# Thread: Martingale doubling chances

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Originally Posted by profloater
I have a friend who uses a gambling site that uses red and black and has the double zero to bet on too. If I got this right, he believes you can influence the number of zeroes by betting all the time on both red and black. I think he believes the other players also influence the frequency and the odds on zero are, I think, 32:1. and thus unpopular. He tells me he wins regularly on that method until he gets bored because its a series of small wins.
Your friend is what a casino would call a "sucker", who either isn't closely tracking his losses, or else is mistaking past good luck for continuation of the luck into the future. He will eventually lose his shirt, his analysis is false. The casino knows the odds of zero or double zero, and it is always in their best interest to keep those odds as they should be-- they could be taken advantage of if they fail to maintain the correct odds, and why should they allow that, when all they have to do is maintain house odds and keep cashing in in the long run.

2. I Know that I wouldn't have any fun gambling in pure chance games, because what's there to be proud of if you come out ahead? So I did a good job of rolling those dice to bring up the good numbers?

3. Originally Posted by Ken G
Your friend is what a casino would call a "sucker", who either isn't closely tracking his losses, or else is mistaking past good luck for continuation of the luck into the future. He will eventually lose his shirt, his analysis is false. The casino knows the odds of zero or double zero, and it is always in their best interest to keep those odds as they should be-- they could be taken advantage of if they fail to maintain the correct odds, and why should they allow that, when all they have to do is maintain house odds and keep cashing in in the long run.
I think you are right, despite his claims I think he is just a gambler who only remembers the wins. I asked him if he believed the casino put in more zeroes than they should, why not just bet on those? But he was convinced he needed the double bets for his system to work. So yes he's a sucker.

4. Originally Posted by roboticmhd
Thank you Ioga for clear answers to my question. I understood ! I knew I was wrong, can't believe the answer is so simple. I knew I was wrong because if the gambler was to play 100 rounds that gives, if to use my analytical method, 100/32 probability of losing which is grater that 1 ! (greater than 100%). This can not be true, since you can't have probability higher than 100%.

Actually I came across this whole gambling thing from here [link removed] (sorry if such links are not allowed here)

This is great site, it gives small amount of bitcoin every hour for free ! Also there is Multiply BTC section where you can play Hi-Lo game, like a casino, and earn a lot of money if you are lucky. I was trying to find a strategy to win, but later I realized it is impossible to increase chances.

The site is also provaby fair, which means that they can prove that the games are fair.

I just can;t believe that highest win on that site is 4500 BTC which is like 90 million \$ right now.
roboticmhd has been infracted for posting a very questionable link. The link has been removed from the post, and this thread is returned to daylight.

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If someone really hates to lose, a slightly different version of the game can be played. Money would be infinitely divisible and there would be no minimum bet. For each roll, the gambler would bet half of his current money supply. He'd quit when he has at least twice what he started with. Since he'd bet only half of what he has on any roll, he could never go to zero.

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Originally Posted by Chuck
If someone really hates to lose, a slightly different version of the game can be played. Money would be infinitely divisible and there would be no minimum bet. For each roll, the gambler would bet half of his current money supply. He'd quit when he has at least twice what he started with. Since he'd bet only half of what he has on any roll, he could never go to zero.
But he would never get out of the hole once deeply down. The strategy fails, as they all do. If you want to double your money, you have a 50% chance with fair odds, period. With house odds, your chances are less than 50%. It doesn't matter your strategy. And if you ever find yourself wagering a fraction of a penny, it's high time to admit you are in denial!
Last edited by Ken G; 2017-Dec-21 at 01:58 AM.

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Originally Posted by Chuck
If someone really hates to lose, a slightly different version of the game can be played. Money would be infinitely divisible and there would be no minimum bet. For each roll, the gambler would bet half of his current money supply. He'd quit when he has at least twice what he started with. Since he'd bet only half of what he has on any roll, he could never go to zero.
He would never go to zero, but the probability that he will double his money is less than one with this strategy.

There are strategies that (contrary to Ken G's claim) allow doubling with certainty - this is not one of them. Such a strategy cannot be bounded below, as this one is. And strategies that are not bounded below are most likely unimplementable in the real world, as they require increasingly large bets, and the casino just won't take the bet when it gets too large.

