- Applying the Martingale strategy to even money bets
- Risk and reward using the Martingale strategy
- Modelling Martingale strategy outcomes
- Understanding the effects of a house edge
First things first, this article comes with a caution. Whenever someone speaks highly of Martingale in the realm of betting, the appropriate action is to correct them and walk away.
Martingale is a notorious progressive betting system that escalates stakes after losses in hopes of recouping them. Such a loss recovery approach should never be taken seriously. I have frequently written about its flaws, including the mathematical evidence proving its inadequacies on Betting Resources.
When dealing with any money management strategy where your perceived edge over the odds (be it positive or negative) remains consistent across all bets, it’s impossible to alter this advantage by merely adjusting stake sizes. All that changes is the risk-reward balance. In Martingale’s case, you’re trading a substantial risk for the possibility of a high reward: potentially catastrophic losses.
Bearing this in mind, I’ve opted to explore Martingale in a controlled manner to illustrate a responsible usage without the usual catastrophic risks. Through this exploration, I aim to highlight how betting is always a trade-off between risk and reward. The greater the reward sought, the higher the risk one must embrace.
How to get the most out of a holiday in Vegas
I've always dreamt of visiting a Las Vegas casino. It’s well-known that casinos primarily offer games with negative expected returns. While techniques like card counting may provide an edge over the Blackjack table, such strategies will likely get you swiftly kicked out.
For instance, all Roulette proposition bets are destined to lose over time, governed by mathematical rules where the house advantage is dictated by the zero. Setting aside tales of biased wheels observed over tens of thousands of spins, it’s safe to assume you won’t maintain a profitable edge.
Acknowledging this, I’ve pondered the optimal approach to maximizing both enjoyment and duration of play during a holiday there. This contemplation will focus solely on even-money bets at the Roulette table, such as odd or even and red or black.
How long will you last with flat staking?
Flat staking, where you wager the same amount on every bet, represents the simplest betting strategy. Suppose we bet a dollar on each spin of the wheel. What might occur over 1,000 spins?
The expected results adhere to a binomial distribution shown below. The blue curve represents outcomes under fair conditions, while the red curve accounts for the 2.7% house advantage on a single zero Roulette wheel:
Even on a fair wheel, the likelihood of significantly increasing our bankroll is slim, at just a 5.7% chance of winning $50. Considering the house edge, the chances worsen (0.74%). However, the risk of losing $50 is comparably modest, with only a 23% chance, and just a 1% chance of losing $100.
While flat staking is safer, it doesn’t offer substantial rewards. Surely, there's a more thrilling way to spend a vacation in Vegas?
Using Martingale and inevitable losses
The basic form of Martingale involves doubling the stake after each loss on an even-money bet until a win is secured, then resetting the stake.
The risks associated with Martingale are clear to all but the overly confident bettors, as losing streaks are a natural outcome of frequent play. The more you bet, the more likely you are to experience extended losses.
As an approximate guide, the expected longest losing streak in a series of n bets can be estimated using the logarithm of n divided by the odds minus one. For even-money bets, you might encounter three consecutive losses within eight bets, four within 16 bets, five within 32 bets, and so forth. In a series of 1,000 bets, you're likely to face a losing streak of about nine to ten bets. 7x7Bets has previously published a comprehensive article on this topic.
Modelling Martingale outcomes
Assuming unlimited funds and an exceedingly accommodating casino that allows any bet size, your projected profit from 1,000 spins would be $500 on a fair wheel, or $486 with a single zero. Realistically, neither scenario is feasible. More crucially, a long losing streak could either deplete your funds or shake your confidence sufficiently to make you quit.
The logical approach involves setting clear objectives and boundaries for your betting while simulating various outcomes, similar to what we did for flat stakes. Consider this scenario:
Starting with a $1 stake for each Martingale cycle, aim to win $500 over 1,000 spins.
We will limit the risk of bankruptcy during the 1,000 plays to 50%.
What is the maximum acceptable loss before deciding to stop, given the above conditions?
