Martingale and Betting Systems: Why They Fail

A betting system is a rule for changing the size of your bets according to what has just happened. The most famous, the martingale, doubles your stake after every loss. None of these systems can beat the house edge, because in a game with independent rounds the expected result of a series of bets is the sum of the expected results of the individual bets, and each one is negative. What a system does change is the shape of your risk: it turns a small steady chance of loss into a high chance of a small win and a low chance of a large loss. This guide shows the working, and a simulation you can reproduce.

How the martingale works

1. Bet a base amount on an event that pays roughly even money, such as red on roulette.
2. If you lose, double the stake. If you win, go back to the base.
3. Because each win pays double the stake, one win recovers all earlier losses in the run and adds one base unit of gain.

It feels like a sure way to gain one unit each time. The catch is in the stakes: after n losses in a row the next stake is 2^n units, and the total you have lost is 2^n - 1.

Every run of losses is rare, but the sums grow faster than the chances fall. A balance of 1,023 units survives ten losses in a row but cannot place the eleventh bet. The chance of a streak that long happening at some point in a long session is far larger than the chance at any single start.

The maths: the edge never goes away

On a single-zero wheel, a bet on red wins 18 times in 37 (48.65%). Here is the key calculation for Cashout Arcade's version, which returns 1.9938 coins per coin staked when it wins. Suppose you win after j losses. The stake on the winning bet is 2^j, you have lost 2^j - 1 before it, and the win returns 1.9938 x 2^j. Your net is:

net = 1.9938 x 2^j - 2^j - (2^j - 1) = 1 - 0.0062 x 2^j

At a casino with a fair even-money payout the last term would be zero (or tiny), and the net would be +1 every time. With any house edge, the shortfall grows with the stake:

So with a 3% edge, a martingale recovery does not even recover after about 8 losses in a row: you win the bet and still end the run behind. This is the edge showing through, not bad luck. In general, the expected loss of any system is the house edge times the expected total you wager. Wagering more (as martingales do after losses) increases the expected loss in coins, even while the chance of ending a short session ahead rises.

What a simulation shows

Here is a simulation anyone can reproduce. Roulette, red, 18/37 chance to win, returns 1.9938 on a win. 100,000 sessions per row, a seeded random generator. Each session starts with a 1,000-unit balance and stops when the balance reaches 1,100 (a goal of +100), when it runs out of coins (a stake larger than the balance is replaced by the whole balance), or after 5,000 bets.

Read the last column. Whatever the strategy, the average loss is about 3% of the total amount staked. The martingale's average loss is smaller than flat betting in coins because it stakes less in total, not because it has a better rate. And look at the shape: 87% of martingale sessions reach their goal, but 12% lose the entire 1,000 balance, and the average still comes out negative.

A smaller goal makes the same point. Aiming for just +10 from a 1,000 balance with a 1-unit base and no cap, the martingale reaches the goal 98.6% of the time and loses everything 1.4% of the time. The average result is -3.9 on 127 units wagered, again about 3%.

A fixed number of rounds shows the same pattern. Over 100 rounds starting at 1,000 with a 1-unit base, 92.7% of sessions end ahead, yet 6% end more than 100 units down, and the 5th percentile is a loss of 977 units. Over 300 rounds, 83% end ahead and 17% end more than 100 down. A system that makes you win most sessions and can erase everything in a minority of them is exactly the profile of a bet with a negative average.

(The simulation uses the exact payout of Cashout Arcade's roulette, 1.9938x. A standard casino wheel at even money, with its 2.703% edge, would show the same pattern with slightly smaller losses.)

Table limits and caps

Real casinos set a maximum bet on every table, and apps do too. A martingale from 1 unit with a limit of 500 can double only nine times (1 + 2 + ... + 256 = 511 lost), and the next stake of 512 is not allowed. Cashout Arcade's caps are payout caps (no round pays more than 50,000 times the bet or 250,000,000 coins), not table limits, but its balance is a limit of its own, and the simulation above shows the capped and uncapped versions give almost the same result. The doubling runs out of money long before it runs out of table.

The mathematical reason is the optional stopping theorem: you cannot turn a sequence of negative-expectation bets into a positive one by choosing when to stop or how much to bet, as long as your bets and balance are bounded. Wikipedia's article on the martingale betting system) cites Mitzenmacher and Upfal's Probability and Computing for the proof, and notes that only with unlimited wealth, bets and time could the system be argued to win.

Other systems, briefly

All of them change how wins and losses are distributed over time. None of them touch the probability of a single bet, so none can change the expected return.

Why the edge stays

The house edge is a property of each bet. A system is only a rule for choosing which bets to make and how big. Adding up negative numbers, however you weight them, gives a negative number. This is also why patterns in the outcomes do not help: Wikipedia's roulette article notes that because each spin is independent, no betting progression changes the negative expectation, and that pattern-based systems lose on average even though players can win in the short run. The streaks guide explains why "wait for a streak" fails.

Why systems are risky

• Exponential stakes. Twenty losses in a row from a 1-unit start means a next stake of 1,048,576 units. It is rare, and it is the case the system exists to hide.
• Small gains, large losses. The reward is capped at one unit per run, but the risk is your whole balance.
• They hide the cost. A string of tiny wins can feel like the system is working, when the edge is accumulating on a growing volume of bets.
• They encourage chasing. Raising a bet to win back a loss is the behaviour most associated with harm, and a doubling system makes it automatic.

If you notice yourself raising stakes to recover, that is a signal to stop. The play-safely page has practical limits, and the house edge guide explains the cost of every coin you stake.