Understanding Martingale
For the martingale betting strategy, see martingale (betting system).
Definitions
A martingale is 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 the stake if a coin comes up heads and loses if it comes up tails. The strategy had the gambler double the bet after every loss, so that the first win would recover all previous losses plus win a profit equal to the original stake. Thus the strategy is an instantiation of the St. Petersburg paradox.
Since a gambler will almost surely eventually flip heads, the martingale betting strategy is certain to make money for the gambler provided they have infinite wealth and there is no limit on money earned in a single bet. However, no gambler has infinite wealth, and the exponential growth of the bets can bankrupt unlucky gamblers who choose to use the martingale, causing a catastrophic loss. Despite the fact that the gambler usually wins a small net reward, thus appearing to have a sound strategy, the gambler's expected value remains zero because the small probability that the gambler will suffer a catastrophic loss exactly balances with the expected gain. In a casino, the expected value is negative, due to the house's edge. Additionally, as the likelihood of a string of consecutive losses is higher than common intuition suggests, martingale strategies can bankrupt a gambler quickly.
The martingale strategy has also been applied to roulette, as the probability of hitting either red or black is close to 50%.
Basics
A random variable is an unknown value or function that takes a specific value for each possible experiment. It can be discrete or continuous. We will consider a discrete random variable below to explain Martingale.
Martingale is a sequence of random variables M1, M2, M3, ..., Mn, where
E [Mn+1|Mn] = Mn, n -> 0, 1, ..., n+1 (1)
and E [|Mn+1|] < ∞,
assuming that the value of Mn equals Mn, it is read as the expectation value of Mn+1, meaning the expectation value remains unchanged.
For a random variable that takes different values x1, x2, x3 with probabilities p1, p2, and p3 respectively, the expectation is calculated as follows:
E[M] = x1 * p1 + x2 * p2 + x3 * p3, in simple terms, the expectation is equivalent to the arithmetic mean.
Note: Martingale is always defined with respect to some information set and probability measure. If the information set or probability related to the process changes, the process may no longer be a Martingale.
Mathematical analysis
Let one round be defined as a sequence of consecutive losses followed by either a win, or 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. Following is an analysis of the expected value of one round.
Let q be the probability of losing (e.g. for American double-zero roulette, it is 20/38 for a bet on black or red). Let B be the amount of the initial bet. Let n be the finite number of bets the gambler can afford to lose.
The probability that the gambler will lose all n bets is qn. When all bets lose, the total loss is
The probability the gambler does not lose all n bets is 1 − qn. In all other cases, the gambler wins the initial bet (B.) Thus, the expected profit per round is
Whenever q > 1/2, the expression 1 − (2q)n < 0 for all n > 0. Thus, for all games where a gambler is more likely to lose than to win any given bet, that gambler is expected to lose money, on average, each round. Increasing the size of wager for each round per the martingale system only serves to increase the average loss.
How does Martingale work?
Let's take a simple coin-flipping game as an example: You flip a fair coin, and if it lands heads, you win one dollar, and if it lands tails, you lose one dollar.
In this game, the probability of heads or tails is always half. Therefore, the average winnings are equal to ½ (1) + ½ (-1) = 0, which means you cannot systematically earn any extra money by playing multiple rounds of the game (although random gains or losses are still possible).
Thus, the scenario described above satisfies the Martingale property, which means that in any given round, regardless of the outcomes of previous rounds, your total earnings are expected to remain unchanged.
Key Takeaways
The Martingale system is a methodology to amplify the chance of recovering from losing streaks.
The Martingale strategy involves doubling up on losing bets and reducing winning bets by half.
It essentially a strategy that promotes a loss-averse mentality that tries to improve the odds of breaking even, but also increases the chances of severe and quick losses.
Forex trading is more well-suited to this type of strategy than for stocks trading or casino gambling.
Conditional Expectation in Martingale
The expression on the left side of equation (1) in the Martingale definition is called "conditional expectation." Conditional expectation (also known as conditional mean) is simply the average calculated after a set of prior conditions has occurred.
The conditional expectation of a random variable X relative to a variable Z is defined as the (new) random variable Y = E(X|Z).
Examples
Common betting systems include:
Martingale Trading Strategy
The Martingale trading strategy involves doubling your risk exposure or investment size in losing trades. Since your expected long-term returns remain the same (losses when prices fall, gains when prices rise), this strategy can be implemented by buying on dips when prices fall and lowering your average entry price. Therefore, even after a series of losing trades, if a profitable trade occurs, it will recoup all the losses, including the initial trade amount, as it generates a profit of $2^p=∑ 2^p-1+1$.
The source of the above information can be referenced from Wikipedia.
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