Martingale System - Proving it doesn't work

[DrDave]

Hello!

Quite new here so hope I'm posting this in the right place!

I got into a debate/argument with a friend recently about the Martingale gambling system (playing roulette specifically), in which you increase your bet each time you lose in order that 'when you eventually win' you have covered all previous losses and get a small amount of profit on top. When you win, the bet is reduced back to the initial bet amount.

He insists it works in theory, but the wily casinos impose table limits to foil the system. I argue that the table limit is irrelevant and that when playing roulette the house wins 1/37 of all bets placed on the table regardless of systems. In fact, since the table has no memory - there can be no system involving roulette.

Can anyone provide a link or show me how to clearly argue the case to debunk this system while making no reference to table limits. He's that convinced that it works that he's about to start playing it on online casinos and I'm sure he'll start to lose a fair amount of money. I'm sure such an argument must exist, as the system is over 200 years old, but googling mainly pulls up dozens of sites selling these systems!

Thanks,

Dr Dave

 

[Tez]

Hello!

Quite new here so hope I'm posting this in the right place!

I got into a debate/argument with a friend recently about the Martingale gambling system (playing roulette specifically), in which you increase your bet each time you lose in order that 'when you eventually win' you have covered all previous losses and get a small amount of profit on top. When you win, the bet is reduced back to the initial bet amount.

He insists it works in theory, but the wily casinos impose table limits to foil the system. I argue that the table limit is irrelevant and that when playing roulette the house wins 1/37 of all bets placed on the table regardless of systems. In fact, since the table has no memory - there can be no system involving roulette.

Can anyone provide a link or show me how to clearly argue the case to debunk this system while making no reference to table limits. He's that convinced that it works that he's about to start playing it on online casinos and I'm sure he'll start to lose a fair amount of money. I'm sure such an argument must exist, as the system is over 200 years old, but googling mainly pulls up dozens of sites selling these systems!

Thanks,

Dr Dave

Someone will write you a proper reply, but at a zeroth order you should check whether your friend thinks that to make the system "work" he has to keep playing at the same roulette table. Or even keep playing roulette, and not gambling on some other (almost) 50:50 game. If he does then (as most people I know who do this think) then he's beyond help...

 

[richardm]

This guy's done some experiments on it. He says:

There is no betting system that will beat a negative expection game. A negative expectation game is one in which the odds are against you, such as all casino games. Obviously if a game could be beaten with any consistency then the casinos would go out of business
...
One reason that people mistakenly think that betting systems work is that they can see how a system could make a short-term win more likely. What they fail to realize is that when losses do occur they will be much larger than normal. So when wins are balanced with losses, you wind up with an overall loss, just as if you had used no betting system at all.

He concludes:

This shows that the Martingale is neither better nor worse than flat betting when measured by the ratio of expected loss to expected bet. All betting systems are equal to flat betting when compared this way, as they should be. In other words, all betting systems are equally worthless.

(emphasis mine).

He backs this up with some experiments, and shows his data and workings. Quite a good read.

 

[a_unique_person]

The empirical proof is in, the Casinos are still making plenty of money.

 

[Rolfe]

And here was me thinking you were talking about this. (These do work.)

Rolfe.

 

[Badly Shaved Monkey]

And here was me thinking you were talking about this. (These do work.)

Rolfe.

Hah! I hope you have some double-blind trial data to prove this wild assertion!!

 

[Rob Lister]

Hello!

Quite new here so hope I'm posting this in the right place!

I got into a debate/argument with a friend recently about the Martingale gambling system (playing roulette specifically), in which you increase your bet each time you lose in order that 'when you eventually win' you have covered all previous losses and get a small amount of profit on top. When you win, the bet is reduced back to the initial bet amount.

He insists it works in theory, but the wily casinos impose table limits to foil the system. I argue that the table limit is irrelevant and that when playing roulette the house wins 1/37 of all bets placed on the table regardless of systems. In fact, since the table has no memory - there can be no system involving roulette.

