Martingales for games of chance

Understand and distinguish winning strategies from others

A bet is by definition risky: you bet and you can lose your stake or earn a profit proportional to the stake made. We're here talking about ONE bet. A martingale is a strategy of betting on several shots, taking into account in particular previous results.
The goal is of course to try to increase your (chances of) winning. The ultimate goal is to win every time.
To achieve this, in a martingale, we correctly adapt the bet at each stage. That's it, everything is said, all the effectiveness, or not, of a martingale lies in this adverb: correctly.

You have to be careful, even suspicious, about the martingales you can find: not all martingales work, and that's what we're going to point out here.

First example of an ineffective martingale

To win the lottery, some say that it is wise to play numbers that have not come up much recently: as the probability of each number coming up is the same, numbers that have not come up very often "will have to catch up" and therefore in the days to come their probability will therefore be greater.
This thought amounts to affirming, for example, that a coin that has been tossed 10 times and getting tails 10 times, will have a greater chance on the 11th toss of coming up with heads. This reasoning is absolutely false: the coin does not remember that it "did that before" (on the other hand, what is remarkable here is to have obtained Tails 10 times consecutively...) Martingales based on this type of reasoning advocating the statistical observation of the results of chance, at best, do not change the math expectation of gain at all.
This reasoning for pure games of chance must on the other hand be adapted to sports betting where the statistics of players and teams provide valuable information (unlike a coin, a player remembers that he has won a lot, or not, previously , and this influences its future results).

Martingales for games of chance

Martingales were initially developed for "pure games of chance", such as casino games. In particular, in these games the rule is simplistic: if you lose, you lose your stake, and if you win, you double your stake.
This amounts to saying that there are only two results, and one of the two with odds of 2.
Many martingales, including the most famous classic doubling martingale we recall below, are ineffective, or even erroneous in a context more complex and varied like that of sports betting where the odds vary and there can be more than two possible outcomes.

Classic martingale (doubling stakes)

A martingale is therefore a betting strategy in which a certain number of games are played consecutively, adapting the stake to each game.
In the classic martingale, this bet is multiplied by 2 after each loss.

We bet at the very beginning for example 1 euro, then at each game:

if we win, we are happy, and we stop there, the martingale is over and we collect our winnings.
if we lose, we wager in the next game double the bet of the previous game.

Which give:

I wager one euro: if I win, I earn 2 euros, or 1 euro net profit.
if I lose, I wager two euros in the next game, so 3 euros in total from the start:
- f I win, I win 2×2 = 4 euros, so 1 euro net profit.
- if I lose, I wager 4 euros in the next game, so 7 euros total bet from the begining:
  - if I win, I earn 4×2 = 8 euros, so always 1 euro net profit.
  - if I lose, I now wager 8 euros, ...

As soon as I win, my net profit is 1 euro, and since, if the game is not rigged, I will inevitably end up winning at some point, this profit is assured. In other words, with this strategy, I definitely win 1 euro.
Unfortunately, this strategy is dangerous because my total bet increases very quickly (after 10 bets for example, I am at a total stake of more than 1000 euros...), and if I stop before having postponed my euro of net profit, I finally simply lost my investment which is also very important for a profit which seems more and more ridiculous: only 1 euro…
Complete mathematical details can be found on this page and where we see (after sophisticated mathematical calculations of probabilities) that the sum to invest to respect our martingale and finally win can not only quickly become astronomical, but also that the probability that this happens is not that weak.

The Fibonacci-Whittacker Martingale

I announced it: these martingales originally designed for games of pure chance are not to be applied directly to the world of sports betting.
The Fibonacci-Whittacker martingale follows this rule. It is interesting and fun to study, hence its place here, but it is not recommended to use it as is. If only because here too, it involves playing a game where the stake is doubled in the event of a win (odds of 2).
I also detail it for those who have esoteric, mystical inclinations, etc.
Indeed, the famous Fibonacci sequence has inspired since many centuries, and we keep finding it everywhere. From there to thinking that it contains a secret, a divine structure (it is linked to the golden ratio: the divine proportion!) which would make it possible to overcome chance...
This famous Fibonacci sequence (dating from the beginning of the 13th century, and stated in a small recreational problem), begins like this

1 - 1 - 2 - 3 - 5 - 8 - 13 - 21 - …

Each number in the sequence is obtained by adding the two previous ones. We use this sequence to bet successively according to the three rules:

in the event of a loss in a round, we wager the next number in the Fibonacci sequence
in the event of a win, we go back two notches for the bet of the next round
if we have a positive overall profit, we stop and we can resume the martingale from the beginning

This is also the idea of the martingale from which we derive in this sequence that of successive bets, summarized in the following table

bet №	stake	totale stake	profit
1	1	1	2
2	2	3	4
3	3	6	6
4	5	11	10
5	8	19	16
6	13	32	26

Profits and risk of the Fibonacci-Whittacker martingale

In the Fibonacci martingale a victory cancel the last two losses generated. The progression of bets in this martingale is therefore less rapid than for the classic martingale in which we systematically seek, at each round, to cancel all the bets made up to that point.
Finally, if the Fibonacci martingale can be very profitable in the short term, it nevertheless remains very risky as soon as a few losses follow one another, and leads like the classic martingale, after a few games, to a profit which ultimately seems quite low compared to the stakes committed.
Finally we cannot leave this martingale without talking about the golden ratio. We find that in the Fibonacci sequence we calculate the ratio of two consecutive terms, "far enough in the sequence" (more precisely as a limit when n tends towards infinity).
This means that as we move forward in this sequence, we tend to multiply a term by the golden ratio φ≃1.61 from one term to the next, therefore from one bet to the next.
For the classic martingale, we multiply by 2 between each bet, which explains that in the Fibonacci martingale the increase in bets is less brutal, more moderate.

Still other martingales

Yes, there are many others: the great martingale, the paroli martingales, the d'Alembert martingale, the Labouchère martingale, etc.
All these more or less famous martingales have in common that they are winning strategies (but nevertheless risky as we have clearly seen) for games of pure chance where the stake is doubled in the event of victory. In sports betting, they therefore only have a very educational use: understanding how they work in order to be able to better adapt.

Building a winning martingale

Two parameters make classic martingales particularly risky:

The gain is "only" doubled in the event of a victory, which requires you to bet increasingly larger to compensate for the losses from previous defeats and therefore quickly increases the total stake.
the probability of winning is low, one chance out of two here

In order not to increase too much the total stake invested in the successive parts of a martingale, it is therefore necessary to concentrate on these two parameters.
Here are several martingale possibilities specifically and mathematically developed for sports betting with varying odds:
The ideal, for martingales, is, once you have fully understood the issues, risks and difficulties, to be able to create your own martingales.
The automatic calculator for building your own martingale is a simple and essential tool to do so.