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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 9️⃣ betting strategies that originated from and were

popular in 18th-century France. The simplest of these strategies was designed for a

9️⃣ game in which the gambler wins the stake if a coin comes up heads and loses if it comes

up 9️⃣ tails. The strategy had the gambler double the bet after every loss, so that the

first win would recover all 9️⃣ previous losses plus win a profit equal to the original

stake. Thus the strategy is an instantiation of the St. 9️⃣ Petersburg paradox.

Since a

gambler will almost surely eventually flip heads, the martingale betting strategy is

certain to make money for 9️⃣ the gambler provided they have infinite wealth and there is

no limit on money earned in a single bet. However, 9️⃣ no gambler has infinite wealth, and

the exponential growth of the bets can bankrupt unlucky gamblers who choose to use 9️⃣ the

martingale, causing a catastrophic loss. Despite the fact that the gambler usually wins

a small net reward, thus appearing 9️⃣ to have a sound strategy, the gambler's expected

value remains zero because the small probability that the gambler will suffer 9️⃣ a

catastrophic loss exactly balances with the expected gain. In a casino, the expected

value is negative, due to the 9️⃣ house's edge. Additionally, as the likelihood of a string

of consecutive losses is higher than common intuition suggests, martingale strategies

9️⃣ can bankrupt a gambler quickly.

The martingale strategy has also been applied to

roulette, as the probability of hitting either red 9️⃣ or black is close to 50%.

Intuitive

analysis [ edit ]

The fundamental reason why all martingale-type betting systems fail

is that 9️⃣ no amount of information about the results of past bets can be used to predict

the results of a future 9️⃣ bet with accuracy better than chance. In mathematical

terminology, this corresponds to the assumption that the win–loss outcomes of each 9️⃣ bet

are independent and identically distributed random variables, an assumption which is

valid in many realistic situations. It follows from 9️⃣ this assumption that the expected

value of a series of bets is equal to the sum, over all bets that 9️⃣ could potentially

occur in the series, of the expected value of a potential bet times the probability

that the player 9️⃣ will make that bet. In most casino games, the expected value of any

individual bet is negative, so the sum 9️⃣ of many negative numbers will also always be

negative.

The martingale strategy fails even with unbounded stopping time, as long as

9️⃣ there is a limit on earnings or on the bets (which is also true in practice).[1] It is

only with 9️⃣ unbounded wealth, bets and time that it could be argued that the martingale

becomes a winning strategy.

Mathematical analysis [ edit 9️⃣ ]

The impossibility of winning

over the long run, given a limit of the size of bets or a limit in 9️⃣ the size of one's

bankroll or line of credit, is proven by the optional stopping theorem.[1]

However,

without these limits, the 9️⃣ martingale betting strategy is certain to make money for the

gambler because the chance of at least one coin flip 9️⃣ coming up heads approaches one as

the number of coin flips approaches infinity.

Mathematical analysis of a single round [

edit 9️⃣ ]

Let one round be defined as a sequence of consecutive losses followed by either

a win, or bankruptcy of the 9️⃣ gambler. After a win, the gambler "resets" and is

considered to have started a new round. A continuous sequence of 9️⃣ martingale bets can

thus be partitioned into a sequence of independent rounds. Following is an analysis of

the expected value 9️⃣ of one round.

Let q be the probability of losing (e.g. for American

double-zero roulette, it is 20/38 for a bet 9️⃣ on black or red). Let B be the amount of

the initial bet. Let n be the finite number of 9️⃣ bets the gambler can afford to lose.

The

probability that the gambler will lose all n bets is qn. When all 9️⃣ bets lose, the total

loss is

∑ i = 1 n B ⋅ 2 i − 1 = B ( 2 9️⃣ 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 − 9️⃣ qn. In

all other cases, the gambler wins the initial bet (B.) Thus, the expected profit per

round is

( 1 9️⃣ − q n ) ⋅ B − q n ⋅ B ( 2 n − 1 ) = B ( 9️⃣ 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 9️⃣ < 0 for all n > 0. Thus, for all games where a gambler is more likely to lose

than 9️⃣ to win any given bet, that gambler is expected to lose money, on average, each

round. Increasing the size of 9️⃣ wager for each round per the martingale system only

serves to increase the average loss.

Suppose a gambler has a 63-unit 9️⃣ gambling bankroll.

The gambler might bet 1 unit on the first spin. On each loss, the bet is doubled. Thus,

9️⃣ taking k as the number of preceding consecutive losses, the player will always bet 2k

units.

With a win on any 9️⃣ given spin, the gambler will net 1 unit over the total amount

wagered to that point. Once this win is 9️⃣ achieved, the gambler restarts the system with

a 1 unit bet.

With losses on all of the first six spins, the 9️⃣ gambler loses a total of

63 units. This exhausts the bankroll and the martingale cannot be continued.

In this

example, the 9️⃣ probability of losing the entire bankroll and being unable to continue the

martingale is equal to the probability of 6 9️⃣ consecutive losses: (10/19)6 = 2.1256%. The

probability of winning is equal to 1 minus the probability of losing 6 times: 9️⃣ 1 −

(10/19)6 = 97.8744%.

The expected amount won is (1 × 0.978744) = 0.978744.