Can anyone provide a link or show me how to clearly argue the case to debunk this system while making no reference to table limits. He's that convinced that it works that he's about to start playing it on online casinos and I'm sure he'll start to lose a fair amount of money. I'm sure such an argument must exist, as the system is over 200 years old, but googling mainly pulls up dozens of sites selling these systems!

Thanks,

Dr Dave

The easy way to prove whether it works or not is to download (or simply link to) any version of a computer roulette game that allows no-limit betting (for fun). There are probably dozens available for free.

ETA: here's one you can play online.

http://www.123games.dk/game/board/roulette/roulette_eng.htm

 

[steenkh]

As I see it, the Martingale scheme is a sure win - provided there is no table limit, and that you have access to unlimited money. In real life, there is a limit, or your means are not umlimited, and the Martingale scheme will make you end up with a nett loss because of it.

 

[drkitten]

He insists it works in theory, but the wily casinos impose table limits to foil the system. I argue that the table limit is irrelevant and that when playing roulette the house wins 1/37 of all bets placed on the table regardless of systems. In fact, since the table has no memory - there can be no system involving roulette.

Just point out to him that unless he's an oil sheik and head of state, his financial resources themselves are limited. Unless he's willing (and able) to dig infinitely deeply into his pockets, he will hit his own personal version of the table limit in how much money he has (and is willing to risk).

 

[CurtC]

He insists it works in theory...

In theory, practice is the same as theory. In practice, they're different.

I'd like to add this little thought experiment. Imagine your friend played his martingale system all night, and kept track of how many times he bet $1, $2, $4, $8, $16, etc. Now imagine that the next night, he went back in and made the same number of each bet, but in a different order. Like he make all the $64 bets first, then the $32 bets, etc., for example. How would his expected winnings differ from the previous night's? Answer: the expected value of his bankroll at the end of the night would be exactly the same. The game of roulette is time-invariant, so carefully betting the different amounts at different times gives you no advantage over someone who bets those same amounts in a random way. Many years ago, this thought was what made me see that the martingale system can't work.

 

[Orangutan]

Most Online casinos let you play thier games for fake money, Assuming that the "free" games arn't rigged to make the player win more often your friend should try his system first. Personally I play the "USA casino" as they have lots of free games. I only ever gamble real money in real casinos and that's not very often.

The problem with his system is that he is the only one who is "breakable". I.e He will never bancrupt the casino but he can run out of money. Do a test like this, the martingale system is basically "double the bet till you win". So you bet like this.

1,2,4,8,16,32,64,128,256,512,1024,2048,5096....

Assuming a $10 bet, does you friend really want to bet $10240 if he loses 10times in a row.

I'm thinking that the table limits might be to protect the consumer. Think about it, if he hit's the limit on a $10 table he can always move to a a $100 table. Hit the limit on that and he goes to the High Stakes table, of course by then he's dropping $10,485,760 because he's lost 20 times in a row, does he have that kind of money?

You should tell him to learn to play black jack, played properly the casino has one of the smallest edges and you get good entertainment (relativly speaking).

O.

 

[Ziggurat]

Re: Re: Martingale System - Proving it doesn't work

I'd like to add this little thought experiment. Imagine your friend played his martingale system all night, and kept track of how many times he bet $1, $2, $4, $8, $16, etc. Now imagine that the next night, he went back in and made the same number of each bet, but in a different order. Like he make all the $64 bets first, then the $32 bets, etc., for example. How would his expected winnings differ from the previous night's? Answer: the expected value of his bankroll at the end of the night would be exactly the same. The game of roulette is time-invariant, so carefully betting the different amounts at different times gives you no advantage over someone who bets those same amounts in a random way. Many years ago, this thought was what made me see that the martingale system can't work.

No, that's wrong. Because the moment you win, you reset - what you bet depends on the results, so it's NOT time invariant.

Edit: also, the length of your betting run is not fixed, it's always truncated at a point where you're ahead. Can't do that with your radom sequence.

 

[drkitten]

Re: Re: Re: Martingale System - Proving it doesn't work

No, that's wrong. Because the moment you win, you reset - what you bet depends on the results, so it's NOT time invariant.

