The Martingale is one of the most extremal and reckless systems. Every beginner invents the same system and gets excited about future gains. It is hard to determine who was the first inventor of Martingale, but surely the term "Martintage" indicating the progressive rising bet system was introduced by mathematician P.Levy at the beginning of the XX century. He was analyzing the paradoxes of gamble games. Another mathematician, Doob also worked with this subject and expanded the Levy's results.
Let's see the most famous variation of Martingale. It is doubling of a wager after every loss in even money game. In case of a win, the serie starts from initial bet. For example, initial bet is one chip. Every serie wins is exactly one chip.
A real game always has the limit of max bet size, it is the restriction of casino or size of player's bankroll (Bill Gates has money only for 24 rounds!). Thus, the serie always has the finite number of steps. Martingale-n is system of series with maximum n steps of bet. For example, Martingale-1 is flat betting, Martingale-2 is doubling only once after lose, etc. The statistics of Martingale system of every size are in the next table (column relations: F=A*C-B*D , G=F/E):
The last row of table shows the general formulas to calculate the statistics for series of any length. p is a probability to lose in one round, for even odds bets of roulette p=19/37. The last column of table displays the players edge in per cent. These values are the same to flat betting or to any other of bet systems.
Although the player edge of Martingale is not differ from a flat betting, but the other parameters vary with length of system. The next figure displays the distribution of net outcomes of flat system and some Martingale systems:
Analysis of this figure allows drawing a next conclusion. Using of Martingale of long length puts the gambling to more unpredictable result. However, it is necessary to note that outcomes distribution of any Martingale system has a form of bell curve in a long period. You can see that after 3000 rounds the Martingale-4 is a bell curve and the Maringale-6 is nearly to this state.
The following figure shows the progress of win\loss probabilities ratio. This ratio shows how many times a probability of net win is more then a probability of net loss. If ratio=1 than these values are equal. If ratio>1 then probability to win is more than probability to lose.
In conclusion it is necessary to summarize the main features of Martingale. The Martingale system has the highest net probability to win the serie. However, from the other point of view, it has the highest total bet amount. It means that Martingale player bet more then flat (or other system) player in the same period. Thus, Martingale player has the worst expected net result (in money, not in per cent!).