What are you? Chicken?

WHAT IS MARTINGALE?

Most people know about the martingale betting system. If you don’t, I’ll explain. The martingale is a betting system where the player doubles his or her bet after a loss, until the player wins a unit. Martingale is usually, but not always, deployed on even money games, where both win/loss outcomes have similar probabilities of occurring.

A unit can be any increment of your total bankroll. For example, if you are flipping a coin, and your bankroll is $100, your number of units can be any amount divisible into your $100. If your betting increments are $1, then you have 100 units. If your betting units are $5, then you have 20 units, and so forth.

If you’re confused about the term ‘unit’, then just think in terms of ‘dollars’.

MARTINGALE WALKTHROUGH

Let’s say your units are $1.

The progression works like this…

Bet 1, Bet 1 unit. Win. Do not increase bet. End series.

Bet 1. Bet 1 unit. You lose. You are now -1 for this series. Double next bet.

Bet 2.  Bet 2 units. You lose. Score: -3 for series. Double next bet.

Bet 3. Bet 4 units. You lose. Score: -7. Double next bet.

Bet 4. Bet 8 units. You lose. Score: -15. Double next bet.

Bet 5. Bet 16 units: You lose. Score -31. Double next bet.

Bet 6. Bet 32 units. You win. Score +1. End series. Next bet is 1 unit.

Bet 1. New series. Bet 1 unit…

Total session win so far: +2 units.

Notice that the next bet after the loss, if there is a win, will always result in a +1 for the series.

With martingale, you will have many little wins. It’s not uncommon in a casino environment to have weeks or months worth of wins, only for the wins to one day be wiped out by one bad session at the baccarat or roulette table betting the even money bets, or any other even money game that is close to 50/50.

Most people who have not worked this out on paper intuitive know that at some point, martingale will fail. For any of us who have been gambling for any period of time, we’ve all seen and lost, ourselves, 10+ hands in a row. You will lose the maximum number of bets and reach the limit, especially if you are in a real world casino with table maximums.

This isn’t just hypothetical. This is reality that you can prove to yourself by deploying martingale in a casino for a week or so.

On its’ face, the flaws of the martingale betting system seem obvious. But are they really what they appear to be?

Let’s discuss…

A PERCEIVED  FLAW OF MARTINGALE THAT’S NOT A FLAW

One of the most cited complaints about martingale is that the bettor, at some point in the progression, must bet a large amount to eventually win a small amount. If you examine the walkthrough above, it seems obvious.

If the bettor loses 10 in a row, by the 11th bet, the bettor will need to bet 1,024 units to win a single unit.

On its’ face, it seems unwise to bet 1,024 units to win a single unit; however, the gross amount bet versus the pay off over the progression is of secondary importance to the overall expected return of the bet. If the bettor has the advantage, it doesn’t matter if the bettor is betting 1 million units to win one unit (assuming a fair game and no discounts due to nonpayment or other reasons).

There are actually many games where the person or entity offering the bet is laying odds. In roulette, the casino is offering to pay 35 units to win a single unit. In Let It Ride, the casino can offer 20,000 units to win a single unit. In other games, the casino is sometimes offering to literally pay 1,000,000 units for a single unit. The casino can make the bets because they have the advantage in the games.

It can be argued that the casino can offer 1 million units to win a single unit because the casino has the bankroll to sustain the bet. That assertion ignores the fact that in a negative expectation proposition, the bankroll size does not change the expected value of the proposition. Even if the casino had an infinite bankroll, if the casino offered only games where players had the advantage, the casino would go broke. The ratio of the amount won versus the amount risked is of secondary importance to the mathematical advantage held by the casino.

The fact that the bettor must risk a large amount to win a small amount does not affect the expected value of the bet.

THE REAL PROBLEM WITH MARTINGALE

Even in a hypothetical casino, with infinite limits, the same problems that I described above would occur. The reason that martingale would fail in a hypothetical casino with infinite limits is because martingale does not change the expected value of the proposition. Therein is the root of the real problem with martingale.

