Martingale (betting system)

A separate article treats the topic of Martingale (probability theory).

Originally, martingale referred to 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 practised it. Of course, none of these practitioners in fact possessed infinite wealth, and the exponential growth of the bets would eventually bankrupt those who choose to use the Martingale. Moreover, it has become impossible to implement in modern casinos, due to the betting limit at the tables. Because the betting limits reduce the casino's short term variance, the Martingale system itself does not pose a threat to the casino, and many will encourage its use, knowing that they have the house advantage no matter when or how much is wagered.

Ben is Wrong

Kevin totally meant over a series.

If He had... He would be right, and I wouldn't have argued with him.

But he didn't.

Effect of variance

As with any betting system, it is possible to have variance from the expected negative return by temporarily avoiding the inevitable 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 their bets, they can still be ahead over-all, since they always win 1 unit when a bet wins, regardless of how many previous losses.

Intuitive analysis

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 all bets is going to be the same, regardless of 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.

Mathematical analysis of one round

Let one round be defined as a sequence of consecutive losses followed by a win, or consecutive losses resulting in bankruptcy of the gambler. After a win, the gambler "resets" and is considered to have started a new round. A continuous sequence of martingale bets can thus be partitioned into a sequence of independent rounds. We will analyze the expected value of one round.

Let q be the probability of losing (e.g. for roulette it is 20/38). Let y be the amount of the commencing bet. Let x be the finite number of bets you can afford to lose.

The probability that you lose all x bets is q^x. When you lose all your bets, the amount of money you lose is

$\sum _{i=1}^{x}y\cdot 2^{i-1}=y(2^{x}-1)$

The probability that you do not lose all x bets is 1 − q^x. If you do not lose all x bets, you win y amount of money (the initial bet amount). So the expected profit per round is

$(1-q^{x})\cdot y-q^{x}\cdot y(2^{x}-1)=y(1-(2q)^{x})$

Whenever q > 1/2, the expression 1 − (2q)^x < 0 for all x > 0. That means for any game where it is more likely to lose than to win (e.g. all chance gambling games), you are expected to lose money on average per round. Furthermore, the more times you are able to afford to bet, the more you will lose.

As an example, suppose you have $10,000 available to bet. You bet $100 on the first spin. If you lose, you bet $200 on the second spin, then $400 on the third, $800 on the fourth, $1,600 on the fifth, and $3,200 on the sixth.

If you win $100 on the first spin, you make $100, and the martingale starts over.

If you lose $100 on the first spin and win $200 on the second spin, you make a net profit of $100 at which point the martingale would start over.

If you lose on the first five spins, you lose a total of $3,100 ($3,100 = $100 + $200 + $400 + $800 + $1,600). On the sixth spin you bet $3,200. If you win, you again make a profit of $100.

If you lose on the first six spins, you have lost a total of $6,300 and with only $10,000 available, you do not have enough money to double your previous bet. At this point the martingale can not be continued.

In this example the probability of losing $6,300 and being unable to continue the martingale is equal to the probability of losing 6 times or (20/38)^6 = 2.1256%. The probability of winning $100 is equal to 1 minus the probability of losing 6 times or 1 - (20/38)^6 = 97.8744%. The expected value is (-$6,300*.021256) + ($100*.978744) = -$36.04.

Anti-martingale

In a classic martingale betting style, gamblers will increase their bets after each loss in hopes that an eventual win will recover all previous losses. The anti-martingale approach instead increases bets after wins, while reducing them after a loss. The perception is that in this manner the gambler will benefit from a winning streak or a "hot hand", while reducing losses while "cold" or otherwise having a losing streak. This general idea of increasing bets when conditions are believed to be favorable can improve the odds in games with a memory by using a strategy like card counting. But in a true random memoryless game there is no such thing as a winning streak or losing streak (these notions are gambler's fallacy) so this strategy can't improve the expected winnings in such situations.

One activity where money management based on an anti-martingale approach has a recognized value is speculation and trading. Many financial markets have some cyclical component to them, and the approach of an individual speculator or trader may only be appropriate for one portion of that cycle. Using an anti-martingale risk management scheme will increase profits during time periods when a trading approach is working well, while automatically decreasing exposure during portions of the cycle where trading is unprofitable. This is believed to decrease the risk of ruin for trading.

In popular culture

In the CSI: Las Vegas episode XX, a character borrows thousands of dollars to test out a brilliant gambling strategy, which turns out to be the Martingale system.