For the proposed strategy, each time you win, you multiply your money by 1.5, and each time you lose, you multiply your money by 0.5. So if you win m times and lose n times, and you started with X, then you have . If we forget (for the moment) about stopping when the money is double, and just keep repeating this strategy forever, regardless of the outcome, the distribution is highly skewed. All of the quantiles approach zero - to take an extreme example, there is a 99.999999999999999999999999999999999999999% chance that your wealth will be below 0.0000000000000000000000000000000000000001 after a sufficiently large number of gambles. This can be probably best be seen by taking the logarithm of the amount of wealth, taking the expected value of the logarithm (this actually goes down for increasing numbers of gambles), and also taking the standard deviation of the logarithm. Then either by Chebychev's inequality or the central limit theorem, you can see that the probability of breaking even (or in fact, the probability of staying above any arbitrary wealth level, however small) goes to zero as the number of gambles goes to infinity. The expected wealth remains at X, but this is because of an extremely low probability of even more extremely high outcomes. To take an example, suppose you gamble 2,000,000 times, and you win 1,000,000 times, and lose the other 1,000,000 times. Then you will have . This is a really small number, and it is the median outcome - half the time you have more, half the time you have less. If you start with , and gamble two million times, then half the time you will have less than , where this number has almost 125,000 zeroes after the decimal point. In other words, a really, really, really small number.

Your strategy also has the feature that the gambler stops when wealth is doubled, and I haven't quite worked out how to analyse that. However, if it does violate Ken G's prescription that the expected gains must be zero, it has to violate it the other way - that is, the strategy is a good way to lose rather than to win. A local martingale (which this gambling strategy is) that is bounded below (and this one also is) is a supermartingale, so the expected wealth cannot increase over time - it can only stay the same or decrease. I think in this case it stays the same, but I'm not completely convinced of this yet.

If you want a strategy that guarantees doubling, this is possible. However, it requires allowing wealth to take on arbitrarily low values before the game is over. Since this strategy has a lower bound, it cannot guarantee doubling. The classic example is to bet 1, then bet 2 if you lose, then bet 4 if you lose again, then 8, and so on, forever. This strategy does guarantee a profit of one. However, wealth could go arbitrarily low before you get the guaranteed profit, and if the casino cuts you off when you make the increasingly large bets that the strategy requires, then it can't be implemented in reality.

Originally Posted by Ken G
If you want to double your money, you have a 50% chance with fair odds, period.
The obvious mistake in this statement (or maybe just a hidden assumption) is that is restricted to strategies that either double or go broke. If you have one, and you gamble 0.5 twice, you have a 25% chance of doubling, a 50% chance of breaking even, and a 25% chance of losing it all.

The much more subtle issue is that before the period, you need to add a comma, and a statement specifying some technical restrictions that preclude strategies such as the unbounded doubling-up one I have described above.

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Originally Posted by Iago
There are strategies that (contrary to Ken G's claim) allow doubling with certainty - this is not one of them.
You are obviously missing that what I am talking about is, you have a stake, and you either double it or lose it. You are not allowed to go arbitrarily in debt, that's a very different issue. It's pretty obvious that I have a very high chance of doubling \$1 if I have access to wagering millions of dollars along the way, but of course all that would happen is I would trade a near certainty of doubling my \$1 for a tiny chance of losing millions. But that's not the issue in this thread, the issue is doubling a stake, or losing it. With fair odds, every single strategy, yes every one, has a 50/50 chance of either result, as long as you recognize that spending your old age betting fractions of a penny does not count as not losing your stake.
The obvious mistake in this statement (or maybe just a hidden assumption) is that is restricted to strategies that either double or go broke. If you have one, and you gamble 0.5 twice, you have a 25% chance of doubling, a 50% chance of breaking even, and a 25% chance of losing it all.
Again you are obviously forgetting that the point of the game is to continue until you have either doubled, or lost all. The 50% of the time you break even, you just keep playing. This is all very obvious, it's just silly to try to create "gambling strategies" for any reason other than keeping yourself from losing more than you can afford to lose. There certainly are no strategies that allow you to always double your stake, unless you are willing to lose an enormous amount of money.
Last edited by Ken G; 2017-Dec-23 at 03:36 AM.

