When Should We Double Up?

The Martingale Betting Strategy

A version of this article appears in my book, “Twisted Logic: Puzzles, Paradoxes, and Big Questions” (Chapman and Hall/CRC Press, 2024).

The Martingale betting strategy is based on the principle of chasing losses through progressive increase in bet size. To illustrate this strategy, let’s consider an example: A gambler starts with a £2 bet on Heads, with an even money pay-out. If the coin lands Heads, the gambler wins £2, and if it lands Tails, they lose £2.

In the event of a loss, the Martingale strategy dictates that the next bet should be doubled (£4). The objective is to recover the previous losses and achieve a net profit equal to the initial stake (£2). This doubling process continues until a win is obtained. For instance, if Tails appears again, resulting in a cumulative loss of £6, the next bet would be £8. If a subsequent Heads occurs, the gambler would win £8, and after subtracting the previous losses (£6), they would be left with a net profit of £2. This pattern can be extended to any number of bets, with the net profit always equal to the initial stake (£2) whenever a win occurs.

CHASING LOSSES AND THE LIMITATIONS

While the Martingale strategy may appear promising in theory, it is important to recognise its limitations and the inherent risks involved. The strategy involves chasing losses in the hope of recovering them and generating a profit. However, it’s crucial to understand that the expected value of the strategy remains zero or even negative.

The main reason behind this lies in the presence of a small probability of incurring a significant loss. In a game with a house edge, such as in a casino, the odds contain an edge against the player. The house edge ensures that, over time, the expected value of the bets is negative. Therefore, even with the Martingale strategy, which aims to recover losses, the expected value of the bets remains unfavourable.

Moreover, in a casino setting, there are structural limitations that impede the effectiveness of the Martingale strategy. Most casinos impose limits on bet size. These limits prevent gamblers from doubling their bets indefinitely, even if they have boundless resources and time, thereby constraining the strategy’s potential for recovery.

THE DEVIL’S SHOOTING ROOM PARADOX

A parallel thought experiment known as the Devil’s Shooting Room Paradox adds an intriguing twist. In this scenario, a group of people enters a room where the Devil threatens to shoot everyone if he rolls a double-six. The Devil further states that over 90% of those who enter the room will be shot. Paradoxically, both statements can be true. Although the chance of any particular group being shot is only 1 in 36, the size of each subsequent group in this thought experiment is over ten times larger than the previous one. Thus, when considering the cumulative probability of being shot across multiple groups, it surpasses 90%.

Essentially, the Devil’s ability to continually usher in larger groups, each with a small probability of being shot, ultimately results in the majority of all the people entering the room being shot.

A key assumption underlying the Devil’s Shooting Room Paradox is the existence of an infinite supply of people. This assumption aligns with the concept of infinite wealth and resources often associated with Martingale-related paradoxes. Without a boundless supply of individuals to fill the room, the cumulative probability of over 90% cannot be definitively achieved.

The Devil’s Shooting Room Paradox serves in this way as another illustration of how probabilities and cumulative effects can lead to counterintuitive outcomes.

CONCLUSION: THE LIMITS OF A MARTINGALE STRATEGY

The Martingale strategy is based on chasing losses, but its expected value remains zero or negative due to the house edge. The strategy’s viability is further diminished by limitations on bet size in real-world casino scenarios. As such, the Martingale system cannot be considered a winning strategy in practical gambling situations. The Devil’s Shooting Room Paradox further demonstrates the complexities and counterintuitive outcomes that can arise when infinite numbers are assumed. Ultimately, a comprehensive understanding of these paradoxes provides valuable insights into the rationality of betting strategies and decision-making in the realm of gambling.