The real problem with martingale is NOT that you will run into a long losing streak that abuts against the table maximum, or that you will incur a giant loss that wipes out your wins, or that you will need to bet 1000+ units to win a single unit. Granted, those are problems, especially if you have an insufficient bankroll; but they’re just symptoms of the real problem.

Because martingale does not change the expected value of the bet, by deploying martingale in a negative expectation bet, you are significantly ballooning your expected loss.

Let’s first explore this further, and then I’ll show you how this knowledge can help you.

ILLUSTRATION OF THE REAL PROBLEM

To illustrate the problem, let’s apply two scenarios: one with martingale and one without martingale. We will now calculate the expected loss for both scenarios.

Martingale with $1 per unit.

Roulette, house edge (HE) of 5.25%

Bet 1. Bet 1 unit. Lose. Total score: -1. Double next bet. Based on HE, expected loss this bet is 5 cents (let’s just round to 5 cents)

Bet 2. Bet 2 units. Lose. Total score: -3. Double next bet. Expected loss this bet based on HE 10 cents.

Bet 3. Bet 4 units. Lose. Total score -7. Double next bet. Expected loss this bet based on HE 20 cents.

Bet 4. Bet 8 units. Lose. Total score: -15. Double next bet. Expected loss this bet based on HE is 40 cents.

Bet 5, Bet 16 units. Win. Total score: +1. Next bet one unit. Expected loss this bet based on HE is 80 cents.

Notice that your expected loss is also doubling on each subsequent bet. By the fifth bet in the series, with the house edge at roulette, you will have theoretically lost $1.55.

Same five bets, flat betting, without martingale…

Roulette, HE of 5.25%

Bet 1. Bet 1 unit. Lose. Total score: -1. Flat bet next. Based on HE, expected loss this bet is 5 cents.

Bet 2. Bet 1 unit. Lose. Total score: -2. Flat bet next. Based on HE, expected loss this bet is 5 cents.

Bet 3. Bet 1 unit. Lose. Total score: -3. Flat bet next. Based on HE, expected loss this bet is 5 cents.

Bet 4. Bet 1 unit. Lose. Total score: -4. Flat bet next. Based on HE, expected loss this bet is 5 cents.

Bet 5. Bet 1 unit. Win. Total score: -3. Flat bet next. Based on HE, expected loss this bet is 5 cents.

With the flat bet (betting same amount on all bets), your theoretical loss is 25 cents.

With martingale, you are making far larger bets and this increasing your expected loss. In the scenario above, your martingale induced loss is $1.55 theoretical, whereas your flat bet loss is 25 cents theoretical. Your martingale theoretical loss more than 6x your flat bet theoretical loss! This is the real reason martingale fails.

Eventually, theory becomes reality, and your theoretical losses manifest themselves as a busted bankroll. That’s how they build all those billion dollar casinos.

Therein lies the real problem with martingale.

MARTINGALE WHEN THE BETTOR HAS +EXPECTATION

Let’s martingale and examine a situation when the player has the advantage, using the same conditions as above…

For purposes of this hypothetical, let’s assume the player has a 5% advantage.

Bet 1. Bet 1 unit. Lose. Total score: -1. Double next bet. Based on advantage, expected win this bet is 5 cents.

Bet 2. Bet 2 units. Lose. Total score: -3. Double next bet. Expected win this bet based is 10 cents.

Bet 3. Bet 4 units. Lose. Total score -7. Double next bet. Expected win this bet based on advantage is 20 cents.

Bet 4. Bet 8 units. Lose. Total score: -15. Double next bet. Expected win this bet based on advantage is 40 cents.

Bet 5, Bet 16 units. Win. Total score: +1. Next bet one unit. Expected win this bet based on advantage is 80 cents.

Theoretical expected win over series is $1.55, or 1.55 units.

So notice that the bettor has the advantage, martingale is actually beneficial for the bettor. If the bettor did not martingale, the bettor’s theoretical win would only be 25 cents (at this point, you can probably work it out yourself).

Granted, the player in a such a situation would not need to martingale. The player could just as efficiently make one or two large bets and realize the positive expected value. However…continue reading…

SO HOW DOES THIS HELP?

