The Martingale strategy, or “catch-up”, is the most common in betting. It has a lot of flaws, and yet it can be useful to players, at least from the point of view of the mathematical principles applied here. What she can teach, we will investigate in this material.

If you hear in a conversation with an “experienced” gambler that he regularly wins decent sums using the Martingale strategy, it is better to end the conversation right away – it is pointless to continue it.

Martingale is perhaps the most aggressive financial betting strategy, which implies an increase in the next bet amount exponentially with each subsequent loss. We will definitely not recommend it to anyone.

As with every method of account management in betting, where the perceived advantage in terms of odds remains the same for all bets, it is impossible to change the advantage by changing the size of the bets within this strategy. You can only change the distribution of risks and rewards. By using the catch-up principle, we are trying to get a large reward, but with a small probability of a catastrophic outcome – a complete bankroll zero.

With this in mind, let’s try to experiment a bit with the Martingale strategy to show how you can use it in a controlled manner without risking colossal losses. Perhaps this article will clearly demonstrate that there is always a balance between risk and reward in betting. The more reward you want to receive, the more you will have to risk your own funds.

## Something about the casino

As you know, casinos offer only games with negative predicted outcomes. Counting cards can give you the edge in blackjack, but you will most likely be quickly blacklisted.

Let’s say that absolutely all roulette bets are losing offers, since they are based on mathematical principles, according to which “zero” gives the casino an advantage. Even if you forget the stories about rigged roulettes, tested by teams of players tens of thousands of times, it can be assumed that you have no advantageous advantage over the casino.

But what method can provide the longest and most interesting casino game, for example, while on vacation. To find the answer to this question, we will only consider one-to-one bets on the roulette wheel, including odd, even, red and black squares.

## Fixed rates

One of the simplest strategies is to use fixed rates, which always remain the same. Let’s say we are betting a dollar on each spin of the roulette wheel. What can you expect by betting on 1000 spins in a row?

The expected profit corresponds to the binomial distribution. Even with a fair wheel, our chances of winning substantially are minimal. The probability of winning $ 50 is only 5.7%. And if we take into account the advantage of the casino, our prospects become completely disappointing (0.74%). Of course, this is offset by the equally small risk of losing $ 50. Even with the casino edge, the probability of losing $ 50 is 23% and the probability of losing $ 100 is 1%.

In general, flat rates are safe, but they will not generate significant returns. It is unlikely that you will like this pastime.

## Unavoidable losses

The simplest Martingale strategy is to double your bet “one to one” after each loss until you win. Then the original bet is restored and the sequence starts over. The dangers of using martingale should be obvious to everyone, but even for the most opinionated players, losing streaks are an inevitable consequence of multiple games. The more you play, the more likely you are losing streak and the risk of losing your bankroll.

The estimated length of the longest losing streak after placing n bets is calculated as the logarithm of n with a base equal to the odds divided by the odds minus one. For one-to-one bets, expect three consecutive losses for eight bets, four for 16 bets, five for 32 bets, and so on. In a 1000-bet series, the longest losing streak should be 9-10 bets. It is unlikely that anyone will like it.

## Modeling possible outcomes

Assuming we have infinite funds or we are playing in a very generous casino where any amount can be wagered, the expected profit after 1000 roulette spins is $ 500 in the case of fair roulette or $ 486 taking into account the casino edge from one zero “. Of course, both of these cases are impossible. Most importantly, sooner or later a losing streak will lead to either complete bankruptcy or large enough losses to shatter your patience and leave the game.

How do you overcome these limitations? To begin with, it makes sense to define goals, as well as set the rules and limits of the game, simulating the likelihood of different outcomes, as we did with fixed rates. Consider the following scenario:

- Using a martingale progression with an initial bet of $ 1, we will try to win $ 500 in 1000 roulette spins;
- We will limit the risk of bankruptcy in any of the 1000 games to 50 percent.
- What is the maximum allowable amount of losses at which we can continue playing, taking into account points 1 and 2?

Let’s try to answer this question using the Monte Carlo model . The Table below shows the results of a Monte Carlo simulation run 10,000 times. In each considered series of 1000 roulette spins, if the bankroll falls below the specified amount, the game stops and the strategy is considered unsuccessful. Otherwise, the game continues up to 1000 roulette spins and the strategy is considered successful. We are supposed to play fair non-zero roulette and thus eliminate the additional advantage of the casino.

