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 if it 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.

Since a gambler will almost surely eventually flip heads, the martingale betting strategy is certain to make money for the gambler provided they have infinite wealth and there is no limit on money earned in a single bet. However, no gambler possess infinite wealth, and the exponential growth of the bets can bankrupt unlucky gamblers who chose to use the martingale, causing a catastrophic loss. Despite the fact that the gambler usually wins a small net reward, thus appearing to have a sound strategy, the gambler's expected value remains 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. Additionally, as the likelihood of a string of consecutive losses occurs more often than common intuition suggests, martingale strategies can bankrupt a gambler quickly.

The martingale strategy has also been applied to roulette, as the probability of hitting either red or black is close to 50%.

Intuitive analysis[edit]

• Roulette is one of the easiest games to play and understand in the casino. As usual the easier a game is to understand the greater the house edge, and roulette is no exception. If you are looking for a easy to understand and slow paced table game, and are willing to sacrifice on the house edge, then you may like roulette.
• The Labouchere method is similar to the Martingale but it moves along in a seemingly better fashion and that’s why many roulette players use it. #3 - THE D’ALEMBERT The D'Alembert method is also known as the Gambler’s Fallacy because its basis is an incorrect idea – that the universe evens things out.
• Best Roulette Strategy – Roulette Betting Systems Explained Being one of the most famous casino games of all time, roulette has been subject to a lot of analysis and odd calculations. The game looks simple enough and has the potential of winning you lots of money with just a single spin of the wheel.
• The real trick to roulette is to get lucky, then quit while you’re ahead. Every spin has a 1 in 37 (for European) or 1 in 38 (for American) chance of landing in a given pocket. Each spin is an individual trial with its own odds but, the longer you play, the more likely it is that you’ll fall into the probabilities that structure roulette.

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]

Essentially, Roulette is all about odds, simple physics, and statistics. The word roulette means “little wheel”, and therefore it shouldn’t surprise you that the spinning roulette wheel was invented by a well-known gambler and math wizard. For the next hundred years or so, the early Roulette wheel remained untouched – until Francois.

However, without these limits, the martingale betting strategy is certain to make money for the gambler because the chance of at least one coin flip coming up heads approaches one as the number of coin flips approaches infinity.

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 10/19 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.

Anti-martingale[edit]

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, also known as the reverse martingale, 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.

So, you enjoy taking the wheel for a spin. We totally get it – it’s no secret that roulette is one of the most exciting and entertaining casino games to be found on the floor. Unfortunately, while it is thrilling, the odds for players are some of the worst of any game. Don’t despair, though – Planet 7 is here to break down the five most common roulette strategies that players around the world have been using to rake in the chips for years. While we recommend using all of these equally, it’s your job to read on and decide which method is the best roulette strategy for you.

Without further ado, let’s break down the first and probably most common roulette winning strategy: the Martingale Strategy.

The Martingale Roulette Strategy

When you see that guy at the roulette table expertly throwing the chips down, he’s most likely using the Martingale Strategy. This method depends on doubling your bets after you take a loss, with the goal of recouping all previous losses and gaining a small profit. In other words, if you place your wagers only on a single color and continue doubling them until you win, you recover your losses. This is provided you keep doubling losing bets.

Martingale Strategy Steps:

1. Find a table with a small minimum bet and a high maximum bet. Starting small is essential, because you want the ability to double your losing bets as much as possible.
2. Place a small wager on black or red; even or odd; or 1-18 or 19-36.
3. If you win, keep the winnings and bet the same small wager again. You could walk away now if you want to, but there isn’t much of a difference between leaving with $2 or $1, though both are better than nothing.
4. If you lose, double your original wager and put it on the same bet again. For example, if you lost $1 on red, place $2 on red again.
5. If you win the second wager, keep your winnings and wager the original small bet – you’ve recouped your losses and can go back to a smaller amount.
6. If you lose again, double the wager and try again.
7. Repeat this process until all the money is gone or you reach the max bet at the table.

Like any gambling strategy, there are some disadvantages to the Martingale. Watch out for that gambler’s fallacy – just because one color has won 100 times in a row doesn’t mean that the other color is more likely to appear on the next spin. Roulette spins have a chance of a little less than 50/50 because of the 0 and 00. In addition, after several consecutive losses you may reach the max bet or run out of money – at this point, you’re in the red whether you win or not. For this roulette winning strategy to work, you need to make larger bets or win in order to recoup losses, and if you can’t do either then you want to walk away.

The Reverse Martingale Roulette Strategy

As might be expected given the name, this method of roulette gameplay is the opposite of the Martingale Strategy. Instead of raising your bets when you lose, the point is to increase them when you win and lower them after losses. The idea is that you’ll capitalize on hot streaks and keep your losses to a minimum during rough patches.

