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A gambling strategy where the amount is raised until a person wins or becomes insolvent

A martingale is 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. Thus the strategy is an instantiation of the St. 🔔 Petersburg paradox.

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 has infinite wealth, and the exponential growth of the bets can bankrupt unlucky gamblers who choose 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 is higher 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 ]

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]

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

∑ i = 1 n B ⋅ 2 i − 1 = B ( 2 🔔 n − 1 ) {\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

( 1 🔔 − q n ) ⋅ B − q n ⋅ B ( 2 n − 1 ) = B ( 🔔 1 − ( 2 q ) n ) {\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 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 does not 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. In fact, while the chance of 🔔 losing 6 times in a row in 6 plays is a relatively low 1.8% on a single-zero wheel, the probability 🔔 of losing 6 times in a row (i.e. encountering a streak of 6 losses) at some point during a string 🔔 of 200 plays is approximately 84%. Even if the gambler can tolerate betting ~1,000 times their original bet, a streak 🔔 of 10 losses in a row has an ~11% chance of occurring in a string of 200 plays. Such a 🔔 loss streak would likely wipe out the bettor, as 10 consecutive losses using the martingale strategy means a loss of 🔔 1,023x the original bet.

These unintuitively risky probabilities raise the bankroll requirement for "safe" long-term martingale betting to infeasibly high numbers. 🔔 To have an under 10% chance of failing to survive a long loss streak during 5,000 plays, the bettor must 🔔 have enough to double their bets for 15 losses. This means the bettor must have over 65,500 (2^15-1 for their 🔔 15 losses and 2^15 for their 16th streak-ending winning bet) times their original bet size. Thus, a player making 10 🔔 unit bets would want to have over 655,000 units in their bankroll (and still have a ~5.5% chance of losing 🔔 it all during 5,000 plays).

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.

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"). This concept is similar to that used in momentum 🔔 investing and some technical analysis investing strategies.