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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

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