Reverse Martingale Strategy Steps:

1. Find a roulette table with a high max bet and a small minimum.
2. Bet a small amount on black or red; even or odd; or 1-18 or 19-36.
3. Keep your wager on the same spot on the table until you hit it. If you keep losing, keep betting a very small amount.
4. When you hit your bet and win, double your bet on the same spot for the next round.
5. If you keep winning, keep doubling your bet.
6. If you lose, move back down to your original bet.

Obviously, the biggest disadvantage to this roulette winning strategy has everything to do with timing. The Reverse Martingale Strategy is really risky because as soon as you lose, you lose your entire earnings. Unfortunately, for this one to work really well you need to hit a hot streak and quit before you lose – something that many gamblers aren’t very skilled at doing.

The D’Alembert Roulette Strategy

For those looking for something a little safer than the Martingale or Reverse Martingale roulette strategies, the D’Alembert Strategy is a perfect alternative. This simple method of gameplay is accomplished by increasing and decreasing bets based by one, which is much safer than doubling.

D’Alembert Strategy Steps:

1. Like the previous roulette strategies, place a small starting wager on black or red; even or odd; or 1-18 or 19-36.
2. Increase your wager by one after a loss, and decrease it by one after a win.
3. Walk away when you’ve had at least as many wins as losses. If you’re on a losing streak, stick tight until the winning picks up and equals your losses. If you’re riding high on a winning streak, continue playing until your wins equal your total losses.
4. When you have an equal number of wins and losses, pick up your winnings and walk away.

If you come out even on total number of wins and losses, using the D’Alembert Strategy will put you in the black. Here’s an example: say you put down a bet of $10 on red. You lose, so you bet $11 on red again. You lose again, so your wager goes up to $12. You win, so you go back down to $11 and win again. You pick up your winnings and walk away. You lost two wagers and won two wagers, so the math comes out like this: – 10 – 11 + 12 + 11 = +2. The disadvantage to this strategy is simply keeping track of your number of wins and losses.

The Fibonacci Roulette Strategy

Leonardo of Pisa, also known as Fibonacci, was a famous Italian mathematician who wrote about a specific series of numbers in the early 1200’s. While the series was around over a thousand years before Fibonacci, the series was dubbed the “Fibonacci sequence” in the nineteenth century. Basically, the sequence is characterized by the fact that every number after the first two is the sum of the two preceding ones. So, the Fibonacci go like this: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc.

When it comes to roulette strategies, the Fibonacci Strategy involves betting by adding the last two bets together. With this method, you can leave with a profit even if you lost more games than you won.

Fibonacci Strategy Steps:

1. Start with a small bet on black or red; even or odd; or 1-18 or 19-36.
2. If you lose, increase your bet by going down the sequence. If you started with a bet of $1 and you lose, bet another $1. If you lose the second bet, increase to $2. ($1 + $1 = $2)
3. When you win, move back two numbers in the sequence and bet that amount.
4. Obviously try to walk away when you’re in the black and not the red – however, don’t rely on your number of wins versus losses to tell you whether you’re ahead, because you could be in the black even if you’ve lost more than won.

An example of how you might bet using the Fibonacci Strategy:

• Bet red, $1 – lose
• Bet red, $1 – lose
• Bet red, $2 – lose
• Bet red, $3 – lose
• Bet red, $5 – win
• Bet red, $2 – lose
• Bet red, $3 – win
• Bet red, $1 – win
• Bet red, $1 – win

So, the total would look like this: – 1 – 1 – 2 – 3 + 5 – 2 + 3 + 1 + 1 = +1

Even though you lost five bets and won only four, you’re still ahead by one. The biggest disadvantage with this roulette winning strategy is that the further you fall down the sequence, the more money you lose. You need to strike quick or risk losing more money than you bargained for.

The James Bond Roulette Strategy

Discover the roulette strategy that Ian Fleming came up with and his famous character used at the table. Fleming told folks that his “foolproof” method could win you the “price of a good dinner” if you used it every night. For this roulette winning strategy, players need at least $200.

James Bond Strategy Steps:

Roulette Method Doubling Calculator

1. Place $140 on the high numbers (19-36).
2. Bet $50 on the numbers 13 through 18.
3. Place $10 on 0 for insurance.

If luck is really not on your side and any number between 1 and 12 shows up, you just lost $200 and should start utilizing the Martingale Strategy that we outlined above. However, if any of the bets comes through, you’ve earned a pretty sweet profit. If 19-36 pockets, you win $80; if 13-18 shows up, you win $100; and if 0 shows up, you’re up $160.

The biggest disadvantage to this one is obviously the risk associated with a number between 1 and 12 showing up. But, who said James Bond wasn’t a risk-taker? Would we expect anything less from Agent 007? I don’t think so.

Roulette Doubling Strategy

Well, there you have it folks – the five best roulette strategies. Which one is right for you? Unfortunately, we can’t answer that question for you – the only way you’ll discover your roulette winning strategy is by trying them all out for yourself. Luckily for you, our real money online casino has an instant play mode that allows players to test all these methods and more for absolutely no risk. All you need to start playing for free is a username and a password – no deposit or download is required. Head over to Planet 7 and play roulette online for real money today!