Given that we have unlimited resources, no betting limits and no casino-rake, shouldn't the Martingale system be profitable in the long run? If no, can someone explain the maths and probability behind it?

/myntomanen

Yes, if by "profitable" you mean guaranteed to reduce the finite wealth of the casino to 0. (It won't increase your own wealth since you already have infinity.)

Also, you mention rake. Did you mean to say edge, or are you talking about poker?

It would also work if there were a house edge*. Without a house edge, you wouldn't need to martingale, you could just flat-bet and still be guaranteed to break the house.

*According to Lebesgue measure theory, which I haven't learned and therefore I don't fully believe it yet :P

In real life, the martingale is so bad, even if you had an edge, you'd be guaranteed to go broke in the long run (though I think it would depend on the parameters). There may be parameters that could allow you to survive, but even then, you'd be making smaller profits than you should.

Actually if that's the question then the answer is yes, but better yet would be to just put the house all-in every time. OP probably meant, pretend infinity+x > infinity. In which case everything I said still applies.

That's incorrect because he didn't say he was planning to live forever. However you define "long run", if it's a finite amount of time, then the expected value of the amount you lose will be the house edge times the total amount bet. If you take the limit of the amount you lose to the casino as time goes to infinity, and the house has an edge, then your loss goes to infinity. The probability that you will win goes to 1. But it will never actually be 1 for any finite amount of time. After any number of bets, there will always be a non-zero probability that you will have lost every bet, and while that probability may be very tiny, the loss in that case will be so large that when multiplied by that tiny probability it causes the expected value of your loss to be exactly the house edge times the total amount bet up to that point. It also isn't true that you are "guaranteed to break the house" even if you do live forever. There is no finite amount of time over which that is true, and it wouldn't be true even if you planned to live forever because of the event where you lose forever. That would have probability 0, but because that event exists, saying it has probability 0 isn't technically the same as saying it's impossible, or that it is guaranteed that you will break the house. If we're going to talk theoretical math, then you need to get the technical terms right and not loosely throw around terms like "guaranteed". There is a difference between probability 1 and a dead certainty. Also, if you aren't putting the house all-in every time, there are infinitely many different outcomes where you never break the house.

Now in measure theory, we can define an expected value of the amount that the casino loses to you, and that expected value is defined in such a way that it is positive if you can live forever. But that's not your grandfather's expected value. The way we normally compute EV wouldn't be defined in that case because we wouldn't know how to deal with the 0*-infinity term for the 0 probability that we lose infinity, so we define a generalization of that EV which is the same as our regular EV for finite games, but it is also defined for infinite games. The way it's defined, it basically takes the 0*infinity term and sweeps it under the rug. It ignores it and calls it 0. What happens is that your EV will get more and more negative as time goes to infinity, but right AT infinity, it jumps discontinuously to a positive value. That isn't useful in any sense that would be important to an actual gambler. It serves primarily as a distraction in these types of discussions, of which there have been many on these forums. Using this as the sole basis to answer questions of this type is misleading and irresponsible.

Too tired to read Bruce' post, but it probably covers everything OP.
I know, you always say this. I did add the follow-up of how bad martingale is in real life. Also, I think it's pretty clear we're not talking about real life practicality when the initial specifications are infinite wealth and infinite betting limits.

Basically measure theory exists in the text books, not at the roulette table.

The short answer is that there is no scenario in games with a house edge, where any betting system can ever have positive expectation. For the 900th time ( that's how many times this thread has been started).

And the corollary to the above, heading off the inevitable claim to the contrary, is that table limits are not there to prevent martingalers. Casinos love them. Their upside is your whole bankroll. Your upside is 1 minimum bet. Duh.

Infinity is nothing but a thought experiment and it does not exist in this universe. So neither do betting scenarios that require infinite time or money.

A series of Martingale bets should come to an end either when the bettor goes broke or when they decide to stop and are in the green. If the bettor can't go broke (which is the scenario posed by the OP) then they are practically guaranteed to profit over any "long run" assuming they have a bit of flexibility with their stopping point. That is, unless you go on a losing streak so long that you are forced to stop while in the red (say you die from old age or something). But at any time that you decide to quit you are about 99.99999% to be able to get into the green within your next 20 bets or so and astronomically likely as the number of bets continues to increase.

EV is not always the best way to think about profitability.

When someone asks about the "long run", they are generally not asking if they can win a dollar and never play again. They are asking if they can get ahead and stay ahead if they continue to play this system long enough. You can actually virtually guarantee the opposite. If you play long enough, you have probability approaching 1 that your bankroll will become -x for x arbitrary large. Systems said to be profitable should remain profitable each time they are played as long as conditions have not changed.

