Martingale

A martingale is a kind of gambling system that gives the illusion of being a "sure thing." The illusion is a complete deception. It is fairly easy to see why the simplest form of martingales do not work, but complicated kinds of gambling system may be harder to see through. Like perpetual motion, a working gambling system--one that guarantees a gambler always wins in the long run--is impossible, if it is a true game of chance and if the gambler is playing a fair game (or playing against a house percentage).

A martingale essentially consists of "double or nothing," doubling up the bet until the gambler wins. The sizes of the bets increase in such a way that when the gambler eventually wins, the bet is so large that it more than recoups all the previous losses. (The name martingale in fact comes from the name of a kind of horse harness in which the straps repeatedly split into two).

For example, consider betting on red or black on a roulette wheel. One form of martingale system works like this:

* Bet #1: bet $10 on red. If red comes up, you've won $10 and the sequence is complete. If black comes up, you've lost a total of $10 so far; make a second bet.
* Bet #2: bet $20 on red. If red comes up, you've won $20 - $10 = $10 and the sequence is complete. If black comes up, you've lost a total of $30 so far; make a third bet.
* Bet #3: bet $40 on red. If red comes up, you've won $40 - $30 = $10 and the sequence is complete. If black comes up, you've lost a total of $70 so far; make a fourth bet.
* Bet #4: bet $80 on red. If red comes up, you've won $80 - $70 = $10 and the sequence is complete. If black comes up, you've lost a total of $150 so far; make a fifth bet...

and so on. Since red must eventually come up, it appears as if the gambler is guaranteed to win $10 on each sequence of bets.

This would seem to be true even when playing against a "house percentage" (e.g. a roulette wheel with 0 and a 00 slots on which nobody wins), which should instantly raise suspicions.

The flaw in the martingale is the assumption that the series of bets can continue for as long as is needed to reach that eventual win. In reality, the run of bets is limited either by the amount of money the gambler has or by casino betting limits.

For example, consider a gambler who has only $70 and tries to follow the system above. If there is an (imaginary!) fair roulette wheel, the probalities are:

* a probability of 1/2 that the first spin comes up red: gambler wins $10.
* a probability of 1/4 that the first two spins come up black, red: the gambler wins $10.
* a probability of 1/8 that the first three spins come up black, black, red: the gambler wins $10.
* a probability of 1/8 that the first four spins come up black, black, black. The gambler loses $70, is broke, and cannot continue.

Thus, playing a martingale sequence, the gambler has a probability of 1/2 + 1/4 + 1/8 = 7/8 of winning $10 and a probability of 1/8 of losing $70.

In other words, by playing a system the gambler can affect the distribution of outcomes, but not the average outcome. A gambler who bets red or black on a single spin of a fair roulette wheel has half a chance of winning and half a chance of losing. By following a martingale, the gambler can convert this into a high chance of winning a small amount, but this is balanced by a small but nevertheless very real chance of losing a very large amount.

The martingale lends itself to a cruel self-deception. It is in the nature of the martingale that there is always a real possibility of huge loss. However, when it occurs, the gambler is likely to feel that he would have won if only he had been able to raise the money for one more bet, and therefore the loss is not the fault of the system. It would have worked, the gambler thinks, had he not had the impossibly bad luck of encountering eight, or nine, or ten "black" spins in a row.