A gambling strategy where the amount is raised until a person wins or becomes insolvent
A martingale is a class of 7️⃣ betting strategies that originated from and were popular in 18th-century France. The simplest of these strategies was designed for a 7️⃣ game in which the gambler wins the stake if a coin comes up heads and loses if it comes up 7️⃣ tails. The strategy had the gambler double the bet after every loss, so that the first win would recover all 7️⃣ previous losses plus win a profit equal to the original stake. Thus the strategy is an instantiation of the St. 7️⃣ Petersburg paradox.
Since a gambler will almost surely eventually flip heads, the martingale betting strategy is certain to make money for 7️⃣ the gambler provided they have infinite wealth and there is no limit on money earned in a single bet. However, 7️⃣ no gambler has infinite wealth, and the exponential growth of the bets can bankrupt unlucky gamblers who choose to use 7️⃣ the martingale, causing a catastrophic loss. Despite the fact that the gambler usually wins a small net reward, thus appearing 7️⃣ to have a sound strategy, the gambler's expected value remains zero because the small probability that the gambler will suffer 7️⃣ a catastrophic loss exactly balances with the expected gain. In a casino, the expected value is negative, due to the 7️⃣ house's edge. Additionally, as the likelihood of a string of consecutive losses is higher than common intuition suggests, martingale strategies 7️⃣ can bankrupt a gambler quickly.
The martingale strategy has also been applied to roulette, as the probability of hitting either red 7️⃣ or black is close to 50%.
Intuitive analysis [ edit ]
The fundamental reason why all martingale-type betting systems fail is that 7️⃣ no amount of information about the results of past bets can be used to predict the results of a future 7️⃣ bet with accuracy better than chance. In mathematical terminology, this corresponds to the assumption that the win–loss outcomes of each 7️⃣ bet are independent and identically distributed random variables, an assumption which is valid in many realistic situations. It follows from 7️⃣ this assumption that the expected value of a series of bets is equal to the sum, over all bets that 7️⃣ could potentially occur in the series, of the expected value of a potential bet times the probability that the player 7️⃣ will make that bet. In most casino games, the expected value of any individual bet is negative, so the sum 7️⃣ of many negative numbers will also always be negative.
The martingale strategy fails even with unbounded stopping time, as long as 7️⃣ there is a limit on earnings or on the bets (which is also true in practice).[1] It is only with 7️⃣ unbounded wealth, bets and time that it could be argued that the martingale becomes a winning strategy.
Mathematical analysis [ edit 7️⃣ ]
The impossibility of winning over the long run, given a limit of the size of bets or a limit in 7️⃣ the size of one's bankroll or line of credit, is proven by the optional stopping theorem.[1]
However, without these limits, the 7️⃣ martingale betting strategy is certain to make money for the gambler because the chance of at least one coin flip 7️⃣ coming up heads approaches one as the number of coin flips approaches infinity.
Mathematical analysis of a single round [ edit 7️⃣ ]
Let one round be defined as a sequence of consecutive losses followed by either a win, or bankruptcy of the 7️⃣ gambler. After a win, the gambler "resets" and is considered to have started a new round. A continuous sequence of 7️⃣ martingale bets can thus be partitioned into a sequence of independent rounds. Following is an analysis of the expected value 7️⃣ of one round.
Let q be the probability of losing (e.g. for American double-zero roulette, it is 20/38 for a bet 7️⃣ on black or red). Let B be the amount of the initial bet. Let n be the finite number of 7️⃣ bets the gambler can afford to lose.
The probability that the gambler will lose all n bets is qn. When all 7️⃣ bets lose, the total loss is
∑ i = 1 n B ⋅ 2 i − 1 = B ( 2 7️⃣ 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 − 7️⃣ qn. In all other cases, the gambler wins the initial bet (B.) Thus, the expected profit per round is
( 1 7️⃣ − q n ) ⋅ B − q n ⋅ B ( 2 n − 1 ) = B ( 7️⃣ 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 7️⃣ < 0 for all n > 0. Thus, for all games where a gambler is more likely to lose than 7️⃣ to win any given bet, that gambler is expected to lose money, on average, each round. Increasing the size of 7️⃣ wager for each round per the martingale system only serves to increase the average loss.
Suppose a gambler has a 63-unit 7️⃣ gambling bankroll. The gambler might bet 1 unit on the first spin. On each loss, the bet is doubled. Thus, 7️⃣ taking k as the number of preceding consecutive losses, the player will always bet 2k units.
With a win on any 7️⃣ given spin, the gambler will net 1 unit over the total amount wagered to that point. Once this win is 7️⃣ achieved, the gambler restarts the system with a 1 unit bet.
With losses on all of the first six spins, the 7️⃣ gambler loses a total of 63 units. This exhausts the bankroll and the martingale cannot be continued.
In this example, the 7️⃣ probability of losing the entire bankroll and being unable to continue the martingale is equal to the probability of 6 7️⃣ consecutive losses: (10/19)6 = 2.1256%. The probability of winning is equal to 1 minus the probability of losing 6 times: 7️⃣ 1 − (10/19)6 = 97.8744%.
The expected amount won is (1 × 0.978744) = 0.978744.
