If It Looks Like a Duck... It Might Not Be a Duck (Part 2): The Martingale Betting System
This is Part 2 of an essay about why it is problematic to infer belief in the gambler’s fallacy from gambling choices. If you haven’t already read Part 1, start there. It covers:
Pascal’s Casino Wager – The idea that once a gambler has committed to playing, even the smallest possibility that the gambler's fallacy isn't a fallacy can justify behaving as though it were true.
Hot Versus Cold Reasoning – The idea that we can both believe and disbelieve something depending on the interplay of rational thought and emotional impulses.
Beliefs Violating the Assumption of Event Independence – The idea that the gambler's fallacy assumes gamblers accept event independence, which is often not the case.
This post—Part 2 of the essay about the challenge of linking behaviors to beliefs—focuses on the Martingale betting system, a strategy often used by beginner roulette players. Superficially, it appears to reflect belief in the gambler’s fallacy. However, it has little to do with that fallacy.
Contents
The Gambler’s Fallacy: A Brief Recap
The gambler’s fallacy is the belief that after a streak of the same outcome—such as five reds in a row in roulette—the likelihood of that outcome occurring again decreases, even for independent events like a fair coin flip or a fair roulette wheel spin. If this belief doesn’t seem like a fallacy to you (if you think, for example, that heads is less likely after multiple heads in a row), you’re not alone. Many people do not believe it’s a fallacy—which, paradoxically, is part of why it's a well-documented fallacy.1
If you're skeptical, start with this essay on the law of large numbers, which explains why chance does not “self-correct.” Then read the first post in the gambler's fallacy series. If you'd prefer other sources, see this ChatGPT query, this Wikipedia page, or this peer-reviewed journal article on belief in the gambler’s fallacy.
The Martingale Betting System
Martingale systems involve increasing bet sizes after losses and decreasing them after wins, much as would be expected with belief in the gambler’s fallacy. The simplest form begins with the table minimum (one unit). After a loss, the player doubles their bet. After a win, they revert to the minimum. This ensures that each win results in a net profit of exactly one unit. To understand why, consider the following examples:
Example 1: W1 – The player bets 1 unit and wins. Net profit: 1 unit.
Example 2: L1, W2 – The player loses 1 unit, then doubles the bet and wins 2 units. Net profit: 1 unit.
Example 3: L1, L2, W4 – The player loses 1, then 2, then wins 4. Net profit: 1 unit.
… Example 11: L1, L2, L4, L8, L16, L32, L64, L128, L256, L512, W1024 – The player loses 10 times in a row before winning 1024 units, having lost a total of 1023 units. Net profit: 1 unit.
This system allows the player to win small amounts consistently—until they hit a table limit or run out of money.
The Appeal of Martingale
The appeal lies in the idea that a win is inevitable, provided there is no limit to the amount that can be bet. Consider, for example, if the gambler’s bankroll and casino limits were large enough to withstand 50 losses in row:
Even in American roulette (which has both a "0" and "00" and an advantage to the casino of 5.26%), 50 consecutive losses will occur less than once for every 8.66 trillion wins.
That’s rare enough that one could use this strategy to win 50 units every day for about as long as the earth has been a planet, and still only lose about once, on average.
The minimum bet in roulette in today’s US casinos is usually $25 or more. With that minimum, players could expect to win $1250 every day, with essentially no chance of losing.
Most players who decide to give martingale a try do not think about it in such specific quantitative terms. Instead they might say something like this: “Come on! How often do you lose 10 times in a row? It never happens.” In either case, on first consideration, martingale can seem like a save method to beat the casinos. And it has nothing to do with belief in the gambler’s fallacy.
Why Martingale Fails
Casino Limits and Bankroll Constraints
Casinos impose table limits. In Las Vegas, the ratio between maximum and minimum bets is usually less than 1000:1. Suppose a casino allows a 1024-unit spread, letting a player withstand 10 losses before reaching the maximum, 11th, bet (as in Example 11 above):
The chance of losing 11 times in a row is 0.086%.
The player will win about 1164 sessions for every one losing session (where each losing session involves a streak of 11 lost bets in a row).
But each win earns 1 unit, while each rare loss costs 2047 units (the sum of all 11 bets from 1 to 1024 units).
Over time, the player would lose about 2047 units for every 1164 they win – a net loss of 883 units!
With a $25 minimum bet, they'd need a $51,175 bankroll and they could expect to lose $22,075 ($25 * 883 units) after combining all the winning sessions with the one losing session.
For a player who tries to win 50 units per playing session and who plays about 23 days per month, that would be a monthly loss.
