8.200 12.125 18.119 31.920
Expected win per session -0.121 -0.222 -0.328 -0.490 -0.863
Ratio money lost to Money bet 2.702% 2.703% 2.702% 2.703% 2.702%

The next simulation is based on any even money bet in double-zero roulette. The simulation size was over 25 billion sessions. As a reminder, the theoretical house edge is 2/38 = 5.2632%.

### Roulette Simulation — Double Zero

Statistic 10 Units 25 Units 50 Units 100 Units 250 Units
Probality winning goal reached 88.68% 94.37% 96.65% 97.91% 98.75%
Average number bet 2.520 3.544 4.112 4.782 5.942
Average total bet 4.660 8.800 13.463 21.083 40.571
Expected win per session -0.245 -0.463 -0.708 -1.109 -2.134
Ratio money lost to Money bet 5.263% 5.263% 5.263% 5.263% 5.261%