The Labouchère system, also called the cancellation system or split martingale, is a gambling strategy used in roulette. The user of such a strategy decides before playing how much money they want to win, and writes down a list of positive numbers that sum to the predetermined amount. With each bet, the player stakes 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 bet is successful, the two amounts are removed from the list. If the bet is unsuccessful, 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 the player runs out of money to wager.
Description of the Gameplay
The theory behind this Labouchère system is that, because the player is crossing two numbers off of the list (win) for every number added (loss) the player can complete the list, (crossing out all numbers) thereby winning the desired amount even though the player does not need to win as much as expected for this to occur.
It should be mentioned that the Labouchère System is meant to be applied to even money Roulette propositions such as Even/Odd, Red/Black or 1-18/19-36. When any of these bets are made in the game of Roulette, a spin resulting in a "0" or "00" results in a loss, so even though the payout is even money, the odds are clearly not 50/50. The Labouchère System attempts to offset these odds.
If a player were to play any one of the above propositions, there are eighteen individual results which result in a win for that player and (for an American Roulette wheel) twenty individual results that result in a loss for that player. The player has an 18/38 chance of success betting any of the above propositions, which is around 47.37%.
Theoretically, because the player is cancelling out two numbers on the list for every win and adding only one number for every loss, the player needs to have his proposition come at least 33.34% to eventually complete the list. For example, if the list starts with seven numbers and the player wins five times and loses three (62.5% winning percentage) the list is completed and the player wins the desired amount, if the list starts with seven numbers and the player wins 43,600 times and loses 87,193 times (33.34% winning percentage) the list completes and the player wins.
A formula to understand this is as follows:
Where x = Number of Wins y = Number of Losses Z = Numbers Originally on the List
( y + z ) / 2 ≤ X
The result is the list being completed.
Assuming a player bets nothing but black (red/black proposition) and black can be expected to hit 47.37% of the time, but the system only requires that it hit 33.34% of the time, it can be said that black only need hit approximately 70.38% of the time (33.34/47.37) it can generally be expected to in order for the system to prevail.
An obvious downfall to the system is bankroll, because the more losses sustained by the player, the greater the amount being bet on each turn (as well as the greater the amount lost overall) is. Consider the following list:
10 10 20 20 20 10 10
If a player were to bet black and lose four times in a row, the amounts bet would be: $20, $30, $40, and $50. By taking these four consecutive losses, the player has already lost $140 and is betting $60 more on the next bet. Consecutive losses, or an inordinate number of losses to wins can also cause table limits to come into play.
Occasionally, a player following this system will come to a point where he can no longer make the next bet as demanded by the system due to table limits. One work-around for this problem is simply to move to a higher limit table, or a player can take the next number that should be bet, divide it by two and simply add it to the list twice. The problem with the latter option is that every time a player commits such a play, it will infinitesimally increase the percentage of spins a player must win to complete the system. The reason this is so is because the player is adding two numbers (which both will be crossed out in the event of wins) where only one loss was sustained.
To prove this, if a player were to play the Labouchère System the same way with the exception being that the player always added half of the wager lost to the bottom of the list twice for every wager lost where:
x = Number of Wins y = Number of Losses Z = Numbers Originally on the List
y + (z/2) ≤ x
The result is the list being completed.
The player would actually have to win in excess of 50% of the time (the actual percentage of wins necessary, given x and y, being dependent on z) in order to complete the list, or more than the player could actually be expected to win.
The algorithm for the Labouchère system can be considered a Las Vegas algorithm since the amount of money a player desires to win will always be a predetermined amount. However, there is no guarantee that the player will reach the desired goal before the bankroll is lost. This is referred to as risk of ruin. For instance, consider the recursive implementation of a round of Labouchère betting in Python.
def gamble(sequence, balance): # Won if len(sequence) < 1: return balance # If the sequence is of length 1, the bet is the number in the sequence. Otherwise, it is the first number # added to the last number. if len(sequence) is 1: bet = sequence else: bet = sequence + sequence[-1] # Lost entire round if bet > balance: return balance won = flip_coin() if won: # Won return gamble(sequence[1:-1], balance+bet) else: # Lost bet return gamble(sequence+[bet], balance-bet)
The recursion of the algorithm terminates when the sequence is empty or when the player possesses insufficient funds to continue betting. When the function is called, the size of the bet made is equal to the sum of the first and last numbers of the sequence. If the length of the sequence is one, then the bet is equal to the sole member of the sequence. If the bet is won, then the first and last members are spliced from the sequence and the next round begins. However, if the bet results in a loss, then an integer equal the size of the lost bet is appended to the sequence and the next round begins. As determined by the parameters for termination of recursion, the only cases in which the algorithm will terminate are those in which the player has either won an amount equal to the summation of the original sequence or has lost all of their available capital. 
The Labouchère system can also be played as a positive progression betting system; this is known as playing the reverse Labouchère. In this version after a win, instead of deleting numbers from the line, the player adds 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, the player deletes the outside numbers and continues working on the shorter line. The player starts their line again if they run out of numbers to bet.
The Reverse Labouchère system is often used because where the Labouchère list represents how much the player wants to win, a reverse Labouchère line represents the most that the player will lose during the betting cycle. It is with this that a player with a bankroll of x can create their own line, or lines, representative of the maximum amount that they can sustain in losses.
Additionally, a player does not necessarily have to continue the system until the table limit is met or exceeded, but could instead pick a single bet that the player does not wish to exceed and make that bet their own personal limit.
Unlike the Labouchère system which (when adhered to strictly) requires a winning percentage of at least 33.34% to complete, 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 line in relation to the table limit.
For example, if a table had a limit of $500 and a player composed a Reverse Labouchère line as follows:
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 player profit of $2,700.
In contrast, if a player composed 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 player 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 example, 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 the player 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 a player's line, the more the player is risking losing. However, the longer the player's line, the better winning percentage the casino need have in order to break the player's line.
Advocates of this system point out that when a player uses the Labouchère System, where a streak in the casino's favor, or many mini-streaks in the casino's favor, will cause the player to sustain a huge loss, a single streak, or a few streaks in the player's favor using the Reverse Labouchère system will cause the player to have a huge gain.
A formula that can be used to determine how this system could fail is as follows:
x = Number of Wins y = Number of Losses Z = Numbers on original List
x + z ≤ y * 2
The system has failed, and all numbers on the line are crossed completely out.
Given an infinite line, the Labouchère System when played by the player requires a winning percentage of at least 33.34% to complete. In contrast, for the Reverse Labouchère to fail requires only that the player lose 33.34% of the time.
Once again, the winning percentage necessary for the system completing to success depends upon a number of variables.