The loser martingale

A martingale is a strategy of successive bets always ensuring a net gain. The major risk of a martingale is a possible long series of losses which would then force you to invest very (too much?) large stackes.
Sports betting is not purely radom game (see also winning at gambling using math ?) and it is possible to combine specific sports knowledge with mathematics. In this article we give such an example, building a mathematical martingale by following a (future) loser.

We focus on a player who is not, for sure, going to win the tournament, or in a team who is not, for sure, going to win all the matches in the championship. Of course zero probability does not exist, but with a minimum knowledge, we can point out a player who is really far from having any chance, for example a tennis player far in the ATP classement, in a grand slam tournament, or a football team of at least (or below) mid-ranking last year.

The idea is, like the classic martingale, to make a first bet, then in case of loss, to make another bet which will at least refund both stakes and ensure a given net profit, ...
The main point in a martingale is to be sure that it will end. This is achieved betting on the particular player or team we focused.
We will always bet on the defeat of our player or our team, who was chosen precisely to be sure that this defeat occurs after a few matches at most. Let's say, for example, that we place a bet of 10 euros on his defeat during the first match, at the very beginning.

In the first match, if the defeat actually occurs, we have won, and we stop there, ending the martingale.
Otherwise, we will bet on defeat in the next match, calculating the stake so that, in the event of defeat this time, our net profit remains 10 euros (redunding losses of the first two bets, plus 10 euros).
If in this second event, our player or team lose: we earn our promise 10 euros, otherwise, if our player or team win once more, we calculate the stake for the 3rd match, with the same objectives as the previous one (refunding previous losses, plus 10 euros).
…

Mathematical details and calculations

We now denote by FL our player or team doomed to lose soon (FL for "Fuzzy Loser", or even less politely for F...in' Loser).
We note each time m_i the wager and c_i the odds for the defeat of FL (or the odds of the opponent, regardless of the possibility of a possible draw: we bet on his defeat).
We have the following partial draw with successive victories of our expected loser, on which are written successive odds c_i and wagers m_i:

m₁ c₁ m₂ c₂ m₃ c₃

We therefore wager, to begin with, m₁ euros.

If FL loses, we win our bet, we're happy and successfull, and we stop there.
If FL wins, we lose our stake, and we will bet again on the defeat of FL, so that, in the event of an actual defeat, we earn a net gain of G euros (in addition to the reimbursement of the two stakes).
In case of defeat, the gross gain is m₂×c₂, from which we subtract the two stakes so as to focus on the net profit:
G_Net = m₂×c₂ − m₂ − m₁
and we want that this net profit to be G euros.
We then have
G_Net = G ⇔ m₂×c₂ − m₂ − m₁ = G ⇔ m₂ = G + m₁/c₂−1
In case of a defeat for FL in this second match, we therefore earn G euros net, and we are happy and successfull and we stop there, the FL martingale ends here.
Otherwise, we have also just lost m₂ and we are preparing to bet next m₃ in the third match on the defeat of FL so as to still earn G euros.
If FL loses, finally, we earn m₃×c₃ from which we subtract our previous lost wagers to get the net profit:
G_Net = m₃×c₃ − m₃ − m₂ − m₁
and we always want to earn G euros, that is
G_Net = G ⇔ m₃×c₃ − m₃ − m₂ − m₁ = G ⇔ m₃ = G + m₁ + m₂/c₃−1
and so on… with the fourth bet
m₄ = G + m₁ + m₂ + m₃/c₄−1

…

Example of martingale bets and loss probabilities

The following table gives an example of the evolution of successive bets in a martingale whose objective is to win 10 euros for sure, according to successive odds.
The probability of loss, loss of all successive rounds until the stake considered, is also given.
The odds of our FL loser decrease, as their probability of losing increases. For example:

round №	odds of	stakes	Probabilities of loss
1	2	10	50%
2	1,8	25	22%
3	1,4	112,5	6%
4	1,3	525	1%

The probability of losing very quickly is very low, but you have to be able to tolerate an unexpected additional victory.
The following calculator

calculates automatically the successive bets for a martingale with other odds/probabilities.

Stakes calculator

A martingale can quickly be doomed to fail, because of the increase in stakes, from one round to the next, which can be dizzying. Complete mathematical details can be found on this page and where we see (after sophisticated mathematical probabilities calculations) that the amount to invest can quickly become astronomical if we lose a few rounds more.
It is therefore very important:

to choose a player/team who will lose quickly (very feasible in practice: few football teams rack up several victories consecutively; many "average" tennis players will certainly not go through several rounds in grand slam tournaments).
estimate the successive bets from the beginning to ensure that you can follow financially: stopping a martingale midway, before winning, means losing several bets, possibly important ones.
The following calculator evaluate the successive bets for the martingale presented above.

The stake to wager depends directly on the odds. For odds of 2, you must double the previous and lost bet in order to ensure a net gain. If the odds are less than 2, which is certainly the case for a future losing player FL or team, then the stake must be more than doubled up.
It can be interesting to follow martingales with lower odds, which is the case for example for a martingale on draws since these are quite often underevaluate and have odds of around 3.