Staking Progressions (Martingale, Fibonacci, d'Alembert)

In one sentence

Staking progressions such as the martingale, Fibonacci and d'Alembert change your stake based on previous results, but they cannot turn a losing or break-even bet into a winning one, and they sharply raise your risk of ruin.

How it works

The martingale doubles the stake after each loss and goes back to the base stake after a win, so one win recovers all losses plus one unit. Fibonacci moves one step up the sequence 1, 1, 2, 3, 5, 8, 13 after a loss and two steps back after a win, while d'Alembert adds one unit after a loss and removes one after a win. Loss-chasing is the informal version: bigger bets to "get back to level".

They feel like they work because most sessions end in a small profit. But the rare losing session is huge, and its size balances its rarity exactly, so the average stays at the edge in the odds.

The reason is simple: every bet has its own expected value, set by the odds and the true probability. Past results do not change the next outcome, so no rule based on past results can change the expected value of the next bet. Changing stakes only changes how much of that expected value you buy.

The maths

• s with subscript i is the stake on bet i, which may depend on earlier results.
• e is the expected profit per £1 staked, the same on every bet at the same odds and probability.
• E means the average over all possible sequences of results.

In plain English: your expected profit is your edge multiplied by how much you stake in total. A progression can make you stake more, but it cannot change the edge, so with no edge you expect zero and with a negative edge you expect to lose more.

Worked betting example

Martingale at evens (2.00) on a fair 50% bet, for example Over 2.5 goals in a match where the price is exactly right. Base stake £10, no commission for now.

1. Seven losses in a row cost £10 + £20 + £40 + £80 + £160 + £320 + £640 = £1,270. The eighth bet needs £1,280.
2. Chance of seven straight losses: 0.5 to the power 7 = 1 in 128, about 0.78%.
3. Expected result per cycle with a £1,270 bank: 127 ÷ 128 × £10 − 1 ÷ 128 × £1,270 = £9.92 − £9.92 = £0.
4. Chance of hitting that losing run at least once in 100 cycles: 54%.

Commission makes it worse. At 2% commission, a win on the £640 stake after six losses pays £627.20, but you had lost £630, so the "guaranteed" cycle ends £2.80 down.

We simulated 20,000 players, each with a £1,000 bank and £10 base stake, for 500 bets at 2.00. "Bust" means the bank could no longer cover the base stake.

Every system returns exactly what the odds allow per £ staked. With the edge against them, average losses were £101 for flat stakes but £197 for martingale, £179 for Fibonacci and £374 for d'Alembert, simply because they staked roughly two to four times as much in total.

Where it's good

• Understanding why these systems fail, so you can spot them in tipster sales pages and trading courses.
• Recognising loss-chasing in your own betting.
• There is no betting situation where a progression improves expected value; if you have an edge, Kelly or level stakes use it better.

Limitations and pitfalls

• They cannot create an edge. Expected value per £ staked is fixed by the odds and the true probability.
• They need an unlimited bank and no maximum stake. A real bank runs out long before the doubling does.
• High session win rates hide the problem until the rare wipeout arrives.
• Commission charged on each winning market means a martingale win does not even recover losses at higher steps.
• Stakes grow fastest exactly when your bank is smallest, the opposite of sensible risk control.
• Loss-chasing is a well-known behavioural trap; big stakes after losses usually reflect emotion, not new information.

How to build it

• The only thing worth building is a simulation to prove the point to yourself: numpy, a loop per system, thousands of runs.
• Record return per £ staked, not just profit, so you see that the edge never changes.
• Practical tip: if you have an edge, use level stakes or fractional Kelly; if you do not, no staking plan will help.

Related methods

• Expected value – the quantity no staking plan can change.
• Risk of ruin – what progressions increase.
• Fixed stakes – the baseline any system has to beat.
• Behavioural biases – why loss-chasing feels right.
• Monte Carlo simulation – the way to test any staking plan.