Casino Roulette Martingale
- The Martingale Roulette system is the most well-known betting strategy when it comes to even odds bets. It is used in Craps as well as in Roulette. For the most part, it is better known for being used as a Roulette strategy. We take an in-depth view at the parole system letting you know how.
- The Philosophy Behind the Martingale Roulette Strategy The Martingale system is well-known for its simplicity and ease of use. In a way, that could be the reason why this betting concept has such a long-lasting presence in the casino world. According to experts, the Martingale was first used in 18th century France.
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%.
There is, perhaps, no faster, easier way to implement the Martingale betting system than in the casino game of roulette. With a simple betting strategy of picking just black or red, then doubling up if you lose, it appears that using the Martingale system at the roulette table is a pretty good way to win. First off, if you want to use the Martingale Betting System when playing roulette, you need to have a casino to play at. Bovada Casino is one of the top casinos out there and they have a good selection of roulette games to play. Plus, they do accept American players, so Bodog Casino is definitely one of the top places to play.
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]
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
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
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 2k 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.
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]
- ^ 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
- ^Lester E. Dubins; Leonard J. Savage (1965), How to gamble if you must: inequalities for stochastic processes, McGraw Hill
- ^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
- ^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.
The experienced roulette player will almost never apply a traditional betting system in the way it is depicted in various sources. The reason for this lies in the fact that all systems have limitations and potential traps, which have to be overcome. We should note that consistent winning is usually achieved with certain adjustments in the methods of play, with additional fine-tuning of system play.
It became clear in the prior article, that the traditional Martingale system and its variations have a robust mathematical logic. However, the traditional system is not immune to drawbacks, which makes it hazardous to play.
One of these drawbacks is the restrained number of opportunities a player has to double his/her losing bets until the required wager grows far too much and reaches the table limit. Another disadvantage lies in that the amount, which is exposed to risk every time a string of losses occurs, grows considerably.
The ”Illusionary Bets”
However, there is a way to cushion the above-mentioned drawbacks and it includes the application of the so-called ”illusionary bets”. The latter represent zero-value bets, which are never placed for real. They are also known as ”null bets”, as a player does not have any money exposed to risk.
If we presume that the minimum bet is 5 units and the table maximum is 1000 units, the following series of losses may be expected before the limit is reached:
5 10 20 40 80 160 320 640
In case the eighth bet loses, it will be quite an unpleasant situation and it may occur once every 170 ball spins, as we said in the previous article. Thus, we need to overcome such a situation.
Let us assume that the moment we take our seat at the roulette table, six even-money outside bets are to be chosen from. During the prior wheel spin, at least two or three of them turned out to be losers. In case a bet is made on one of the prior losers, this would mean it is the second bet in a potential series of losses. This way the first bet was an illusionary one, a null bet. It just extended the theoretical series of losses to nine bets, or a bit longer than before. We now have the following string:
0 5 10 20 40 80 160 320 640
Such a series of losses (9) may occur on average once every 323 spins. Thus, there has been an improvement in the situation.
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We may not make the first bet until one of the even-money bets lost two or three consecutive times. Let us presume that we abstain from placing the first bet until there have been three losses in a row. The theoretical series of losses will now include three null bets and will be as follows:
0 0 0 5 10 20 40 80 160 320 640
Such a series of losses (11) may occur on average once every 1 165 wheel spins, or it would be equal to 12-14 hours of play.
Playing the Martingale System
The D’Alembert System
The Labouchere System
Playing the Labouchere System
Roulette Inside Bet System
The ”CYA Bets”
Cunning players have come up with another bet modification, known as the CYA bet. It represents a bet covering past losses and does not provide a player with an opportunity to score gains. The CYA bet is a more logical action than a double bet because, after a string of several consecutive losses, a player will look for compensation instead of scoring a meager 5-unit gain.
When playing the Martingale system, a player should profit on winning bets and after a series of no more than three losses. Once a fourth or a fifth consecutive loss has been registered, he/she will need to shift its objective from making a profit to compensating for the string of losses. In doing so, a player will have a complimentary wheel spin before the table limit is reached.
If we have three null bets and a CYA bet after the fourth loss, a table limit of 1000 units will not be encountered until the 12th loss. Or, we have the following series:
| ”CYA Bets” | ||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Bet Number: | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| Bet Amount: | 0 | 0 | 0 | 5 | 5 | 10 | 20 | 40 | 80 | 160 | 320 | 640 |
The probability of twelve consecutive losses occurring is 1/2213 (once every 2213 ball spins). However, we should not forget that it may occur on the very next twelve ball spins.
In the past few years, a number of casinos have raised their maximum outside bet limits to $10 000 or even more. Given that figure, if we have three null bets and a CYA bet after the fourth loss, we would be able to double our losing bets 16 times. The probability of sixteen consecutive losses occurring is 1/28 844 (once every 28 844 ball spins). However, it may occur on the very next sixteen ball spins.
The methods we discussed above are valid for the traditional Martingale system only. As far as the grand and the reverse variations of the Martingale are concerned, average players and, especially beginners, should avoid using them, as there is no certain way to cushion the high levels of risk.