D'Alembert Betting System

The D'Alembert system is a betting system most commonly associated with roulette and is similar to the more well-known Martingale system. Here we take a detailed look at this betting strategy, focusing on primarily on what it is and how it works, but also explaining the history of the concept and why it is, ultimately, flawed.

History of the D'Alembert Betting System

The beginning is invariably a fine place to start and so let us first of all consider the history of the D'Alembert and how it came into being. The system is named after the 18th century French mathematician, philosopher and physicist, Jean le Rond d'Alembert.

Born in Paris in 1717, d'Alembert has made a great many contributions to the world of maths, science and philosophy. However, the great polymath’s contribution to the world of gambling actually came about due to an incorrect argument he made in Croix ou Pile (which translates as “Heads or Tails”). In this treatise he argued that a coin that has previously landed on heads or tails is more likely to land on the other side on the next toss.

This argument, now universally accepted to be incorrect, formed the basis of the D'Alembert system which is, in essence, a variation of the Martingale system. The exact time at which this betting system or staking plan was first used or written about is, as with much in the murky world of gambling history, unclear.

However, as with the Martingale system itself, we can be fairly confident that it was probably “invented” towards the end of the 1700s. Whether it was coined as the name for a codified staking plan during the life of d’Alembert is also unclear, the Frenchman dying of a bladder infection in 1783 at the age of 65.

What is the D'Alembert Betting System?

As said, this system is essentially a form of Martingale system and, like the Martingale itself, it is a negative progression staking plan that sees gamblers increase their stake following a loss. “Negative progression” refers to the fact that the result is negative, that is to say the bettor loses, whilst the stake increases, or progresses.

Martingale, as we explain in our dedicated article on that particular system, dictates that the user should double their stake following a losing bet. After a winning bet the stake drops to the original unit, be that £1, £10, £100 or whatever.

The D'Alembert system works differently in that you only increase your stake by a single unit after each loss, as opposed to doubling it. Similarly, after a win you reduce the stake by a single unit, as opposed to reverting to the initial wager amount.

So, if you decide to play roulette and opt to bet on black with a £10 stake, one loss sees your bet increase to £20, exactly as it would with Martingale. However, should you lose the second spin, your stake increases to just £30, whereas with Martingale it would double once more to £40. Lose again and D'Alembert sees you staking £40, as opposed to £80 on Martingale. The more consecutive losses suffered, the greater the divergence between the stake required for the two staking plans.

If we imagine two roulette-loving friends, Martin Gale and Alan Bear, visit the casino using the two different staking plans. Based on an initial bet of just £10, the Martingale system would dictate a bet of £640 following a sixth consecutive loss, whilst the D'Alembert system would have you betting a far more manageable £70.

However, if we assume the seventh spin under each system is a winner, the Martingale bettor would be sitting on a £10 profit (one staking unit) whilst the player using the D'Alembert system would actually still be down by £140.

It’s also worth noting that whilst Martin Gale would then bet £10 on his next spin, his friend, Alan Bear, would have to bet £60. The two systems deliver very different returns, with much greater volatility experienced by Martin than by Alan.

The risk of ruin, of going broke, is considerably higher for Martin, because the doubling of the stake very quickly creates a huge bet. As we saw, just six straight losses sees him having to bet £640 at the risk of ending that spin a massive £1,270 down. In contrast the figures for Alan are a bet of £70 that risks overall losses reaching £280.

Pros and Cons of the D'Alembert Betting System

Ultimately the D'Alembert system is based on an incorrect principle, that being the notion that a win is essentially more likely following a loss. Because a coin will, theoretically, land on tails as often as it will land on heads, D'Alembert argued that a head is more likely after a tail, as, in order to reach the natural, theoretical equilibrium, a head is required.

However, if we rule out the notion of an imperfect coin or coin toss, or – in roulette – a biased wheel or corrupt croupier, the simple fact is that what has gone before has no bearing on what will happen next. The coin, the wheel, or whatever else you try to use D'Alembert to bet on, has no “memory”. The notion that there will be an equal number of both outcomes, be that heads and tails, reds and blacks, high and low or anything else, is only theoretical.

The closer you were to go to an infinite number of roulette games, the more closely that theory would resemble reality but you can’t pay for drinks with a theory and you can’t eat infinity. At least that’s what a wise old man told us and the wise old man was a bookmaker.

Anyway, this section was supposed to be about the pros and cons of the D'Alembert betting system before we got side tracked. However, the point is this: can we talk about pros of a betting system that is essentially, fundamentally flawed? Well, to a degree, we can, we will and here it comes:

Pros

  • Lower variance - Martingale will deliver lots of small losses and the occasional huge loss whereas this staking plan is less likely to cause such a huge loss
  • Longer game time - Because the stakes alter by smaller amounts, D'Alembert will tend to deliver longer sessions at the wheel
  • Risk of ruin reduced - Risk of ruin is reduced due to the flatter progression of the stakes
  • Less chance of hitting house limits - For the same reason you are less likely to be forced to diverge from the recommended stake due to reaching the table or house limit

Having listed these as benefits of the D'Alembert betting system, we must add that these are, as you can see, really only “relative pros”. We have listed them as advantages but that is largely in relation to the Martingale system.

As we have said, like the Martingale system itself, the D'Alembert is badly flawed if it is to be viewed in its own right as a staking plan and especially if it is to be viewed as some form of magical casino-busting secret.

Why the D'Alembert Betting System is Flawed

As we have already said, the D'Alembert staking plan is founded on a principle that is simply false. Time and time again maths experts and computer simulations have proven that the odds and probability of one outcome are not altered by previous outcomes when it comes to games such as roulette. This is also very strongly backed up by anecdotal evidence from croupiers and dealers, many of whom have stories to tell of unfeasibly long runs of reds or blacks, of odds or evens, or of high or low numbers.

The other fundamental issue with the D'Alembert system, as with Martingale and just about any other casino staking plan, secret system or betting scheme, is that how much you bet – in other words, a staking plan – cannot turn a bad bet into a good bet. Another way of phrasing that is to say that a staking plan cannot overturn the house edge.

All casino games have a house edge and in roulette this means that the odds paid for any of the bets do not reflect the fact that there is a zero (or two if you are so foolish as to play American roulette) on the table. There are 18 reds, 18 blacks and a single zero, giving 37 possible slots for the ball to land in. The probability of any single number is one in 37, meaning that “fair” odds, with zero house edge, would be 36/1.

That the casino pays out at odds of only 35/1 creates a house edge of 2.70%, meaning that in the long term, regardless of how you arrange your stakes, you can expect to lose 2.70% of all that you wager.

D'Alembert, Martingale and other staking plans can never alter that. All that they alter is the short term variance and how you can reasonably expect each session to play out. If we imagine a roller coaster with a start and a finish. The ride always starts and ends in the same place. But perhaps there are two or three different tracks, some with huge highs and huge lows, others with more gentle curves, climbs and falls. The ride may differ but the start will always be the same and the end will always be, in the case of the roller coaster of roulette, 2.70% lower.