The Martingale strategy is one of the most famous methods in sports betting. Read on to find out what players can learn from the Martingale strategy.
First, this article is given with a warning. Whenever you find that someone is talking about Martingale in a positive light in the context of betting, the correct answer is to tell them that they are wrong and walk away from the conversation.
Martingale is a well known progressive financial betting plan that aggressively raises amounts after losses in an attempt to recover them. This kind of loss recovery is never seriously recommended. We have already written why this is so, including demonstrating a mathematical proof of why it is wrong on bookmaker resources.
For any money management plan where your perceived advantage over odds (positive or negative) is constant across all bets, it is not possible to change that advantage by changing the size of the bet. All you can do is change the distribution of risks and rewards. In the case of Martingale, you are trying to buy a large reward, but at the cost of a small probability of very, very high risk: a complete collapse.
With this in mind, we wanted to show you how you can use Martingale in a controlled way without risking the destruction it can cause. That being said, hopefully, this article demonstrates well how betting, in general, is always a balance of risk and reward. The more reward you want, the more risk you must take in order to achieve it.
How long will you last at roulette?
The simplest form of casino betting is flat staking, which uses the same amount for each bet. Let's bet a dollar on red for each wheel spin. What can we expect if we bet on 1000 wheel spins?
Our expected returns will match the binomial distribution shown below in Chart 1 . The blue distribution shows the range of possible outcomes assuming fair odds, while the red distribution also takes into account the 2.7% house edge for the zero sector:
Even without zero, our chances of seeing any noticeable accumulation of our bankroll are minimal. We only have a 5.7% chance of making $ 50. With the zero sector, our prospects are even worse (0.74%). Of course, this is counterbalanced by the equally small risk of losing $ 50. Even with the house edge, there is only a 23% chance of losing $ 50 and only 1% of losing $ 100.
Flat bets may be safe, but they will not contribute to any significant bankroll increases. Surely there must be a more fun way to spend your time gambling?
Martingale use and inevitable losses
The simplest version of the Martingale strategy doubles the bets after each bet loss at odds of about 2.00 before winning. The bet is then reset to the original amount and the progression starts again. The dangers of Martingale should be obvious to all but the most opinionated players, losing rounds are an inevitable consequence of the repeated play. The more you play, the more likely you are to have a long losing streak.
Roughly speaking, the expected longest losing sequence that you are likely to see in a sequence of n bets is the logarithm of n to the base of the odds divided by the odds minus one. For red bets, you probably see a losing sequence of three in eight bets, four in betting 16, five in betting 32, and so on. In a 1000-bet streak, your expected longest losing streak is between nine and ten bets.
Modeling Martingale Results
If the casino suddenly allows you to bet any amount, your expected profit from 1000 spins of the wheel is $ 500 on no zero roulette or $ 486 on zero. Of course, neither one nor the other is impossible. Most importantly, there will come a point when a losing streak will either destroy your funds or cause significant damage to them to make you lose confidence.
To deal with this, a smart approach is to define goals and set rules and limits for your game, simulating the probabilities of different outcomes, just as we did for flat bets. Let's consider the following scenario:
- Using $ 1 for the initial bet in any Martingale progression, we will aim to win $ 500 after 1000 spins of the wheel.
- We will limit the risk of going bankrupt at any time for 1000 bets to 50%.
- What is the maximum bankroll loss we can accept before we decide to abandon the game for points 1 and 2?
To answer this question, we can turn to Monte Carlo simulations . The table below shows the results of a series of 10,000 Monte Carlo simulations. For each simulated 1000 spin series, if the bankroll falls below a certain threshold, the game stops and the strategy fails. Otherwise, the game continues for 1000 wheel spins and the strategy is considered successful. Here the roulette wheel was considered without the zero sector:
Bankroll threshold |
Average bankroll win |
Average bankroll loss |
Bankruptcy |
Offer Rates |
-fifty |
503 |
-106 |
82% |
5.69 |
-one hundred |
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 |
fifty% |
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 |
Reformulated Bid Offer
Unsurprisingly, we see that the more we are willing to lose at any point in the series, the more likely we are to be successful in the end. A threshold of about minus $ 300 gives us about a 50% chance that we will get our $ 500 after 1000 spins of the wheel. At other times, we can expect an average loss of about $ 500. In fact, we've reformulated the bidding proposal: risk $ 500 to win $ 500, which gives us a bid ratio of about 2.00 (or 1: 1 in a fractional system).
