Martingale game

A game of Pharo, Johann Baptist Raunacher (1729–1771), Eggenberg Palace near Graz

Roulette game around 1800

As Martingale or short Martingale is known since the 18th century a strategy in gambling , especially when Pharo and later at roulette where the insert is increased in case of loss.

Classic martingales

description

The classic and simplest form of the Martingale , the Martingale classique , is doubling or doubling and is illustrated using the roulette game.

The player starts with a bet of one unit (one piece ) on a simple chance, e.g. B. Rouge or Noir .

The Martingale player mostly relies on the Perdante (see Marche ), which is the last chance that lost: if the ball last fell on Rouge , he therefore relies on Noir .

If he loses, he bets two pieces in the next coup, if he loses again, he bets four pieces, etc. As soon as he wins, all losses that have occurred up to that point are canceled and the player can look forward to a total profit of one piece. After winning, he continues his attack on the casino again with one piece.

However, this apparently safe system does not work - which countless players are not convinced of despite their own experience to the contrary: Many players overlook the fact that continued doubling is no longer possible at the latest when the maximum stipulated by the casino (i.e. the maximum bet) is reached, but mostly it is fails much earlier due to the limitations of one's own gambling capital.

Who z. B. consistently plays Martingale with a playing capital of 1000 pieces, which will with a relatively high probability complete a certain game stretch (e.g. 50 coups) with a modest profit of about 25 pieces, but he will bear the usually completely underestimated Risk of losing your entire fortune: Overall, profit expectations are negative; d. H. in the long run the casino wins.

Precisely in the fact that a Martingale player achieves small winnings relatively often and has to wait a long time for the (in the mathematical sense) certain total loss to occur, the explanation for the phenomenon of the proverbial beginner's luck lies : Who is not constantly with in the course of a game evening plays the same high stakes, but increases the stakes in whatever way, has - like a Martingale player - relatively good chances of winning back any losses and ultimately finishing with a positive balance.

example

A player should play the Martingale for Impair at roulette. To illustrate this, a few simplifying assumptions are made:

Zéro means loss like any other pair number (i.e. there is no prison)
The starting stake, a "piece", is 10 €, after a loss, 20, 40, 80 € etc. are placed one after the other.
The maximum set by the casino is € 20,480, so the Martingale can contain a maximum of 12 games.

The probability of losing a single game is 19/37 ≈ 51%.

The probability of losing 12 games in a row:

{\ displaystyle \ left ({\ frac {19} {37}} \ right) ^ {12} \ approx 0 {,} 0336 \, \%}

(i.e. approx. 1: 2974)

Accordingly, the probability of ending a Martingale with a profit is:

{\ displaystyle 1- \ left ({\ frac {19} {37}} \ right) ^ {12} \ approx 99 {,} 9664 \, \%}

.

That really seems to be an excellent chance, but: In the event of a lucky conclusion, the player only wins just € 10, while in the event of a loss he loses € 40,950 (namely ). ${\ displaystyle 10 + 20 + 40 + \ dotsb + 20480 = 40950}$

For the casino, it is not so much the probability of winning as the expected profit that is important. However, the expected value for the player is negative:

{\ displaystyle 0 {,} 999664 \ cdot 10 \, \ mathrm {EUR} -0 {,} 000336 \ cdot 40950 \, \ mathrm {EUR} \ approx -3 {,} 77 \, \ mathrm {EUR}}

d. H. approx. 37.7% of the initial sentence in a game series.

Apart from that, it can hardly be assumed that a player who enters the casino with a capital of € 40,950 will start playing with a stake of only € 10, so that for him a much shorter series of losses means the loss of the entire gaming capital (i.e. ruin in the mathematical sense).

Generally it can be calculated from the maximum number of game (in this case ) and the start of use (here the (always negative) compute the expected value): ${\ displaystyle n}$ ${\ displaystyle n = 12}$ ${\ displaystyle S}$ ${\ displaystyle S = 10 \, \ mathrm {EUR}}$

{\ displaystyle E (n, S) = \ left (1- \ left ({\ frac {38} {37}} \ right) ^ {n} \ right) \ cdot S}

The fact that a player only loses half the stake in the event of Zéro can be depicted in such a way that a player loses his stake with a probability of 18.75 to 37 and with a probability of 18.25 to 37 wins a unit - the bank advantage in a single game this corresponds to just 0.5 / 37 ≈ 1.35%, as is also the case in the actual game with Prison.

The following values are then obtained:

With a probability of ≈0.0287% (i.e. 1: 3487) the martingale will be lost,
the Martingale will be won with a probability of ≈99.9713%.

The expected loss is therefore approx. € 1.75, ie approx. 17.5% of the initial set per game series.

Variants of the martingale game

Aside from doubling , a large number of other Martingale strategies have been developed, the most important examples - because on the one hand historically interesting and on the other hand widely used

the Montante Américaine or Labouchère , Labby , Annulation Américaine ,
the Montante Hollandaise or Annulation Hollandaise ,
the Progression d'Alembert , named after the mathematician Jean Baptiste le Rond d'Alembert ,
the Fitzroy system ,
the Montant et Démontant system .

All of these game strategies, whether it is that the stake is increased in the event of a loss, or that it is increased in the event of a win (see parole game ) or that consistently played with the same stake ( mass égale ), are actually not promising: The mathematical proof for the non-existence of safe winning strategies can be provided with the help of the martingale theory.

etymology

The word "Martingale" comes from Provencal and is derived from the French town of Martigues in the Bouches-du-Rhône department on the edge of the Camargue , whose inhabitants used to be considered somewhat naive. The Provencal expression jouga a la martegalo means something like "to play very daring".

Since the martingale was and is the best known game system, the term was also used as a synonym for "game system", and so this game system gave the martingale theory, a branch of probability theory, its name.

The equally "Martingale" said Reins in equestrian sport should be also named after the town of Martigues, but this meaning of the word "Martingale" with the game system to do anything.

Giacomo Casanova

Giacomo Casanova, 1788

One of the many Martingale players was Giacomo Casanova , he notes in the story of my life :

Before that, MM asked me to go to her casino, get some money there and play at half-game with her. I did and took whatever money I found. With this I played the martingale, always doubling the sentences; I won daily until the end of the carnival. I was fortunate enough to never lose the sixth card; and if that had happened to me, I would have run out of gambling capital; for this sixth rate was two thousand zechines . I was glad to have increased the treasure of my dear lover.

After some initial luck, Casanova notes a little later:

I was still playing my martingale, but so unhappy that I soon ran out of Zechine. Since I was playing on joint account with MM, I had to give her an account of my finances.

At her urging, I gradually sold all of her diamonds. I lost the proceeds again; she kept only five hundred zechines for herself in case of need. There was no longer any talk of kidnapping; for how could we have made our way through the world penniless?

Because of this literary evidence, doubling is also called Martingale de Casanova .

Individual evidence

↑ The Origins of the Word "Martingale" (PDF; 1.3 MB). In: Jehps.net , June 2009.

[1] The Origins of the Word "Martingale" (PDF; 1.3 MB). In: Jehps.net , June 2009.