1. Martingale Gambling System
2. Reverse Martingale System

What is the Martingale betting system? The Martingale betting system is a mathematically-proven way to win casino games against the house. Each time you lose, you double your bet to ensure that your losses are earnt back. It’s most applicable to games with a 50% chance of winning, like the top games at roulette casinos. There are a huge number of casino gaming strategies that have been used over the years, but one of the most proven strategies is the Martingale system, the lucky finger system. Img source: chartattack.com. Martingale system is a system that is popular with fans of casino. This system is very popular because it offers many possibilities.

A martingale is any of a class of betting strategies that originated from and were popular in 18th-century France. The simplest of these strategies was designed for a game in which the gambler wins the stake if a coin comes up heads and loses it if the coin comes up tails. The strategy had the gambler double the bet after every loss, so that the first win would recover all previous losses plus win a profit equal to the original stake. The martingale strategy has been applied to roulette as well, as the probability of hitting either red or black is close to 50%.

Since a gambler with infinite wealth will, almost surely, eventually flip heads, the martingale betting strategy was seen as a sure thing by those who advocated it. None of the gamblers possessed infinite wealth, and the exponential growth of the bets would eventually bankrupt 'unlucky' gamblers who chose to use the martingale. The gambler usually wins a small net reward, thus appearing to have a sound strategy. However, the gambler's expected value does indeed remain zero (or less than zero) because the small probability that the gambler will suffer a catastrophic loss exactly balances with the expected gain. In a casino, the expected value is negative, due to the house's edge. The likelihood of catastrophic loss may not even be very small. The bet size rises exponentially. This, combined with the fact that strings of consecutive losses actually occur more often than common intuition suggests, can bankrupt a gambler quickly.

Intuitive analysis[edit]

The fundamental reason why all martingale-type betting systems fail is that no amount of information about the results of past bets can be used to predict the results of a future bet with accuracy better than chance. In mathematical terminology, this corresponds to the assumption that the win-loss outcomes of each bet are independent and identically distributed random variables, an assumption which is valid in many realistic situations. It follows from this assumption that the expected value of a series of bets is equal to the sum, over all bets that could potentially occur in the series, of the expected value of a potential bet times the probability that the player will make that bet. In most casino games, the expected value of any individual bet is negative, so the sum of many negative numbers will also always be negative.

The martingale strategy fails even with unbounded stopping time, as long as there is a limit on earnings or on the bets (which is also true in practice).^[1] It is only with unbounded wealth, bets and time that it could be argued that the martingale becomes a winning strategy.

Mathematical analysis[edit]

The impossibility of winning over the long run, given a limit of the size of bets or a limit in the size of one's bankroll or line of credit, is proven by the optional stopping theorem.^[1]

Martingale Gambling System

Mathematical analysis of a single round[edit]

Let one round be defined as a sequence of consecutive losses followed by either a win, or bankruptcy of the gambler. After a win, the gambler 'resets' and is considered to have started a new round. A continuous sequence of martingale bets can thus be partitioned into a sequence of independent rounds. Following is an analysis of the expected value of one round.

Let q be the probability of losing (e.g. for American double-zero roulette, it is 20/38 for a bet on black or red). Let B be the amount of the initial bet. Let n be the finite number of bets the gambler can afford to lose.

The probability that the gambler will lose all n bets is qⁿ. When all bets lose, the total loss is

${\displaystyle \sum _{i=1}^{n}B\cdot 2^{i-1}=B(2^n-1)}$

The probability the gambler does not lose all n bets is 1 − qⁿ. In all other cases, the gambler wins the initial bet (B.) Thus, the expected profit per round is

${\displaystyle (1-q^n)\cdot B-q^n\cdot B(2^n-1)=B(1-(2q{)}^n)}$

Whenever q > 1/2, the expression 1 − (2q)ⁿ < 0 for all n > 0. Thus, for all games where a gambler is more likely to lose than to win any given bet, that gambler is expected to lose money, on average, each round. Increasing the size of wager for each round per the martingale system only serves to increase the average loss.

Suppose a gambler has a 63 unit gambling bankroll. The gambler might bet 1 unit on the first spin. On each loss, the bet is doubled. Thus, taking k as the number of preceding consecutive losses, the player will always bet 2^k units.