The expected

amount lost is (63 × 9️⃣ 0.021256)= 1.339118.

Thus, the total expected value for each

application of the betting system is (0.978744 − 1.339118) = −0.360374 .

In 9️⃣ a unique

circumstance, this strategy can make sense. Suppose the gambler possesses exactly 63

units but desperately needs a total 9️⃣ of 64. Assuming q > 1/2 (it is a real casino) and

he may only place bets at even odds, 9️⃣ his best strategy is bold play: at each spin, he

should bet the smallest amount such that if he wins 9️⃣ he reaches his target immediately,

and if he does not have enough for this, he should simply bet everything. Eventually 9️⃣ he

either goes bust or reaches his target. This strategy gives him a probability of

97.8744% of achieving the goal 9️⃣ of winning one unit vs. a 2.1256% chance of losing all

63 units, and that is the best probability possible 9️⃣ in this circumstance.[2] However,

bold play is not always the optimal strategy for having the biggest possible chance to

increase 9️⃣ an initial capital to some desired higher amount. If the gambler can bet

arbitrarily small amounts at arbitrarily long odds 9️⃣ (but still with the same expected

loss of 10/19 of the stake at each bet), and can only place one 9️⃣ bet at each spin, then

there are strategies with above 98% chance of attaining his goal, and these use very

9️⃣ timid play unless the gambler is close to losing all his capital, in which case he does

switch to extremely 9️⃣ bold play.[3]

Alternative mathematical analysis [ edit ]

The

previous analysis calculates expected value, but we can ask another question: what is

9️⃣ the chance that one can play a casino game using the martingale strategy, and avoid the

losing streak long enough 9️⃣ to double one's bankroll?

As before, this depends on the

likelihood of losing 6 roulette spins in a row assuming we 9️⃣ are betting red/black or

even/odd. Many gamblers believe that the chances of losing 6 in a row are remote, and

9️⃣ that with a patient adherence to the strategy they will slowly increase their

bankroll.

In reality, the odds of a streak 9️⃣ of 6 losses in a row are much higher than

many people intuitively believe. Psychological studies have shown that since 9️⃣ people

know that the odds of losing 6 times in a row out of 6 plays are low, they incorrectly

9️⃣ assume that in a longer string of plays the odds are also very low. In fact, while the

chance of 9️⃣ losing 6 times in a row in 6 plays is a relatively low 1.8% on a single-zero

wheel, the probability 9️⃣ of losing 6 times in a row (i.e. encountering a streak of 6

losses) at some point during a string 9️⃣ of 200 plays is approximately 84%. Even if the

gambler can tolerate betting ~1,000 times their original bet, a streak 9️⃣ of 10 losses in

a row has an ~11% chance of occurring in a string of 200 plays. Such a 9️⃣ loss streak

would likely wipe out the bettor, as 10 consecutive losses using the martingale

strategy means a loss of 9️⃣ 1,023x the original bet.

These unintuitively risky

probabilities raise the bankroll requirement for "safe" long-term martingale betting to

infeasibly high numbers. 9️⃣ To have an under 10% chance of failing to survive a long loss

streak during 5,000 plays, the bettor must 9️⃣ have enough to double their bets for 15

losses. This means the bettor must have over 65,500 (2^15-1 for their 9️⃣ 15 losses and

2^15 for their 16th streak-ending winning bet) times their original bet size. Thus, a

player making 10 9️⃣ unit bets would want to have over 655,000 units in their bankroll (and

still have a ~5.5% chance of losing 9️⃣ it all during 5,000 plays).

When people are asked

to invent data representing 200 coin tosses, they often do not add 9️⃣ streaks of more than

5 because they believe that these streaks are very unlikely.[4] This intuitive belief

is sometimes referred 9️⃣ to as the representativeness heuristic.

In a classic martingale

betting style, gamblers increase bets after each loss in hopes that an 9️⃣ eventual win

will recover all previous losses. The anti-martingale approach, also known as the

reverse martingale, instead increases bets after 9️⃣ wins, while reducing them after a

loss. The perception is that the gambler will benefit from a winning streak or 9️⃣ a "hot

hand", while reducing losses while "cold" or otherwise having a losing streak. As the

single bets are independent 9️⃣ from each other (and from the gambler's expectations), the

concept of winning "streaks" is merely an example of gambler's fallacy, 9️⃣ and the

anti-martingale strategy fails to make any money.

If on the other hand, real-life stock

returns are serially correlated (for 9️⃣ instance due to economic cycles and delayed

reaction to news of larger market participants), "streaks" of wins or losses do 9️⃣ happen

more often and are longer than those under a purely random process, the anti-martingale

strategy could theoretically apply and 9️⃣ can be used in trading systems (as

trend-following or "doubling up"). This concept is similar to that used in momentum

9️⃣ investing and some technical analysis investing strategies.

See also [ edit ]

Double or

nothing – A decision in gambling that will 9️⃣ either double ones losses or cancel them

out

Escalation of commitment – A human behavior pattern in which the participant takes

9️⃣ on increasingly greater risk

St. Petersburg paradox – Paradox involving a game with

repeated coin flipping

Sunk cost fallacy – Cost that 9️⃣ has already been incurred and

cannot be recovered Pages displaying short descriptions of redirect targets

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