In 19th-century casinos, most common betting took the form of even-money bets placed on red or black in games such as Roulette. Strategy lay in deciding how much to wager on each bet, and when to quit playing. Many gamblers of the era held strong beliefs that, with a clever betting system, they could guarantee a certain profit. One prominent system, known as the d’Alembert, calls for a player to begin with a unit bet, and then increase the size of the bet by one unit after every loss. Meanwhile, he should decrease the bet by one unit after every win, unless the bet is already at one unit. In short, increase bets following losses, decrease them after wins.
In a recent paper, LML External Fellow Harry Crane and Glenn Shafer of Rutgers University explore what makes such betting systems so seductive. Part of the answer, they argue, is that these systems, when they do win, give bettors a high 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. The d’Alambert (erroneously attributed to the mathematician Jean Le Rond d’Alembert) and other related systems offer a high probability of winning a small or moderate amount, while others offer a very small probability of making a large gain. In the paper, Crane and Shafer compare a variety of such systems on both their return on investment and their success in hiding their pitfalls – no system really does offer a certain profit. Among these systems, they observe, the d’Alembert seems especially good at quickly delivering an impressive return on investment, thereby encouraging gamblers’ hopes that they can use it so gingerly as to avoid the possible large losses. This may explain why its popularity was so durable.
Crane and Shafer conclude by suggesting that these systems offer some useful perspective for evaluating success in business and finance. As they note, corporate executives are also surely not immune to the similar temptations when making investment decisions. How often has a dashing corporate executive, celebrated for their exceptional success, simply played a strategy which offers a low probability of a large win? How often, when a corporate executive succeeds by risking more when investments go bad, are we seeing the seduction of the d’Alembert system? The public, including investors and politicians, might be better served if journalists reporting on these successes understood this aspect of the mathematics of gambling.
The paper is available at https://researchers.one/articles/20.08.00007