You misunderstand. Think of it, perhaps, as two separate nights.

On Monday night, you go in and play your favorite Martingale system. I'm watching you very closely and I observe that you place 65 $10 bets, 32 $20 bets, 20 $40 bets, and so forth.

On Tuesday, you go in and you place exactly the same number of bets, in exactly the same distribution, but in a different order. Your first 65 bets might all be $10, your next 32 might all be $20, and so forth.

We can repeat the same experiment on Wednesday, Thursday, Friday, and so forth.

Because each spin of the roulette wheel is independent, your expected winnings each night (including Monday) are the same. But it's obvious that you are expected to lose on Tuesday, Wednesday, Thursday, and so forth.... so you are also expected to lose on Monday.

 

[steenkh]

You are arguing from a finite run. The Martingale system only works with possibly infinite runs: You bet until you win, and you only quit if you have won.

It is interesting to notice that even if you have literally unlimited funds, you might still lose with the Martingale system if the Casino closes ...

 

[Iconoclast]

Can anyone provide a link or show me how to clearly argue the case to debunk this system while making no reference to table limits.

The definitive mathematical proof is in Allan Wilson's "The Casino Gambler's Guide", available all over the net including here. Note however that this is a general proof that can be used to disprove the worth of any progressive system against any acausal game where all payoff schedules have negative expectation and is probably too difficult for your friend to understand.

A simpler proof involving JUST the even money bets on single-zero blackjack would be:

p = probability of winning a single round is 1 in 37 or 0.027
q = probability of losing a single round is 36/37 or 0.973
B = amount bet per round
M = payoff multiplier which is 35 to 1 for the even money bets.

Since any win is multiplied by the payout multiplier, but a loss isn't, we get as an expected value E per hand:

E = B(Mp - q)

You''ll find the above equation in any introductory statistics textbook, though I've probably screwed up the symbols for some of the terms. In english, the amount we will win on any hand is the probability of winning times any payoff multiplier MINUS the probability we will not win times the loss multiplier (in our case 1 so it's not shown), and that's ALL multiplied by the amount we bet in the first place.

So,

E = B(0.945 - 0.973)
E = B * -0.028 as expected, the difference of 0.001 is due to rounding.

So, for every bet your friend makes, he can expect to win -0.028 times the amount he bets, which is to say, every round has a negative expected outcome for him. So, his session win will be:

Sum (B_n * -0.028) where B_n is the amount he bets on the n^th round.

It is important to note that if your friend bets a positive amount on each round, every term in this sum will necessarily be negative, so we see that the session expected value MUST be negative if all bets in the session are positive, no matter the size of each individual bet. And I never mentioned table limits once.

---------------------------------------------------------------

Or, you could put it to him another way: I realise you're working on this progression because you want to win a little bit of money, but do you realise that if you ever come up with a betting progression for roulette that shows a positive return, you will have successfully falsified an entire branch of mathematics.

And you are correct about table limits, they do not exist to foil Martingale betters, they are put in place merely to prevent a given table from running out of chips too often.

 

[Ziggurat]

Re: Re: Re: Re: Martingale System - Proving it doesn't work

You misunderstand. Think of it, perhaps, as two separate nights.

On Monday night, you go in and play your favorite Martingale system. I'm watching you very closely and I observe that you place 65 $10 bets, 32 $20 bets, 20 $40 bets, and so forth.

On Tuesday, you go in and you place exactly the same number of bets, in exactly the same distribution, but in a different order. Your first 65 bets might all be $10, your next 32 might all be $20, and so forth.

We can repeat the same experiment on Wednesday, Thursday, Friday, and so forth.

Because each spin of the roulette wheel is independent, your expected winnings each night (including Monday) are the same. But it's obvious that you are expected to lose on Tuesday, Wednesday, Thursday, and so forth.... so you are also expected to lose on Monday.

I understand perfectly well. But this analysis is still completely wrong, and it's completely wrong precisely because, although the roulette wheel results are independent of the bets, the reverse is not true. The bets that first night are dependent of the roulette wheel. There is extra information there (namely how much you've won/lost so far) that is used to decide the bet amounts. Information changes probabilities - in this case, expected payoffs.