## Other bid offer

The more we are ready to lose at one point or another in the series, the more likely it is to succeed in the end. With a threshold amount of about –300 dollars, the probability of winning 500 dollars per 1000 roulette spins is approximately 50%. In the remaining half of the cases, an average loss of about $ 500 is expected. Basically, we have reformulated the bet proposal: we risk losing $ 500 to win $ 500, which is about 2.00 odds.

These odds are close to the average win-loss ratio. Essentially, this ratio is an indication of the actual ratio. On fair roulette without zero, these two series of odds must be the same.

However, if the threshold is $ 1000, we can afford longer losing streaks. Consequently, bankruptcy occurs less frequently and the supply ratio drops to 1.29. Of course, if you don’t want to risk losing $ 1,725, you can simply decrease your initial bet size in progression. With an initial bet of $ 0.29, we are asked to risk losing $ 500 to win about $ 145. An extremely dubious proposal, agree.

Compared to the more conservative fixed-betting strategy, there is a much greater risk of a big loss in this case. But the chance of a big win is also significantly increased. You won’t lose $ 500 for 1000 $ 1 bets, but you won’t win $ 500 either. The probabilities of both outcomes tend to zero.

## Impact of the casino edge

Taking into account the advantage of the casino of 2.7% due to the use of “zero”, the situation is changing. To keep the probability of winning at the 50% level, you will need to change the threshold amount of acceptable losses by –440 dollars. In this case, you risk losing approximately $ 670 to win $ 486. Remember that since the expected win rate is now 48.6%, the projected profit on a successful streak will be approximately $ 486, not $ 500.

This ratio implies a coefficient of 1.73, which is significantly lower than the coefficient of 2.00 calculated on the likelihood of failure. In essence, this illustrates a loss of profit. Please note that this ratio is significantly higher than the casino margin. The implied profit is 1.73 / 2.00, or 0.865. It looks like there is a high price to pay for using Martingale. And the risks are not justified at all.

The expected profit for one-to-one roulette is 36 to 37, or 0.973. But what happens if you repeat this for 1000 bets? At fixed rates, the probability of any profit is 20% and the probability of any loss is 80%. Thus, the implied success rate is 5.00 and the predicted profit is 2.00 / 5.00 (that is, 0.40) on fair non-zero roulette.

The loss of profit when using such a controlled Martingale strategy depends on the loss threshold and the odds offered. The lower the supply ratio and the failure rate, the less profit loss. With a threshold of $ 100, the supply ratio implied by the failure rate is about 5.00 on roulette with one zero and 3.68 on roulette without zero, which is 0.74 profit.

But with a threshold of $ 1000, the odds are 1.4 and 1.29, respectively, which implies a profit of 0.92. At the $ 10,000 threshold, the profit approaches 0.99. This attitude is suspiciously reminiscent of the bias effect in judging favorites and outsiders .

## What happens if you change the odds?

And while we looked at this supervised Martingale strategy using simple one-to-one sentences, it can be applied to any odds in any betting market, including sports betting. It is enough to change the Martingale progression multiplier. To determine it, divide the coefficient by the coefficient minus one. Thus, with the odds of 3.00 after the loss, the bet increases by 1.5 times. At a factor of 1.50, the multiplier is three.

The higher the odds, the more the actual amount you will have to risk to reformulate the offer as a one-to-one bet. Naturally, the higher the odds, the longer the losing streak.

So, by placing bets at fair odds of 5.00, you will have to risk losing $ 800 to win $ 800. On the other hand, at odds of 1.50, it is suggested to risk losing $ 333 to win $ 333. Of course, you can always change your initial bet.

## Conclusion

We have considered an interesting experiment, and we do not want to recommend the Martingale strategy to anyone, like any other progressive betting system. However, this strategy served as yet another illustration of the fact that betting is a game of risk and reward.

In sports betting, unlike casinos, there is the possibility of positive expectations. If you happen to be one of the lucky owners of this advantage, then you don’t need any of the financial strategies at all. In this case, you will become a professional tipper, and sports betting will bring you regular profit.