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Originally Posted by Ken G
You are obviously missing
It seems we both agree that someone is obviously missing something, just not on who that is. But since you have decided to go against conventional advice and keep on digging, please allow me to help you out with that.

Originally Posted by Ken G
that what I am talking about is, you have a stake, and you either double it or lose it.
OK, so that's what you're talking about.

Originally Posted by Ken G
You are not allowed to go arbitrarily in debt, that's a very different issue.
Au contraire, that is a highly relevant issue if we want to take your broken claim that every strategy has expected winnings of zero, and turn it into a true claim.

In fact, it is one of the two standard assumptions used to outlaw strategies such as the infinite doubling up strategy - simply betting 1, 2, 4, 8, 16, 32, 64, and so on, until you win. Bounded wealth below kills that one off. It doesn't outlaw so-called "suicide strategies", such as betting 1, 2, 4, 8, 16, 32, 64, and so on, until you lose. So even if we outlaw strategies that do not have wealth bounded below, the claim that expected winnings are always zero, regardless of strategy, is still false.

The other commonly used restriction on gambling strategies outlaws both the guaranteed winning strategy and the guaranteed losing strategy. It's a bit more complicated, though, so I think it best to focus on the easier of the two common restrictions, even though it doesn't quite deliver the 50/50 result you are clearly pulling for.

These two types of restrictions are so commonly used in any kind of analysis of gambling strategies that anyone who has studied the area would have instantly known what I was talking about as soon as I referred to the technical restrictions needed to make your claim true. Did you obviously forget them, or did you obviously never know them?

Originally Posted by Ken G
It's pretty obvious that I have a very high chance of doubling \$1 if I have access to wagering millions of dollars along the way, but of course all that would happen is I would trade a near certainty of doubling my \$1 for a tiny chance of losing millions.
If you were to impose some restriction such as wealth bounded below (I think you said "that's a very different issue" on this point), then that would be true. If you don't have such a restriction, or an alternate restriction on the allowable gambling strategies, then you don't have a tiny chance of losing millions - just keep going until you win. Then you have zero chance of losing millions. I would have thought that would be "pretty obvious".

Originally Posted by Ken G
But that's not the issue in this thread
It most definitely is the issue if we're talking about why your claim about every strategy, with no qualification, is wrong.

Originally Posted by Ken G
With fair odds, every single strategy, yes every one, has a 50/50 chance of either result,
There it is again! False before, false now, and false if it comes up again in the future.

Originally Posted by Ken G
as long as
Well wonders never cease. A statement like "every single strategy, yes every one" is followed by "as long as"? So there might be exceptions to such a seemingly absolute statement? Should I be expecting a breakthrough here?

Originally Posted by Ken G
you recognize that spending your old age betting fractions of a penny does not count as not losing your stake.
What a disappointment, no breakthrough. This is the wrong thing that needs to appear after the "as long as". Chuck's strategy is a martingale. (OK, I'm partly at fault here - I should have said this in my earlier post. I did say it was a supermartingale, which is true, but I should have made the stronger statement that it was a martingale.) You don't need to add any qualifier for Chuck's strategy, because Chuck's strategy does have expected winnings of zero. It's not one of the counter-examples to your claim, that needs to be ruled out. (Strictly speaking, getting a 50/50 result for Chuck's strategy requires us to talk about convergence of an infinite sequence of probability measures, which is why I stated the result differently than you did.)

The statement "With fair odds, every single strategy, yes every one, has a 50/50 chance of either result" is indeed wrong unless we add an "as long as" qualifier to it, but you added the wrong one above. You don't need to outlaw Chuck's strategy - you need to outlaw strategies like the infinite doubling-up one, betting 1, 2, 4, 8, 16, 32, and so on until you win. Placing a lower bound on wealth (when you lose a certain amount, you have to stop) is one of the two common methods for outlawing such strategies. Do you still say "that's a very different issue"?

Originally Posted by Ken G
Again you are obviously forgetting that the point of the game is to continue until you have either doubled, or lost all.
No, I am not "obviously forgetting" this, I am "obviously remembering" this. If I had "obviously forgotten" this, then I wouldn't have been able to point out that it was missing from your earlier statement.