But Betting Limits Are Not the Only Problem
People who accept the above shortcoming of the martingale system often still believe it would work, if only the casinos did not cap the maximum bet. That view is also flawed. Imagine, for example, if the player had a bankroll that could withstand 20 losses in a row instead of 11, and the casino was willing to allow it. Even with American roulette’s two zeroes, just adding those extra 9 losses would reduce the frequency of a bankroll-maximizing loss from once in every 1165 tries to once in every 375,900 tries, making the risk of losing trivial. The problem is that the bankroll required to cover that losing streak is 1,048,575 units (or $26.2 million given a $25 minimum bet). Players could earn significantly more by putting that money in a conservative interest-bearing government treasury bond while sipping daiquiris at the beach, rather than sitting for hours everyday at the roulette table.
Given the Costs, Why Do Gamblers Use Martingale?
The short answer is that experienced gamblers do not use martingale.
Beginners often do, however, for a few reasons:
The face-value appeal of the system: The most obvious reason was already discussed: on first consideration, the martingale system can sound like a winning strategy. Few players sit down and work out the numbers and realize how high the risks are relative to the gains, even after just a month of play. They often have to learn that the hard way.
Failure to commit: Many beginner players continue to use the martingale system for some time even after suffering extensive losses. The first reason for this is that they often get cold feet, failing to commit to the system till the very end, once their streak of losses gets close. Imagine, for example, that you’ve just lost the 256 unit bet (so perhaps $6400 given a $25 minimum bet). That’s already a total loss of $12,775. You have two more bets to go before you’re finished. That’s another $38,400 at risk, and now there’s more than a one in four chance it will happen (a 27.7% chance, to be exact). It is common at such a point for players to lose confidence in the system and stop betting, only to see that they would have won had they stuck with it to the end. That is enough for some to give the system another try.
Using “bad luck” to rationalize a loss: Even players who do stick to the system to the end often rationalize their loss with reference to bad luck. The losing streak needed to lose with martingale is longer than most beginner roulette players have ever seen, much less experienced themselves. And—as all gamblers who have had a winning session are aware—luck matters. Even gamblers with authentically winning systems—such as skilled card counters in blackjack or professional poker players—sometimes have losing streaks. It is not unreasonable for a person convinced that the martingale system works to dismiss a loss as an unfortunate blip in an otherwise winning strategy.
That said, many beginners who try out the martingale system are fortunate enough to learn about its shortcomings before they every experience that first major loss (much less a second or third). That is because roulette, like other casino games, is played in a social setting, allowing the opportunity to learn from others. Players can learn vicariously by watching other gamblers try the system and fail. More often they learn from they advice of more experienced roulette players, nearly all of whom are familiar with the system and can tell compelling stories about its dangers.
In any case, whether players learn about the dangers of the martingale system from others or learn about it the hard way, with enough experience, roulette players eventually learn that martingale is to be avoided. Because of the harsh impact of this strategy, even though martingale can be confused for belief in the gambler’s fallacy, it tends to have far worse consequences. Most people who believe in the gambler’s fallacy just change their bet from one color to another (assuming the choice is between black and red in roulette). Perhaps they’ll increase their bets, too, but not so dramatically as with the martingale system. As such, belief in the gambler’s fallacy tends not to be so dramatically extinguished or adamantly opposed as belief in martingale betting systems.
A Time When Martingale Makes Sense
There is one situation where Martingale can be a rational choice: when a player desperately needs a small win with minimal risk. Suppose someone needs exactly $50 to afford a lifesaving medical procedure. Martingale will almost always succeed in this case. By contrast, switching between red and black based on recent outcomes (the gambler’s fallacy) provides no advantage.
Conclusion
While martingale can look like a strategy based on the gambler’s fallacy, its appeal is rooted in a different kind of misunderstanding of probability and risk. Unlike the gambler’s fallacy, it promises small, frequent wins with little chance of losing. The rare catastrophic loss, however, more than makes up for the small wins. Attempts to minimize the likelihood of such losses to make the risk seem trivial require bankrolls so large that the money could be invested elsewhere with less work and greater returns. The gambler’s fallacy, on the other hand, can coexist with betting strategies that do not have any impact on the frequency or size of wins relative to losses, as explained in a recent essay, “Belief in the Gambler’s Fallacy Is Trivial.”
The next essay will continue the discussion about why it is so difficult to infer belief in the gambler’s fallacy from behavior, starting with a discussion of chasing. Chasing behavior looks similar to martingale and has similar consequences (usually working, but occasionally having disastrous consequences). But its motivation is quite different from martingale, and is again unrelated to belief in the gambler’s fallacy.
Understanding how it can be true that many people believe in the gambler’s fallacy and that the gambler’s fallacy is a myth, the final point of this series of essays, will have to wait until that final essay.