From inception, the martingale system was referred to as a class of betting strategies popular in 18th century France. The simplest of these strategies was designed for a game in which the gambler wins his stake if a coin comes up heads and loses it if the coin comes up tails. The strategy had the gambler double his bet after every loss, so that the first win would recover all previous losses plus win a profit equal to the original stake.

Since a gambler with infinite wealth will, with probability 1, eventually flip heads, the Martingale betting strategy was seen as a sure thing by those who advocated it. Of course, none of the gamblers in fact possessed infinite wealth, and the exponential growth of the bets would eventually bankrupt those who chose to use the Martingale. It is widely believed that casinos instituted betting limits specifically to stop Martingale players, but in reality the assumptions behind the strategy are not sound. Players using the Martingale system do not have any long-term mathematical advantage over any other betting system or even randomly placed bets.

The Effect of Variance

As with any betting system, it sometimes happens that one achieves a better result than the expected negative return, by temporarily avoiding a losing streak. Furthermore, a straight string of losses is the only sequence of outcomes that results in a loss of money, so even when a player has lost the majority of his bets, he can still be ahead overall, since he always wins 1 unit when a bet wins, regardless of how many previous losses.

Intuitive Analysis

When the expected value of the stopping time is finite (which is true in practice), the following argument explains why the betting system fails: Since expectation is linear, the expected value of a series of bets is just the sum of the expected value of each bet. Since in such games of chance the bets are independent the expectation of each bet does not depend on whether you previously won or lost. In most casino games, the expected value of any individual bet is negative, so the sum of lots of negative numbers is also always going to be negative.

The martingale system fails even with unbounded stopping time, as long as there is a limit on earnings or on the bets (which are also true in practice). It is only with unbounded wealth, bets and time that the martingale system can succeed.

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