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.

Mathematical analysis [ 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.

( 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})}

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).