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. Unfortunately, none of these practitioners in fact possessed infinite wealth, and the exponential growth of the bets would eventually bankrupt those foolish enough 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.

Example

Suppose that someone applies the martingale betting system at an American roulette table, with 0 and 00 values; on average, a bet on either red or black will win 18 times out of 38. If the player's initial bankroll is $150 and the betting unit is $10, he can afford 4 losing bets in a row (of $10, $20, $40, and $80) before he runs out of money. If any of these 4 bets wins he wins $10 and wins back any past losses. The chance of losing 4 bets in a row (and therefore losing the complete $150) is (20/38)⁴ = 7.67%. The remaining 92.3% of the time, the player will win $10. We will call this one round (playing until you have lost 4 times or until you win, whichever comes first). If you play repeated rounds with this strategy then your average earnings will be (0.923·$10) − (0.0767·$150) = −$2.275 per round. Therefore, you lose an average of $2.275 each round. However, if the gambler possesses an infinite amount of money, the expected return is (18/38)*b per roll (where b is the initial bet). With an initial bet of $10, the expected return is thus $4.736 per roll..

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

Let $q$ be the probability of losing (e.g. for roulette it is 20/38). Let $y$ be the amount of the commencing bet (e.g. $10 in the example above). 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. 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. Furthermore, the more times you are able to afford to bet, the more you will lose.

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 (Alain J.) borrows thousands of dollars to test out a brilliant gambling strategy, which turns out to be the Martingale system.