The Labouchère Progression


What the Labouchère Progression is about

The Labouchère Betting Strategy, also known as Cancellation Strategy or Split Martingale, is a staking strategy used for even money bets such as Even/Odd, Red/Black or High/Low. It was named after its creator, Henry Du Pré Labouchère, (1831 – 1912) an English politician, writer, publisher and theatre owner. He inherited a large fortune and engaged in a number of occupations and was a gambler who loved to bet on horse races.

If you use this strategy you decide before playing how much money you want to win, and write down a list of positive numbers that sum up to the predetermined amount. With each bet, you stake an amount equal to the sum of the first and last numbers on the list. If only one number remains, that number is the amount of the stake. If a bet is won, the two amounts are removed from the list. If a bet is lost, the amount lost is appended to the end of the list. This process continues until either the list is completely crossed out, at which point the desired amount of money has been won, or until you run out of money to wager.

You are crossing two numbers off of the list on a win for every number added on a loss. So the theory behind the strategy is that you can complete the list by crossing out all numbers and thereby winning the desired amount even though you do not need to win as many times as you may expect. The more losses you experience while playing the list the closer the win/lose ratio required to eventually complete the list comes to 1:1.

Example 1: If the list starts with seven numbers and you lose three times then the list is completed when you win five times. This is a required win/lose ratio of 5:3 which is a 66.67% win rate in a 48.65% chance game.

Example 2: If the list starts with seven numbers and you lose seven times then the list is completed when you win seven times. This is a required win/lose ratio of 1:1 which is a 50% win rate in a 48.65% chance game.

The following formula describes this in mathematical terms:

x = Number of wins to complete the list
y = Number of losses
z = Numbers originally on the list

( y + z ) / 2 ≤ x

How you play the Labouchère

First you'll need to set a goal for how much money you'd like to win. It's usually simplest to express this in a certain number of 'units' – where the unit value can be any amount you like. For instance, if you'd like to win $200 you could set the unit value at $20, meaning your goal is to win ten units.

Once you know how many units you want to win, you'll have to break that down into some smaller numbers and write those numbers down in a line. For example, you could use the following number sequence:

1 | 2 | 3 | 3 | 1

The key is that those numbers add up to ten. If you are able to cross all five of those numbers off your sheet, you have reached your goal.

In order to determine the size of your bet, you should add the first and last numbers of your line together and wager that many units. For our example, the numbers at the ends of our line are 1 and 1. That totals two units, meaning your first bet would be $40. If you win your bet, you get to cross out both numbers you used to make that bet. Your line would now look like this:

X | 2 | 3 | 3 | X

Your next bet would now be for $100 (2 + 3 = 5 units). For the sake of this example, let's say you lose this bet. Now, the number of units you just lost goes to the end of your line, making it look like this:

X | 2 | 3 | 3 | X | 5

Your next bet would now be for 7 units, or $140. Let's say you win and cross off the two numbers for your new line looking like this:

X | X | 3 | 3 | X | X

Your next bet would now be for 6 units, or $120. Let's say you win again and cross off the two remaining numbers having cleared your line and made your $200 profit.

X | X | X | X | X | X

Advantages and Disadvantages to the Labouchère Strategy

The obvious advantage to the Labouchère betting strategy is that you do not need to score constant wins in order to eventually meet your goal. Since you only add one number to the line each time you lose, but cross off two numbers each time you win, winning just 33.3% of the time is enough to maintain your line. Since you will win more than 1/3rd of the time in the long run, you can feel confident that you will tend to shrink your line over time and eventually clear it to meet your goal. There's also a certain excitement that comes as you get close to meeting your goal, making the Labouchère a lot more fun to play. If you'd like to play an enjoyable strategy that may help you stay focused on meeting a specific goal while you play, the Labouchère is definitely right for you.

Unfortunately, the odds aren't always as simple as they seem, and fluctuations happen frequently. Even a seemingly safe Red/Black wager can quickly become costly as players add their cumulative losses to every bet. While your bets won't increase as quickly as in the Martingale, they can get quite large. Also, since a single win won't bring you back down to your minimum betting amount, it will often take several large bets to get you back on track after a losing streak. In the long run the Labouchère system doesn't change the casino's house edge either, so the casino still has an advantage over a player. However, this is a system that requires you to think a little bit more than others, making it feel just a bit more interactive than most other betting strategies.

The Reverse Labouchère Strategy

The Labouchère system can also be played as a positive progression betting system. This is known as playing the Reverse Labouchère. Instead of deleting numbers from the line, you add the previous bet amount to the end of the line. You continue building up your Labouchère line until you hit the table maximum. After a loss, you delete the outside numbers and continue working on the shorter line. You have to start the line again if you run out of numbers to bet.

The Reverse Labouchère system is often used because while the Labouchère list represents how much you want to win, a Reverse Labouchère line represents the most that you will lose during the betting cycle. With your bankroll you can create your line, or lines, representative of the maximum amount that you can sustain in losses. Additionally, you do not necessarily have to continue the system until the table limit is met, but could instead pick a single bet amount that you do not wish to exceed and make that bet your own personal limit.

Unlike the Labouchère system the winning percentage needed to complete a Reverse Labouchère line is going to be dependent on both the table limit or the maximum single bet a player is willing to make as well as the numbers on the initial list in relation to the respective limit. For instance, if a table had a limit of $500 and you compose a Reverse Labouchère line such as:

50 | 50 | 50 | 50 | 50

Nine consecutive wins (100, 150, 200, 250, 300, 350, 400, 450, 500) would cause the next bet in the system to exceed the table limit, and thus the line would be completed with a profit of $2700. In contrast, if you compose a Reverse Labouchère line such as:

25 | 25 | 25 | 25 | 25

Nineteen consecutive wins (50, 75, 100, 125, 150, 175, 200, 225, 250, 275, 300, 325, 350, 375, 400, 425, 450, 475, 500) would cause the next bet in the system to exceed the table limit, thus the line would be completed with a profit of $5,225.

The length of the line in the Reverse Labouchère system is also important as it relates to the percentage of wins necessary to complete the system. For instance, if a line of

50 | 50 | 50 | 50 | 50

suffers three consecutive losses as soon as the system begins, then the line is completed and a new line must be started, or you may choose to quit. In contrast, if a line of

50 | 50 | 50 | 50 | 50 | 50 | 50 | 50 | 50 | 50 | 50 | 50

suffers three consecutive losses, then there are still six numbers remaining on the list. In the line immediately above, it would take an opening streak of six consecutive losses for the line to be completed. All other things remaining the same, the longer your line is, the more you are risking losing, however, the longer your line is, the better winning ratio the casino needs to have in order to break your line.

The following formula can determine when the strategy has failed, and all numbers on the line are crossed completely out:

x = Number of wins
y = Number of losses
z = Numbers on original list

( x + z )/ 2 ≤ y

RNG and Live Dealer Tests of the Labouchère Strategy for 32000 bets

For demonstration purpose we have tested a simple roulette system. The rules are just to bet High and Low even chances following the previous spin result using the following Labouchère line in order to win 15 units.

1 | 3 | 4 | 4 | 3

As you can see in the charts most of the time the bet amounts stay relatively moderate but at times they increase up to a maximum of 10554 units in the RNG and 23721 units in the Live Dealer test run. The average bet amounts are 87 units and 96 units respectively. The longest game, that is how many bets have been placed in order to clear the line, took 154 bets in the RNG and 115 bets in the Live Dealer test. The longest observed losing streaks are 16 spins 15 spins long respectively.

RNG Tests Results

Live Dealer Tests Results