Roulette winning Strategies
A martingale is any of 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 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. The martingale strategy has been applied to roulette as well, as the probability of hitting either red or black is close to 50%.
Since a gambler with infinite wealth will, almost surely, 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 "unlucky" gamblers who chose to use the martingale. The gambler usually wins a small net reward, thus appearing to have a sound strategy. However, the gambler's expected value does indeed remain zero (or less than zero) because the small probability that he will suffer a catastrophic loss exactly balances with his expected gain. (In a casino, the expected value is negative, due to the house's edge.) The likelihood of catastrophic loss may not even be very small. The bet size rises exponentially. This, combined with the fact that strings of consecutive losses actually occur more often than common intuition suggests, can bankrupt a gambler quickly.
Assuming that the win/loss outcomes of each bet are independent and identically distributed random variables, the stopping time has finite expected value. This justifies the following argument, explaining 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 strategy 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 it could be argued that the martingale becomes a winning strategy, except that you cannot increase unbounded wealth.
One round of the idealized martingale without time or credit constraints can be formulated mathematically as follows. Let the coin tosses be represented by a sequence 0, 1, … of independent random variables, each of which is equal to with probability , and with probability = 1 – . Let be time of appearance of the first ; in other words, 0, 1, …, –1 = , and = . If the coin never shows , we write = ∞. is itself a random variable because it depends on the random outcomes of the coin tosses.
In the first – 1 coin tosses, the player following the martingale strategy loses 1, 2, …, 2–1 units, accumulating a total loss of 2 − 1. On the th toss, there is a win of 2 units, resulting in a net gain of 1 unit over the first tosses. For example, suppose the first four coin tosses are , , , making = 3. The bettor loses 1, 2, and 4 units on the first three tosses, for a total loss of 7 units, then wins 8 units on the fourth toss, for a net gain of 1 unit. As long as the coin eventually shows heads, the betting player realizes a gain.
What is the probability that = ∞, i.e., that the coin never shows heads? Clearly it can be no greater than the probability that the first tosses are all ; this probability is qk. Unless = 1, the only nonnegative number less than or equal to qk for all values of is zero. It follows that is finite with probability 1; therefore with probability 1, the coin will eventually show heads and the bettor will realize a net gain of 1 unit.
This property of the idealized version of the martingale accounts for the attraction of the idea. In practice, the idealized version can only be approximated, for two reasons. Unlimited credit to finance possibly astronomical losses during long runs of tails is not available, and there is a limit to the number of coin tosses that can be performed in any finite period of time, precluding the possibility of playing long enough to observe very long runs of tails.
As an example, consider a bettor with an available fortune, or credit, of (approximately 9 trillion) units, roughly half the size of the current US national debt in dollars. With this very large fortune, the player can afford to lose on the first 42 tosses, but a loss on the 43rd cannot be covered. The probability of losing on the first 42 tosses is , which will be a very small number unless tails are nearly certain on each toss. In the fair case where , we could expect to wait something on the order of tosses before seeing 42 consecutive tails; tossing coins at the rate of one toss per second, this would require approximately 279, 000 years.