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D'Alembert Betting System

Introduction

D'Alembert Betting System

The d'Alembert system is a well-known approach to betting that has stood the test of time. While it typically leads to minor gains, it carries the risk of significant losses at times. Like all betting methods, it cannot overcome the built-in house advantage, and in fact, is unable to significantly reduce it.

Like the Martingale, Labouchere and Fibonacci betting In the d'Alembert system, players tend to increase their bets after experiencing losses, yet it is not as aggressive as other progressive systems, which allows for longer playing times and less fluctuation in results. However, this can result in a reduced chance of reaching overall success.

Rules

One clear consensus about the d'Alembert method is that players adjust their bets by one unit upwards following a loss and lower their bet by one unit after a win. Other guides seldom focus on the initial bet amount or the criteria for winning or losing. In my assessment, I will set the player’s initial bet to one unit, aiming for a win of one unit as the target. Here's a more structured breakdown of the system.

  1. The player must establish both their desired winning target and the amount they are willing to gamble.
  2. The player’s 'unit size' will align with their winning goal.
  3. The player starts with a one-unit bet.
  4. If the player results in a tie, they shall place the same bet again.
  5. If the player wins and successfully reaches their goal, they can walk away satisfied.
  6. Should the player win but their current bet is one unit, they will keep it the same. If the bet was more than one unit prior to the win, they will reduce their stake by one unit.
  7. In contrast, following a loss, the player will raise their bet by one unit.
  8. The player bets.
  9. Return to rule 4 until the player either meets their winning target or exhausts their entire bankroll.

*: If increasing the bet would exceed the winning goal upon a win, adjust the bet down to the amount that would precisely achieve the winning goal on the next bet.
**: If the player lacks sufficient funds for the next bet, reduce the bet to whatever amount they have remaining.

General Comments

An intriguing aspect of the d'Alembert system is that the sequence of wins and losses isn't critical, similar to flat betting. The primary factor affecting session outcomes is the total number of wins compared to losses. The system can yield profits even when losses exceed wins, provided the difference isn't excessive. The subsequent table illustrates the net gains based on various win-loss scenarios. For instance, a player could secure 22 wins and 28 losses while still achieving 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

When the losses equal or outnumber the wins, the formula for calculating the net gain becomes W - D*(D+1)/2, where:

W = Number of wins
D represents the difference between wins and losses, specifically calculated as losses minus wins.

Using the example of 22 wins against 28 losses, the net gain can be calculated as 22 - 6*7/2 = 21.

Even with the potential to profit during sessions characterized by more losses than wins, this comes with the trade-off of typically small gains and potentially substantial losses during very poor sessions, which can erase those minor profits over time.

Simulation Results

To provide insight into potential outcomes when using the d'Alembert system, I created a simulation adhering to the aforementioned rules across various betting games. This simulation employed a Mersenne Twister random number generator. In each instance, both the initial bet and the winning goal were set to one unit. I conducted simulations across several bankroll scenarios: 10, 25, 50, 100, and 250 units.

The first simulation focused on making bets on the Player bet in the game. baccarat This simulation comprised over 73 billion individual sessions. For context, the theoretical house edge for the Player bet stands at 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%

My initial simulation was predicated on placing bets on the pass bet option. craps This simulation included more than 65 billion sessions. Notably, the theoretical house edge associated with the pass bet is 1.41%.

Craps Simulation — Pass Bet

Statistic 10 Units 25 Units 50 Units 100 Units 250 Units
Probability 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%

Following that, I ran a simulation targeting the don’t pass bet. craps This experiment included over 76 billion betting sessions. The house edge for the don’t pass bet is recorded at 1.364%.

Craps Simulation — Don't Pass Bet

Statistic 10 Units 25 Units 50 Units 100 Units 250 Units
Probability 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%

Next, I executed a simulation for even money bets in single-zero roulette.

Roulette Simulation — Single Zero

Statistic 10 Units 25 Units 50 Units 100 Units 250 Units
Probability 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%

This one amassed over 25 billion sessions, where the theoretical house edge comes to 1/37 = 2.7027%.

Roulette Simulation — Double Zero

Statistic 10 Units 25 Units 50 Units 100 Units 250 Units
Probability 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%

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Discussion Another simulation concentrated on even money bets in double-zero roulette.