The expected amount lost is (63 × 7️⃣ 0.021256)= 1.339118.
Thus, the total expected value for each application of the betting system is (0.978744 − 1.339118) = −0.360374 .
In 7️⃣ a unique circumstance, this strategy can make sense. Suppose the gambler possesses exactly 63 units but desperately needs a total 7️⃣ of 64. Assuming q > 1/2 (it is a real casino) and he may only place bets at even odds, 7️⃣ his best strategy is bold play: at each spin, he should bet the smallest amount such that if he wins 7️⃣ he reaches his target immediately, and if he does not have enough for this, he should simply bet everything. Eventually 7️⃣ he either goes bust or reaches his target. This strategy gives him a probability of 97.8744% of achieving the goal 7️⃣ of winning one unit vs. a 2.1256% chance of losing all 63 units, and that is the best probability possible 7️⃣ in this circumstance.[2] However, bold play is not always the optimal strategy for having the biggest possible chance to increase 7️⃣ an initial capital to some desired higher amount. If the gambler can bet arbitrarily small amounts at arbitrarily long odds 7️⃣ (but still with the same expected loss of 10/19 of the stake at each bet), and can only place one 7️⃣ bet at each spin, then there are strategies with above 98% chance of attaining his goal, and these use very 7️⃣ timid play unless the gambler is close to losing all his capital, in which case he does switch to extremely 7️⃣ bold play.[3]
Alternative mathematical analysis [ edit ]
The previous analysis calculates expected value, but we can ask another question: what is 7️⃣ the chance that one can play a casino game using the martingale strategy, and avoid the losing streak long enough 7️⃣ to double one's bankroll?
As before, this depends on the likelihood of losing 6 roulette spins in a row assuming we 7️⃣ are betting red/black or even/odd. Many gamblers believe that the chances of losing 6 in a row are remote, and 7️⃣ that with a patient adherence to the strategy they will slowly increase their bankroll.
In reality, the odds of a streak 7️⃣ of 6 losses in a row are much higher than many people intuitively believe. Psychological studies have shown that since 7️⃣ people know that the odds of losing 6 times in a row out of 6 plays are low, they incorrectly 7️⃣ assume that in a longer string of plays the odds are also very low. In fact, while the chance of 7️⃣ losing 6 times in a row in 6 plays is a relatively low 1.8% on a single-zero wheel, the probability 7️⃣ of losing 6 times in a row (i.e. encountering a streak of 6 losses) at some point during a string 7️⃣ of 200 plays is approximately 84%. Even if the gambler can tolerate betting ~1,000 times their original bet, a streak 7️⃣ of 10 losses in a row has an ~11% chance of occurring in a string of 200 plays. Such a 7️⃣ loss streak would likely wipe out the bettor, as 10 consecutive losses using the martingale strategy means a loss of 7️⃣ 1,023x the original bet.
These unintuitively risky probabilities raise the bankroll requirement for "safe" long-term martingale betting to infeasibly high numbers. 7️⃣ To have an under 10% chance of failing to survive a long loss streak during 5,000 plays, the bettor must 7️⃣ have enough to double their bets for 15 losses. This means the bettor must have over 65,500 (2^15-1 for their 7️⃣ 15 losses and 2^15 for their 16th streak-ending winning bet) times their original bet size. Thus, a player making 10 7️⃣ unit bets would want to have over 655,000 units in their bankroll (and still have a ~5.5% chance of losing 7️⃣ it all during 5,000 plays).
When people are asked to invent data representing 200 coin tosses, they often do not add 7️⃣ streaks of more than 5 because they believe that these streaks are very unlikely.[4] This intuitive belief is sometimes referred 7️⃣ to as the representativeness heuristic.
In a classic martingale betting style, gamblers increase bets after each loss in hopes that an 7️⃣ eventual win will recover all previous losses. The anti-martingale approach, also known as the reverse martingale, instead increases bets after 7️⃣ wins, while reducing them after a loss. The perception is that the gambler will benefit from a winning streak or 7️⃣ a "hot hand", while reducing losses while "cold" or otherwise having a losing streak. As the single bets are independent 7️⃣ from each other (and from the gambler's expectations), the concept of winning "streaks" is merely an example of gambler's fallacy, 7️⃣ and the anti-martingale strategy fails to make any money.
If on the other hand, real-life stock returns are serially correlated (for 7️⃣ instance due to economic cycles and delayed reaction to news of larger market participants), "streaks" of wins or losses do 7️⃣ happen more often and are longer than those under a purely random process, the anti-martingale strategy could theoretically apply and 7️⃣ can be used in trading systems (as trend-following or "doubling up"). This concept is similar to that used in momentum 7️⃣ investing and some technical analysis investing strategies.
See also [ edit ]
Double or nothing – A decision in gambling that will 7️⃣ either double ones losses or cancel them out
Escalation of commitment – A human behavior pattern in which the participant takes 7️⃣ on increasingly greater risk
St. Petersburg paradox – Paradox involving a game with repeated coin flipping
Sunk cost fallacy – Cost that 7️⃣ has already been incurred and cannot be recovered Pages displaying short descriptions of redirect targets
