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Martingale System

Traditionally, martingale was a popular betting strategy class in France during the 1900s. The simplest martingale strategy was made for games where the player wins whenever a coin falls on heads and actually loses whenever a coin falls on tails. With this strategy, the player would double bets after each loss, so any win after that could get back any previous loss along with a real profit that equals the first bet. Because players with an infinite amount of money will eventually land on heads, this particular strategy of betting was a sure win for those who used it - or so they though.

Naturally, though, nobody has an infinite amount of money and such an exponential betting growth could turn users of the martingale system bankrupt in no time. Because of this, it serves as the Taleb distribution example: the player wins small profits and looks like he has a steady strategy, but his bet still gets a negative value of expectations because there is a small chance of him suffering catastrophic losses in the end. There is a wide belief that casinos have betting limits just to stop these Martingale players. However, the truth is that those strategy assumptions are completely unsound as martingale players don't have a mathematical edge in the long term over other systems of betting or random bets.

The Effects of Variance

Like with any other betting system, one can get better results than negative returns sometimes just by avoiding losing streaks on a temporary basis. Plus, straight losses are the single sequence of results that end in money losses, so if players lose most of their bets, they can still stay ahead because they win a unit with every win, no matter how many times they lost in the past.

An Intuitive Analysis

If there is a finite expected value when it comes to stopping time, the system of betting fails because the expectation tends to be linear and the value that is expected out of the betting series is a mere sum of every bet's expected value. Because these bets tend to be independent in games of luck, every bet's expectation will not rely on previous wins or losses. Instead, in the majority of casinos games, each individual bet's expected value is negative. Therefore, a negative number will always be negative.

This strategy actually fails with unbounded time of stopping as well, provided a limit exists on the wagers or on the earnings. Only with infinite wealth, time and bets can this strategy truly succeed.

 

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