The doubling bet that inspired a branch of probability
Eighteenth-century French gamblers loved a system called the martingale: double your stake after every loss, and the first win recovers everything plus a profit. Given unlimited money and time it cannot fail. Real bankrolls are finite, and mathematicians later built a whole theory showing why no such system beats a fair game.
The simplest version was played on a coin toss. Heads, you win your stake; tails, you lose it. Doubling after each loss means one head wipes out the run of tails and leaves you ahead by the original bet. As wealth and time head towards infinity, the chance of eventually seeing heads approaches certainty, so the scheme looks foolproof. The catch is the exponential growth of the bets, which sooner or later bankrupts anyone with a limited purse.
In modern probability, the word names a type of random process rather than a strategy. A martingale is a sequence in which the expected next value, given everything that has happened so far, equals the current value. History offers no edge: on average, tomorrow looks exactly like today. A gambler's fortune is a martingale if every game played is fair. Win a dollar on heads and lose one on tails, and the expected fortune after the next toss is simply what you hold now. An unbiased random walk, stepping left or right with equal odds in any number of dimensions, behaves the same way.
Paul Lévy introduced the idea in 1934 without giving it a name. Jean Ville supplied the term in 1939 and extended it to processes in continuous time, and Joseph Leo Doob did much of the early development. Part of the motivation was to prove mathematically that successful betting systems in games of chance are impossible.
Being a martingale depends on the probability measure used to compute expectations, so a process can qualify under one measure and fail under another. The Girsanov theorem gives a way to find a measure that turns certain processes into martingales, and stopped Brownian motion can model the path of a doubling gambler.
Source: Martingale (probability theory)
