The Martingale system
The Martingale system says to double your stake after every loss, so that the first win gets back everything you lost plus one small profit. It wins a little very often and loses a lot now and then. The arithmetic says the losses cost more than the wins pay, and the book's limit makes sure of it.
The idea
You start with a base bet. Win and you pocket it and start again. Lose and you double, and keep doubling until a win comes, which pays off the whole run of losses and leaves you one base bet ahead. It looks like a machine for turning patience into money, which is why every generation of gamblers rediscovers it.
The short version, at even money, is on what a betting system is. Sports betting makes it worse, because the bet you are doubling on is not even money.
A run of losses at -110
At -110 you have to stake $11 to win $10. So to recover your losses and still finish $10 ahead, each bet has to win the total lost so far plus $10, and at -110 that means staking 1.1 times that amount. Here is how it grows, with the chance of getting that far on a game that is a true coin flip.
| Bet | Stake | Lost so far if it loses | Chance of losing this many in a row |
|---|---|---|---|
| 1 | $11.00 | $11.00 | 1 in 2 |
| 2 | $23.10 | $34.10 | 1 in 4 |
| 3 | $48.51 | $82.61 | 1 in 8 |
| 4 | $101.87 | $184.48 | 1 in 16 |
| 5 | $213.93 | $398.41 | 1 in 32 |
| 6 | $449.25 | $847.66 | 1 in 64 |
| 7 | $943.43 | $1,791.09 | 1 in 128 |
| 8 | $1,981.20 | $3,772.29 | 1 in 256 |
By the eighth bet you are putting up almost $2,000 to win $10. I wrote bets like that in my time, and I can tell you the man making them never looked like a man who expected to win $10.
The wall
Now give the book a limit. Say it will take $1,000 on this game, which is a plain example, not any particular book's figure. Bet 7 at $943.43 fits under it. Bet 8 at $1,981.20 doesn't. After seven straight losses you are down $1,791.09 and the book won't write the bet that was supposed to get it back.
That is what a limit does, and the Martingale is one of the reasons it exists. The same wall stands in a casino. Say a roulette table runs from $10 to $500 and you double on red. After six losses you are down $630, and the next bet would be $640, which the table won't take. On a double-zero wheel red wins 18 times in 38, so six in a row against you comes about once in 47 tries, and seven about once in 89.
Your own bankroll is the other wall, and it usually arrives first. To survive seven losses at -110 you need $1,791 behind a system that wins $10 at a time.
The sum on one run
Take the seven-bet run with the $1,000 limit. On a coin flip, 127 times in 128 a win arrives somewhere in those seven bets and you are $10 up. Once in 128 all seven lose and you are $1,791.09 down.
- 127/128 × $10 = $9.92.
- 1/128 × -$1,791.09 = -$13.99.
That is -$4.07 every time you start a run. Now the part people don't expect: on average a run puts $89.56 in action, and the vig at -110 costs 4.55 percent of whatever you wager, which comes to the same $4.07. The doubling didn't change anything. It moved all the cost into one rare, enormous loss, and left you paying exactly the book's price on every dollar.
Once in 128 sounds safe. Play 100 runs, which is two or three months of betting for some players, and the chance of hitting the wall at least once is 54 percent. At 200 runs it is 79 percent. The expected value page has the general sum behind this.
Why it feels like it works
Most nights it does. You start a run, you win within two or three bets, you are up $10, and you do it again. A player can go weeks without a long losing streak and decide he has found something. I watched that happen more than once from the book's side, and the book never minded, because it knew how the story ended.
The trouble is that a staking plan can't change the price. If each bet loses money on average, a clever way of sizing them only changes when you lose it. Reversing it, doubling after wins instead of losses, has the same problem from the other end: lots of small losses and an occasional big hit, still at the book's price.
Flat and dull
Flat stakes, 1 or 2 percent of your bankroll on every bet, win or lose. Never raise the stake to get back what you lost, which is the Martingale in a single sentence and the habit that has emptied more accounts than any bad pick. If you want to win, the work is in the price: line shopping, and bets that pay more than their chance of winning is worth.
Doubling up feels like control. It is the same bet, bigger.