By realizing the actual problem with martingale, you can now use it to your advantage. How? By using martingale when you have the advantage in a game or situation.

Martingale is not an inherently flawed system of betting. It’s only a flawed system if used in conjunction with a negative expectation game because it significantly increases your hourly or daily theoretical loss (the casinos track daily increments).

There are rare times in a casino where you will have the advantage (that’s a discussion for later). When you believe that you have the advantage, you want to martingale. There are also times outside of a casino where you will run into a game of chance, which involves several trials/rolls/flips, where you have the advantage. Use martingale.

Here is a common play involving martingale, outside of a casino: you are at a local poker game or at the bar with the local drunk, and someone is prop betting you on a coin flip (assuming you aren’t being scammed, which is a distinct possibility). If you pick the side facing up, you will have the advantage (coin flips are not 50/50; https://www.smithsonianmag.com/science-nature/gamblers-take-note-the-odds-in-a-coin-flip-arent-quite-5050-145465423/). In those situations you want to martingale.

Note: I’m the RoadGambler, not just the ‘CasinoGambler’.

Assuming you have an adequate bankroll, in situations where you have the advantage, you want to martingale. Even if you don’t have the bankroll, it’s ok. Whether you decide to deploy an infinite step martingale or a mere two step martingale, your win-loss expectation will remain the same.

THE POWER OF MARTINGALE AS COVER

Why not make large flat bets, if you have the edge? Because martingale has the power of cover. Cover is a concept that many successful gamblers lose sleep over. Cover is the ability to disguise what you are doing.

For example, knowing this, if let’s say you run into a biased roulette wheel, you can use martingale. Sometimes, you can’t just start betting 1 unit, and then all of a sudden move to 20 units. That will draw attention. In those cases, if the advantage is around long enough, you can martingale your way up to larger bets. Martingale, even used at a starting point of 1 unit, will naturally cause the bettor to have a larger average bet. Also, with martingale, you will just look like a lucky gambler who will eventually hit the martingale wall (actually, you may go broke, even with an edge).

For the times you can’t just bust out the big bet immediately, use martingale.

Most people do not understand the real problem with martingale, but they’re very aware of martingale and it’s perceived flaws as a betting system. So anytime people see martingale being used, they will assume that you are a foolish gambler. But who is fooling whom?

MARTINGALE AS PROTECTION AGAINST MALFEASANCE  OR MISTAKE

Let’s say you run into a betting situation where you estimate that you have an advantage. You can attempt to make  bets large and take full advantage immediately. Or you may dip your feet in the water slowly, and test to see if you really have the advantage.

Sure, you may have found a biased roulette wheel. But what if you are mistaken? Do you want to risk it with big bets upfront?

This is where starting at one unit may be beneficial.

Or what if you run into a group of frat kids who want to bet you their drinking money on a coin flip? How do you know they aren’t running a scam on you. By making small bets which are consistent with martingale early bets, you have a chance to gauge the fairness of the game. If you later believe that the game is fair, you can spread your bets under the cover of martingale.

By starting small, you are protecting yourself against mistake or malfeasance. I like to call this a ‘reduction in the discount’.

That’s another powerful benefit to martingale, if used properly.

A WORD OF CAUTION

When I started the RoadGambler website, I wanted to stay away from any advantage play talk. I wanted to talk about casual gambling because that’s what I do noways. Also, AP discussions can be controversial. Finally, many people find AP talk to not be fun. But enough people have asked me to expand on AP. So I’ve decided that I am going to expand on gambling theory and AP issues, every now and then. Not regularly, but sometimes.

If anything, it gives me more to write about.

Having said that, let me conclude by saying that if you are one to skim over an article, DO NOT go to and start martingaling games. That’s not what I said.

TL;DR

The real problem with martingale is that it does not change the expected value of the bet. Thus in a negative expectation game, you will have theoretical  larger losses. Conversely, if you can find a game with a positive expectation, then martingale will create a larger theoretical win.

 

 

Posted in: Gambling

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