With a win on any given spin, the gambler will net 1 unit over the total amount wagered to that point. Once this win is achieved, the gambler restarts the system with a 1 unit bet.

With losses on all of the first six spins, the gambler loses a total of 63 units. This exhausts the bankroll and the martingale cannot be continued.

In this example, the probability of losing the entire bankroll and being unable to continue the martingale is equal to the probability of 6 consecutive losses: (10/19)⁶ = 2.1256%. The probability of winning is equal to 1 minus the probability of losing 6 times: 1 − (10/19)⁶ = 97.8744%.

The expected amount won is (1 × 0.978744) = 0.978744.
The expected amount lost is (63 × 0.021256)= 1.339118.
Thus, the total expected value for each application of the betting system is (0.978744 − 1.339118) = −0.360374 .

In a unique circumstance, this strategy can make sense. Suppose the gambler possesses exactly 63 units but desperately needs a total of 64. Assuming q > 1/2 (it is a real casino) and he may only place bets at even odds, his best strategy is bold play: at each spin, he should bet the smallest amount such that if he wins he reaches his target immediately, and if he doesn't have enough for this, he should simply bet everything. Eventually he either goes bust or reaches his target. This strategy gives him a probability of 97.8744% of achieving the goal of winning one unit vs. a 2.1256% chance of losing all 63 units, and that is the best probability possible in this circumstance.^[2] However, bold play is not always the optimal strategy for having the biggest possible chance to increase an initial capital to some desired higher amount. If the gambler can bet arbitrarily small amounts at arbitrarily long odds (but still with the same expected loss of 1/19 of the stake at each bet), and can only place one bet at each spin, then there are strategies with above 98% chance of attaining his goal, and these use very timid play unless the gambler is close to losing all his capital, in which case he does switch to extremely bold play.^[3]

Alternative mathematical analysis[edit]

The previous analysis calculates expected value, but we can ask another question: what is the chance that one can play a casino game using the martingale strategy, and avoid the losing streak long enough to double one's bankroll.

As before, this depends on the likelihood of losing 6 roulette spins in a row assuming we are betting red/black or even/odd. Many gamblers believe that the chances of losing 6 in a row are remote, and that with a patient adherence to the strategy they will slowly increase their bankroll.

In reality, the odds of a streak of 6 losses in a row are much higher than many people intuitively believe. Psychological studies have shown that since people know that the odds of losing 6 times in a row out of 6 plays are low, they incorrectly assume that in a longer string of plays the odds are also very low. When people are asked to invent data representing 200 coin tosses, they often do not add streaks of more than 5 because they believe that these streaks are very unlikely.^[4] This intuitive belief is sometimes referred to as the representativeness heuristic.

Reverse Martingale System

Anti-martingale[edit]

This is also known as the reverse martingale. In a classic martingale betting style, gamblers increase bets after each loss in hopes that an eventual win will recover all previous losses. The anti-martingale approach instead increases bets after wins, while reducing them after a loss. The perception is that the gambler will benefit from a winning streak or a 'hot hand', while reducing losses while 'cold' or otherwise having a losing streak. As the single bets are independent from each other (and from the gambler's expectations), the concept of winning 'streaks' is merely an example of gambler's fallacy, and the anti-martingale strategy fails to make any money. If on the other hand, real-life stock returns are serially correlated (for instance due to economic cycles and delayed reaction to news of larger market participants), 'streaks' of wins or losses do happen more often and are longer than those under a purely random process, the anti-martingale strategy could theoretically apply and can be used in trading systems (as trend-following or 'doubling up'). (But see also dollar cost averaging.)

See also[edit]

References[edit]

1. ^ ^abMichael Mitzenmacher; Eli Upfal (2005), Probability and computing: randomized algorithms and probabilistic analysis, Cambridge University Press, p. 298, ISBN978-0-521-83540-4, archived from the original on October 13, 2015
2. ^Lester E. Dubins; Leonard J. Savage (1965), How to gamble if you must: inequalities for stochastic processes, McGraw Hill
3. ^Larry Shepp (2006), Bold play and the optimal policy for Vardi's casino, pp 150–156 in: Random Walk, Sequential Analysis and Related Topics, World Scientific
4. ^Martin, Frank A. (February 2009). 'What were the Odds of Having Such a Terrible Streak at the Casino?'(PDF). WizardOfOdds.com. Retrieved 31 March 2012.

