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A martingale is a class of betting strategies that originated from and were popular in 18th-century France. The simplest of0锔忊儯 these strategies was designed for a game in which the gambler wins the stake if a coin comes up heads0锔忊儯 and loses if it comes up tails. The strategy had the gambler double the bet after every loss, so that0锔忊儯 the first win would recover all previous losses plus win a profit equal to the original stake. Thus the strategy0锔忊儯 is an instantiation of the St. Petersburg paradox.
Since a gambler will almost surely eventually flip heads, the martingale betting strategy0锔忊儯 is certain to make money for the gambler provided they have infinite wealth and there is no limit on money0锔忊儯 earned in a single bet. However, no gambler has infinite wealth, and the exponential growth of the bets can bankrupt0锔忊儯 unlucky gamblers who choose to use the martingale, causing a catastrophic loss. Despite the fact that the gambler usually wins0锔忊儯 a small net reward, thus appearing to have a sound strategy, the gambler's expected value remains zero because the small0锔忊儯 probability that the gambler will suffer a catastrophic loss exactly balances with the expected gain. In a casino, the expected0锔忊儯 value is negative, due to the house's edge. Additionally, as the likelihood of a string of consecutive losses is higher0锔忊儯 than common intuition suggests, martingale strategies can bankrupt a gambler quickly.
The martingale strategy has also been applied to roulette, as0锔忊儯 the probability of hitting either red or black is close to 50%.
Intuitive analysis [ edit ]
The fundamental reason why all0锔忊儯 martingale-type betting systems fail is that no amount of information about the results of past bets can be used to0锔忊儯 predict the results of a future bet with accuracy better than chance. In mathematical terminology, this corresponds to the assumption0锔忊儯 that the win鈥搇oss outcomes of each bet are independent and identically distributed random variables, an assumption which is valid in0锔忊儯 many realistic situations. It follows from this assumption that the expected value of a series of bets is equal to0锔忊儯 the sum, over all bets that could potentially occur in the series, of the expected value of a potential bet0锔忊儯 times the probability that the player will make that bet. In most casino games, the expected value of any individual0锔忊儯 bet is negative, so the sum of many negative numbers will also always be negative.
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