Reverse Labouchere: Disproved
Assumptions
The program is designed to mimic real life as closely as possible. Twelve players with an initial bankroll of 1233 francs play 180 spins each, with 6 to a table. Each player is assigned one of the even chances to cover, and does not change throughout the day. All the rules of European roulette are implemented (although see the note on la partage and en prison below) and the stakes are calculated precisely as Norman Leigh indicates in his book.
If a player is unable to reach the required stake the player will cease play immediately - note that a new line will not be started with lower stakes. Norman Leigh does not describe how this situation would be dealt with in his system.
Results
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The reverse labouchere program available below simulates a team of twelve players playing for one day on two tables and calculates the result of the day's play. A sample of 1,000,000 daily profits was generated using the reverse labouchere simulation, which can be used to estimate the underlying sampling distribution.
The histogram of results to the right provides a useful approximation of the sampling distribution of daily profits. The vertical red line separates the losses from profitable samples. It should be noted that the distribution is asymmetric since maximum losses are capped at the team's combined initial bankroll for the day, 14,796 francs. Without this limit I am fairly certain that the results would be normally distributed. This could be checked by giving the players infinite bankrolls, however this is of no real significance.
The mean of the samples is -1055 francs and the standard deviation is approximately 7840 francs. The histogram indicates leptokurtosis, with much of the data concentrated about the mean while a small amount is distributed along the long right tail. This is a dangerous property for a roulette system; not only is the expected result a loss, it seems that it is difficult to achieve any profit at all.
Analysis
Only 21.85% of the sampled days were profitable, with the average result being a loss of 1055 francs. These two fundamental facts immediately indicate the failure of the reverse labouchere system and attention shifts to determining exactly how unlikely were the results claimed by Norman Leigh.
In Thirteen against the Bank Leigh claims to have achieved daily profits of over 100,000 francs on a number of occasions. The simulation results show a maximum profit of 86,578 francs, with only 0.00005% of the sample exceeding a profit of 80,000 francs. The table below compares Leigh's results with the sample and gives an estimate of the probability of observing a result at least as good as Leigh's on each day.
Day |
Leigh (francs) |
Proportion of sample at least as good |
|---|---|---|
1 |
78,000 |
0.0007% |
2 |
130,450 |
0 % |
3 |
66,165 |
0.004 % |
4 |
67,000 |
0.0033 % |
5 |
55,385 |
0.0216 % |
6 |
123,003 |
0 % |
7 |
159,666 |
0 % |
8 |
119,595 |
0 % |
Roulette spins can reasonably be modeled as independent, and so the total probability of observing results at least as good as Norman Leigh over the eight days combined can be calculated by multiplying the above probabilities. This is clearly zero, however even if we ignore the most exceptional days with zero terms the estimated probability of observing four days as good as those remaining is 1.99584e-18, roughly a 1 in 5,000,000,000,000,000,000 chance (this is one in five quintillion).
Download
The simulation can be downloaded either as a compiled package or as C# source.
Further Development
Currently there is some confusion regarding the la partage and en prison choices which occur on the even chances when zero is spun. Norman Leigh does not indicate which rule the team were instructed to use, although it should not matter as the expected profits using either option are identical. The simulation applies the la partage rule only, resulting in a loss of half the stake while returning the rest to the player.