Martingale betting system is one of the oldest systems, having started in the 18th century. Early descriptions of the system are found in books dedicated to casino games across Europe. However, the name of the system is credited to John Martingale, a British casino owner, who encouraged players to double up after every loss to recover their money when they won. However, this was long after the method was first used in betting. This method differs from others in that it requires players to spend more after a loss and specifically double their stake. The system has been used in both online and land-based casinos for the long term. In this review, you are going to learn how it works, and the way to use it on an online roulette betting session.

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How Does The Martingale System Work?

The Martingale roulette is a progressive negative system where the stake is only increased when you lose your bet. The idea behind it is to make the most out of the winning streaks by hoping that you would win the next round after a loss. In this case, it instructs players to double their bets when they lose but use the base stake when they win. In an ideal system, this would allow you to recover your cash and make some short-term profit. The system chases losses and uses it as a base to determine your next bet. However, it may not work if you are looking for long-term profits. The system is usually used for even money bets such as blackjack, roulette and baccarat.

For you to use the system effectively, you must have enough stake to keep doubling up until your next win. Otherwise, if you are unable to double up the stake, you will not recover the accumulated losses over several rounds. You may use one of the free Martingale system calculators on the internet to determine your wagering progression. However, they only represent the ideal situation, which may be different from the actual gameplay.

Pros of the Martingale Betting System:

• Easy to Use: The system is as easy as doubling up whenever you lose and using base stake when you are winning
• You can change your base stake: You can determine your base stake, depending on the amount of cash you have put as the base stake
• It allows you to generate short-term profits if you use the method correctly
• Applying the Martingale system makes the game interesting and enhances your ability to manage your bankroll

Cons of the Martingale Betting System:

• It does not improve the house edge: In contrast to the myth, using this system does not affect the return to player rates in the market.
• Bet rates can increase fast as you will be required to play with double the stake every time you lose
• You need unlimited bankroll for the method to be successful. Otherwise, you may not be able to retrieve all the money that you have staked.

Martingale strategy in Canadian Roulette Casino

You may use the systems effectively to play your favourite roulette variant. The rules for the betting system are easy to follow. Here is the step-to-step guide for using the Martingale roulette for your bets.

1. Start by choosing the roulette variant that you would like to play. Once it loads, determine the amount of stake that you would like to play with as the minimum stake. This system is best used to play on even-money bets. These bets are usually the outside bets, such as 1-18, 19-36, black, red, odd or even bets. These have an almost 50% chance of winning. This means that you can win the amount of cash that is equal to the bet amount in every spin.
2. Keep playing with the minimum bet until you lose.
3. After the loss, double the amount of your bet in the next spin. In case you win in this round, you will get your lost stake back and win some extra cash. However, if you keep losing your bet, continue doubling your bet until the next win.
4. When than win occurs, start with the base amount that you initially used for betting. Follow the procedure stated above for another cycle.

Martingale Variants

As usual, people adapt to the betting system as new betting methods come into being. You may use any variant to formulate Martingale roulette strategy, depending on your goals and specific requirements of the variant. Here are some of the commonest systems.

Anti-Martingale

The anti-Martingale system is the opposite of the Martingale system. In this case, you double your bet after every win. The system is also called reverse Martingale and is a positive progression system.

Triple Martingale System

The triple Martingale system a progression of the standard Martingale system. In this case, you triple your stake, instead of doubling it, when you lose a spin. The idea is to recover your stake and make an additional profit when you win.

Quadruple Martingale

This system is just like the original one, except that in this case, you will be betting four times your minimum stake when you lose. This is to increase your chances of earning huge profits with every win. The Martingale roulette calculator, in this case, would go like 1, 4, 14, 64, 256.

Super Martingale

This variant is less aggressive than those discussed above. However, you can ramp higher wins with the system than all the other systems. In this case, you double the bet after every loss. You also add one unit to give you higher profits if you are able to strike a lucky win. On the calculator, the profile would go like this; 1, 3, 7, 15, 31, 63.

Some people play a reverse of all of the above systems to increase the stake after wins rather than after losses. Other people mix the bet systems after every win to increase their odds of making a profit.