It's quite true that if you pick a number of dollar amounts for bets beforehand, then the expected outcome doesn't depend on their order. But this system doesn't simply determine the order of the bet amounts, it also determines WHAT the amounts are, and that's the part you're not figuring in with your comparison. The amount of your next bet is determined by the outcome of your current bet - if you win, you bet less, and if you lose, you bet more. If you keep winning when amounts are small, you never end up making any big bets to lose on. If you keep losing while the bets are small, you bet bigger until you win on a big enough bet to offset the losses. This is decidedly UNlike the scenario you propose, in which the bet amounts are all determined a priori.

 

[Ziggurat]

Re: Re: Martingale System - Proving it doesn't work

The definitive mathematical proof is in Allan Wilson's "The Casino Gambler's Guide", available all over the net including here. Note however that this is a general proof that can be used to disprove the worth of any progressive system against any acausal game where all payoff schedules have negative expectation and is probably too difficult for your friend to understand.

I looked over that proof. Basically, it begins with the assumption that there's a maximum number of bets the bettor can make before going bankrupt, and then takes the limit of this as it goes to infinity. In the case of ANY limited number of bets, you always expect to lose out over time. As you increase the number of maximum bets alowable, the increasing probability of a positive but essentially limited payout is offset by the increasingly rare but increasingly costly results of running out at the end. But if you can really set the number of bets you can make as high as you want, you can also set the odds of this disastrous outcome as arbitrarily small as you want, which in practical terms means that you pretty much can count on a finite win (with arbitrarily small odds of unbounded catastrophy). It becomes sort of like a bet where 999,999 times out of 1,000,000, you win $1, but 1 time out of 1,000,000 you lose $1,000,000. Mathematically speaking, it's a losing bet, but practically speaking, if you only play once it's an almost certain $1.

 

[jj]

As I see it, the Martingale scheme is a sure win - provided there is no table limit, and that you have access to unlimited money. In real life, there is a limit, or your means are not umlimited, and the Martingale scheme will make you end up with a nett loss because of it.

I think you need to calculate the handle and the expectation before you pronounce this a "sure win" because it's a "sure lose".

 

[CurtC]

You are arguing from a finite run. The Martingale system only works with possibly infinite runs:

It doesn't even work with infinitely many runs. How many negative-expectation bets must you sum to achieve a positive-expectation overall? The answer sure ain't infinite.

Originally posted by Ziggurat
I understand perfectly well. But this analysis is still completely wrong, and it's completely wrong precisely because, although the roulette wheel results are independent of the bets, the reverse is not true.

I think you understand the martingale "puzzle," but I haven't communicated what I was trying to say. I'm trying to say that the expected value for placing the bets in the order that he did is the same expected value for placing the bets in any order. The expected value is the average of all the possible outcomes. It is a little pointless to talk about the expected value for a sequence of bets that have already occurred, but you'd have to consider doing that same sequence again and again, and comparing that.

 

[CBL4]

This is a time when proof get in the way, look at like this:
Each time you go to a casino, you start by betting a dollar. If you win, you leave. If you lose, you double your bet. If you win, you leave. If you lose, you double your bet. ...

Let's look at the possibilities: (W1 means win 1 dollar, L4 means lose 4 dollars
W1 - Leave with a dollar (+1)
L1, W2 - Leave with a dollar. (-1 + 2)
L1, L2, W4 - Leave with a dollar (-1 + -2 + 4)
L1, L2, L4, W8 - Leave with a dollar (-1 + -2 + -4 + 8)
L1, L2, L4, L8, W16 - Leave with a dollar (-1 + -2 + -4 + -8 + 16)
The pattern is clear. If you lose 20 in a row,
L1, L2, L4 .... L1048576, W2097152 (leave with a dollar)

As long as you have infinite money and infinite time, you always win a dollar. The problem is one of the days, your lack of funds or closing is going to get you. You may win a single dollar the first 200 times you come in but eventually you will lose a bundle.

CBL