Note that these bid ratios closely reflect the average win/loss ratio; in fact, this ratio is a measure of your real odds. For a wheel without zero, these two sets of odds must be the same.
If the threshold is instead at minus $ 1000, we can adjust to longer losing runs. Therefore, we fail less often and our chances of being offered are less (1.29). Of course, if you don't want to risk losing a total of $ 1,725, you can simply reduce the initial progression bet size. Using $ 0.29 will give you the opportunity to risk $ 500 to win $ 145 or so.
Compared to the more conservative flat betting strategy, there is now a much greater risk of much larger losses. However, this gives a much better chance of a much larger win. You may never lose $ 500 in 1000 roulette games by betting $ 1 per game, but you can never win $ 500 either. The likelihood of both is incredibly small.
Understanding casino and bookmaker margins
The introduction of a 2.7% casino edge using the zero sector really makes a difference. Chart 2 below compares the abandonment (or bankruptcy) rates between wheels with and without zero. With zero entered at the threshold of minus $ 300, you can expect to fail about 58% of the time.
To ensure that you still have 50% supply, you will need to increase your maximum allowable loss threshold to about minus $ 440. In this scenario, you risk losing an average of about $ 670 to win $ 486. Remember, since your expected win rate is now 48.6%, your expected profit for a successful streak will be around $ 486, not $ 500.
This ratio now implies a factor of 1.73, significantly less than the 2.00 supply factor, calculated from the probability of failure. In essence, this illustrates a loss in value. Please note that it is significantly higher than the casino margin. Your implied value is 1.73 divided by 2.00, or 0.865. This may seem like a high price to pay for using Martingale.
The expected value of the roulette bet is 36 out of 37 or 0.973. However, please note that this is repeated in over 1000 bets. With flat bets, you have about 20% chances of making some kind of profit, compared to 80% for some kind of loss. You can see this by comparing the areas under the orange curve in Chart 2 above to the left and right of the zero profit line.
Then your implied odds of success are 5.00 and your implied value is 2.00 divided by 5.00, or 0.40 relative to a zero-zero wheel.
In fact, your loss of value when using this managed Martingale strategy varies depending on your minimum bankroll loss threshold and your odds of being offered. The less likely you are to be offered (and the lower the chance of rejection), the less value you lose. For a threshold of minus $ 100, the odds of supply implied by the bounce rate are about 5.00 with zero compared to about 3.68 without it, which means a value of 0.74.
In contrast, with a threshold of minus $ 1000, the odds are about 1.4 and 1.29, respectively, which gives a value of 0.92. If you had used the $ 10,000 threshold, it would have been almost 0.99. This connection seems oddly reminiscent of the favorite betting bias.
Allocation of risks and rewards
We can visualize how using a guided martingale strategy like this modifies the bid offer, plotting the distribution of possible outcomes. Chart 3 below shows the distribution of 10,000 Monte Carlo simulation runs for the minus $ 300 bankroll threshold scenario using the no-zero roulette wheel. You can see how it is divided into distinctive zones of success or failure. Compare this to the original distribution from flat bets (shown in the dotted orange curve).
While this discussion of the Managed Martingale strategy deals with simple betting proposals (“red” or “black”), you can apply it to any odds and any betting market, including sports. All that is required is resizing the value of the Martingale progression. This is given by the factor divided by the factor minus one. Thus, for 3.00 odds, the bets after a loss increase by 1.5 times, and for 1.50 odds - three times.
Unsurprisingly, the lower the odds you bet on, the more effective the bet you have to risk to make sure you reformulate your offer as an even money bet. Obviously, betting on higher odds means longer losing streaks.
For example, a fair 5.00 bet means you have to risk about $ 800 to win around $ 800. In contrast, betting at odds of 1.50 results in an effective offer to risk $ 333 to win $ 333. Of course, you can always adjust the initial progression rate to accommodate this, as described earlier.
Do not try to do this in a casino (or a bookmaker's office)
This article is another way to demonstrate that betting is a game of risk and reward, and how money management can be used to balance and distribute your risks and rewards.
In sports betting, unlike casinos, there is room for positive expectations. If you are one of the few players who is truly a pro, you won't need to worry about Martingale or any progressive betting system at all. Just let the law of large numbers slowly work for you.
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