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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.

See also [ edit ]

Double or nothing – A decision in gambling that will 💸 either double ones losses or cancel them out

Escalation of commitment – A human behavior pattern in which the participant takes 💸 on increasingly greater risk

St. Petersburg paradox – Paradox involving a game with repeated coin flipping

Sunk cost fallacy – Cost that 💸 has already been incurred and cannot be recovered Pages displaying short descriptions of redirect targets

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tirada 28 de Junho 2024 mostra uma vista da planta central do campo gasolina, fornecimento na província chinesa 💶 por investimento no Iraque 2011. O Campo petróleo a Ahdab e o primeiro projeto para cooperação financeira internacional (foi in 💶 primeiro projecto projedo dado à empresa internalizada pelo país) 2011

No verão escaldante, fileiras de imponentes tamareiras e vibrantees tamargueira do 💶 deserto cria uma cena que não tem vida no campo gasolina da Ahdadeb na província iarquinana das Wasit um forte 💶 contraste com o terreno baldio salino branco.

O campo gasolina de Ahdab, o primeiro projeto da cooperação petrolífera internacional do Iraque 💶 no pós-guerra e foi inaugurado 2009 começou a produção in 2011. Operado pela Al Waha Petroleum Company of China 💶 deste seu inicial.

"Nossa empresa não apenas se esforça para sountar à produção de petróleo, mas também presta muita atenção á 💶 proteção do meio ambiente local", disse Jiang Ming diretor geral da Al-Waha Petroleum.

Irakue é um país rico petróleo 💶 e gás, mas há muito tempo Luta contra as consequências ambientais de seu estador da energia. A prática do queima 💶 das coisas associadas -queimar o produto natural conjunto com a pecríleo –tems os problemas dele mesmo

No final de 2011, uma 💶 planta do processo central no campo gasolina, foi incluído e rasgado-se a primeira instalação da Iraque um abranger totalmente os 💶 processos para o comércio petroleo.

A planta converte esse valor crescente recurso enxufre, gas seco y gás liquefeito de petróleo 💶 (GLP), rede convertido significativamente os resíduoes and as emissões. O que seco é usado para alimentar o campo gasolina tecífero 💶 a também está exportado nos

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tira 💶 30 de junho com 2024 mostra a vegetação no campo gasolina, na província do Wasit e Iraque. O Campo 💶 petróleo para Ahdab o primeiro projeto da cooperação petrolífera internacional pelo iraquiano pela guerra 2011 foi inaugurado Company projeto por 💶 ocasião 2009

Funcionário mão na planta de processamento central do campo gasolina, da província operação industrial no Iraque (28o mês) 💶 2011. O acampamento petróleo e serviços Serviços Empresariais 2009 - Foi inaugurado Company

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