This query can be addressed using a Monte Carlo simulation. The following table displays outcomes from 10,000 simulations. If the bankroll dips below a set threshold during the series, play is halted, and the strategy is deemed unsuccessful. Otherwise, play continues through 1,000 spins, marking the strategy as successful. Here, the roulette wheel is presumed fair, with no zero:
Bankroll threshold | Average bankroll win | Average bankroll loss | Bankruptcies | Proposition odds |
-50 | 503 | -106 | 82% | 5.69 |
-100 | 501 | -188 | 73% | 3.68 |
-150 | 500 | -270 | 65% | 2.82 |
-200 | 500 | -341 | 59% | 2.47 |
-250 | 500 | -443 | 54% | 2.15 |
-300 | 499 | -496 | 50% | 1.99 |
-400 | 498 | -608 | 45% | 1.81 |
-500 | 498 | -774 | 40% | 1.65 |
-750 | 498 | -1088 | 31% | 1.45 |
-1000 | 496 | -1725 | 22% | 1.29 |
A reformulated betting proposition
It's apparent that the more loss we're willing to tolerate at any point in the series, the higher our chances of ultimately reaching our $500 goal. A loss limit of about $300 provides a 50% probability of success after 1,000 spins. Conversely, half the time, the average loss approaches $500. Essentially, we've redefined the betting scenario: risk $500 to gain $500, translating to odds of about 2.00 (or even odds in fractional terms).
These odds closely mirror the ratio of average gains to losses, effectively representing your true odds. For a fair wheel without a zero, these odds should match.
The element of randomness in sports betting Setting the threshold at minus $1,000 allows for longer losing streaks, reducing the likelihood of failure and thus offering shorter odds (1.29). However, if you're reluctant to risk a total loss of $1,725, you could reduce the initial stake size in the progression. A stake of $0.29 would then equate to a bet of risking $500 to win approximately $145.
Compared to the more conservative flat staking method, this approach significantly increases the risk of a larger loss but also raises the potential for a larger win. You may not lose $500 in 1,000 roulette spins betting $1 per spin, but you’re also unlikely to win $500. The chances of either scenario are exceedingly low.
Understanding the effects of a house edge
Introducing a 2.7% house edge through the zero alters the dynamics. The chart below compares the bankruptcy rates between wheels with and without a single zero. With the introduction of a zero at a threshold of minus $300, you can expect to fail approximately 58% of the time.
To maintain a 50% chance of success, you would need to increase your maximum acceptable loss threshold to about minus $440. Under such conditions, you're facing an average loss of about $670 to win $486. Remember, since your bet win expectation is now 48.6%, your actual profit for a successful series would be closer to $486, not $500.
This adjustment suggests odds of 1.73, significantly lower than the initial 2.00 calculated by the failure probability. This discrepancy illustrates a loss in value, far exceeding the house margin. Your implied value ratio is 1.73 divided by 2.00, or 0.865, a steep price for employing Martingale.
What distinguishes winning from losing bettors? The expected value for an even-money roulette bet over 1,000 bets with flat stakes gives you around a 20% chance of turning a profit, versus an 80% chance of a loss. This is evident from the areas under the orange curve in the binomial chart, to the left and right of the zero-profit line.
Your implied odds of success are then 5.00, and your value ratio is 2.00 divided by 5.00, or 0.40, relative to a fair wheel without a zero.
In reality, your loss of value using this managed Martingale strategy varies depending on your minimum bankroll loss threshold and your proposition odds. The shorter the odds (and the lower the failure rate), the less value you lose. For a threshold of minus $100, the implied odds by the failure rate are about 5.00 with a single zero wheel, compared to about 3.68 without, implying a value of 0.74.
Conversely, with a threshold of minus $1,000, the odds are approximately 1.4 and 1.29 respectively, suggesting a value of 0.92. If the threshold were $10,000, the value would be nearly 0.99. This relationship is curiously reminiscent of the favorite–longshot bias.
The distribution of risks and rewards
We can depict how a managed Martingale strategy like this alters the betting scenario by illustrating the distribution of potential outcomes. The chart below displays the distribution from the 10,000 Monte Carlo simulations for the minus $300 bankroll threshold scenario on a fair roulette wheel. It shows a clear division between zones of success or failure, contrasting sharply with the original distribution from flat staking (shown in the dotted orange curve).
What happens when you change the odds?
While this discussion revolves around simple even-money bets, the principles of Martingale can be applied to any odds and any betting market, including sports. Adjustments to the size of the Martingale progression are calculated by the formula: odds divided by odds minus one. Thus, for odds of 3.00, stakes after losses increase by a factor of 1.5; for odds of 1.50, by a factor of three.
Predictably, the longer the odds you bet on, the larger the stake you must risk to reformulate your bet as an even-money proposition. For example, betting at fair odds of 5.00 means you would need to risk about $800 to potentially win $800. Conversely, betting at odds of 1.50 results in an effective proposition of risking $333 to win $333. However, you can always adjust the initial progression stake size to account for this, as discussed earlier.
Now you know what can the Martingale strategy teach us about betting and how to apply it into money betting. Sign Up Now or click HERE to play at 7x7Bets, the most reliable and trustworthy online casino in India. Don't forget to claim your withdrawable real money welcome deposit bonus, weekly cashback bonus and referral bonus!
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