It is not often that I recommend a research paper on voodoo gambling ideas, especially a paper on progressions. Then again, it is very rare for a quality research paper on the topic to be written. Last November (2019), I visited Rutgers University as an invited speaker in their probability and statistics seminar. During the visit, I met the authors of this paper, professors Harry Crane and Glenn Shafer. They are first-rate scholars and are among the very few who are finding original angles on ancient ideas. After reading this paper, I felt it was important to pass it along. The paper I am recommending is:
If you read this paper and want to play around with the d’Alembert progression in roulette, I’ve crated a Monte Carlo spreadsheet simulation of 100k roulette spins that you can download:
Here is a simulation of 1000 shoes of baccarat using the d’Alembert progression. I assume that you restart the progression with each new shoe.
With the author’s permission, I am reprinting the abstract of their paper:
The most common bets in 19th-century casinos were even-money bets on red or black in Roulette or Trente et Quarante. Many casino gamblers allowed themselves to be persuaded that they could make money for sure in these games by following betting systems such as the d’Alembert. What made these systems so seductive? Part of the answer is that some of the systems, including the d’Alembert, can give bettors a very high probability of winning a small or moderate amount. But there is also a more subtle aspect of the seduction. When the systems do win, their return on investment — the gain relative to the amount of money the bettor has to take out of their pocket and put on the table to cover their bets — can be astonishingly high. Systems such as le tiers et le tout, which offer a large gain when they do win rather than a high probability of winning, also typically have a high upside return on investment. In order to understand these high returns on investment, we need to recognize that the denominator — the amount invested — is random, as it depends on how successive bets come out.
In this article, we compare some systems on their return on investment and their success in hiding their pitfalls. Systems that provide a moderate gain with a very high probability seem to accomplish this by stopping when they are ahead and more generally by betting less when they are ahead or at least have just won, while betting more when they are behind or have just lost. For historical reasons, we call this martingaling. Among martingales, the d’Alembert seems especially good at making an impressive return on investment quickly, encouraging gamblers’ hope that they can use it so gingerly as to avoid the possible large losses, and this may explain why its popularity was so durable.
We also discuss the lessons that this aspect of gambling can have for evaluating success in business and finance and for evaluating the results of statistical testing.