That's right, the correct way to think about profitability is to use expected utility. Advantage gamblers have diminishing marginal utility functions, meaning that they may not make a bet just because it has +EV, but they will certainly not make a bet that has -EV. Their utility function is concave down and always below the linear utility of basing their decisions on EV. Winning a certain amount of money means less to them than losing an equivalent amount, and this depends on the amount to be won or lost relative to their bankroll. Only a risk seeker (gambler) would have a utility function that is concave up, meaning that winning an amount would mean more to them than losing the equivalent amount, so they make -EV bets. The person with diminishing marginal utility will decide that even though his risk of losing is miniscule, the amount that he will win when he wins is also miniscule compared to the size of his bankroll, and he would have negative expected utility by playing the system. So by any reasonable economic theory, the system is not profitable.

I think this forum needs an Official Martingale Thread. Maybe as a sticky.

Using American Roulette as an example, betting a color the chance to lose 20 in a row sometime during your next 10,000 bets is 1.25% or about 1 in 80 chance. If we spin 100K times, then the chance for a 20 loss streak is 12%.

Astronomical isn't a good description, and only looking at the chance for your very next 20 bets in isolation isn't really a valid criteria for risk. For that to be useful we would need to include the Magic Casino Door fallacy too.

But in this scenario the bankroll is unlimited, not "arbitrarily large". If we played for 1 trillion bets using the Martingale system with initial bets of $1, for example, and then picked the next time we won as a stopping point, then we should expect to be "up" (debatable whether that makes sense since we already have infinite money) by about half a trillion dollars at the end.


You misunderstood my post. The next 10,000 or 100,000 bets is irrelevant to what I was talking about. The scenario I'm describing is one in which you've been using the Martingale system for an arbitrary number of bets and you want to stop. You can stop the next time you win a bet. So the only thing preventing you from ending up as a winner at the point when you decide to stop is if you go on a losing streak so long that you die or something along those lines. 99.9999% of time time you'll be able to stop within your next 20 bets. 99.999999999999% of the time you'll be able to stop within your next 40 bets. And so on. I said it becomes "astronomical" as your number of bets goes up beyond 20. Remember we have unlimited bankroll.

Are you going to stop for the rest of your life? That's the only scenario in which your point is valid.

As I mentioned in the post you responded to, you would need a Magic Casino Door otherwise.

Anybody have a link (or explanation) of the "Magic Casino Door fallacy"?

(searching google only finds this thread...)

Juk

I may have coined the name myself a while back on this forum, but the idea comes up a lot and it's a key part of many betting systems. It's the theory that you can quit when ahead and walk out of the casino, making every session (or most) a winning one. Coming back later apparently starts your life over. Variations include changing casinos, changing tables, etc. When you do these things then the law of large numbers and the central limit theorems no longer apply to you, because you have somehow interrupted the continuum and jumped through a wormhole or something.

Ah thanks!

Juk

That was the scenario I was describing, yes. But I think it's valid either way. When you have an unlimited bankroll and are doing Martingale you actually can make every session a winning one as long as you aren't interrupted against your will. It really doesn't matter if you have a losing streak of 20 hands at some point during your next 10,000 hands.

These kinds of "if the world had no constraints or physical laws then..." are annoyingly pointless and useless, as everything is possible in this fairyland, where nothing means anything.

As BruceZ made clear early on, the answer to the OP question is No. It is not profitable. If you disagree then please define infinity +1.

If you just want a 99.9999% chance of winning half a trillion dollars, it would be stupid to play a trillion times. You can do in that in no more than 21 plays if you just start by betting half a trillion. You can do that with a finite bankroll.

Playing a trillion times, it would be true that your probability of being ahead would increase with time even as your EV goes more negative with time. Playing forever, there will be times where you are negative with probability 1 after any amount of time.

Yeah, surely with an infinite bankroll your first bet would be "the casinos whole bankroll" which you would walk away with if you won. If you lost, the casinos bankroll has doubled so now you're betting 2x and so forth.

Starting with a small amount is only a feature of martingaling with table and bankroll limits. If you started too high in the real world you wouldn't be able to withstand as many losing turns.

Yes, of course. I mean there's no point playing at all, really, if you had an infinite bankroll. I just picked one trillion as an arbitrary large number and 1$ as the initial bet size for simplicity's sake. It could be any amount.

Yeah and I'm not sure it really makes sense to think of using the Martingale system forever, because it relies on quitting while ahead.

You don't need an infinite bankroll to make any given amount with any given probability < 1. The infinite bankroll is only relevant if you plan to play forever. Whatever your goal is, with a finite bankroll you're most likely to get there by making the largest bet possible to reach your goal on any given bet, and if that's not possible, then you bet your entire bankroll. Betting less requires a larger bankroll for a given probability of success.


You just play the martingale system repeatedly. If it's profitable, then it must be profitable every time you play it, so why wouldn't you?

This makes me think...maybe there is a sort of "gambler's hell" where one is forced to martingale at roulette for eternity.

No, they have to use that chart from the other thread

That's only hell if there are two zeros, no dancing girls, and a homely dealer.