Originally Posted by Ken G
The 50% of the time you break even, you just keep playing. This is all very obvious, it's just silly to try to create "gambling strategies" for any reason other than keeping yourself from losing more than you can afford to lose.
Like I said, "hidden assumption".

I think it is better to state such assumptions explicitly, which is why I brought up this point (somehow, despite "obviously forgetting" it). If all the assumptions are stated clearly, then people who understand basic stochastic process theory won't have to guess what is or is not "obvious" or "silly" to you.

Originally Posted by Ken G
There certainly are no strategies that allow you to always double your stake, unless you are willing to lose an enormous amount of money.
As this statement says "always double your stake", I have to interpret "unless you are willing to lose an enormous amount of money" to mean "unless you are willing to lose an enormous amount of money along the way while executing the strategy, before its final victorious conclusion when you win all those losses back and finally double your initial stake". If that's what you mean, then this statement is actually true, which is a bit surprising, as you seemed to spend most of the post rejecting the need for such a qualifier. (If what you mean is "There certainly are no strategies that allow you to double your stake with very high probability, unless you are willing to accept a small probability of losing an enormous amount of money", this remains as false as it ever was, without the type of restriction on gambling strategy that I say you need, but you seem to be resisting.)

The statement "With fair odds, every single strategy, yes every one, has a 50/50 chance of either result" with no further restriction, is simply false, all of the times it has come up so far (and any times it may come up again in the future). Even if we add a restriction that wealth is bounded below, it's still not true. But, if we consider the modified statement "With fair odds, every single strategy, yes every one, in which the two possible outcomes are doubling initial wealth or losing it all, has at most a 50% chance of doubling" - well, this statement is actually true if we interpret "every single strategy, yes every one" to mean "every single strategy, yes every one, except for those that violate a lower bound on wealth with some probability". (Is that one of those "obvious" hidden assumptions?)

So the last statement in your post is actually causing me to wonder whether maybe you finally understand why "With fair odds, every single strategy, yes every one, has a 50/50 chance of either result" is wrong, but I got my hopes up around the middle of your post, only to have them quickly dashed. I hope I won't be let down again.

If you really want to understand this very fundamental issue, get a stochastic process book and look up the difference between "martingale" and "local martingale".

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I thought that doubling your money or being done when losing it all was a basic assumption of this thread and wouldn't need to be stated in every post.

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Originally Posted by Chuck
I thought that doubling your money or being done when losing it all was a basic assumption of this thread and wouldn't need to be stated in every post.
If that's the case, I'm surprised to find mentioning it once again causes so much controversy.

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Originally Posted by Chuck
I thought that doubling your money or being done when losing it all was a basic assumption of this thread and wouldn't need to be stated in every post.
I want to modify the post I just made a minute ago, but I can't, because I'm in some sort of moderation gaol, and my post isn't visible for me to edit. So I'll just make a new one.

Within the class of strategies that are bounded below (for example, once your wealth hits zero at any point during the game, you have to stop) and also bounded above (for example, once your wealth doubles, you take your profits and go home), then Ken G's claim is actually true - every strategy within this class has expected winnings of zero. (Relax either of the bounds, and it stops being true.) This includes the strategy you propose - the problem with it is, the longer the game goes on, the more lopsided a win/lose ratio you need to double your money. If you win once and lose once, your wealth has declined to 3/4 of what it was before. If you win a thousand times and lose a thousand times (which is the median outcome for two thousand gambles), your wealth has diminished to a tiny fraction of what it was, and you need a large number of wins to crawl all the way back to the eventual doubling. The probability of eventually doubling does not go to one in your example.

In a larger class of strategies (for example, those that temporarily allow you to go negative), the 50/50 claim is simply not true - there are definitely strategies that produce a doubling with probability one. One extraordinarily simple one is just to double up until you win. Bet 1, 2, 4, 8, 16, 32, 64, and so on, until you eventually win. The probability of doubling is one, and the probability of losing all your money is zero. Not 50/50. The reason you can't rack up guaranteed profits with this strategy in the real world is because there is a lower bound on wealth - if you run out of money, the casino won't let you bet any more.

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Originally Posted by Iago
Within the class of strategies that are bounded below (for example, once your wealth hits zero at any point during the game, you have to stop) and also bounded above (for example, once your wealth doubles, you take your profits and go home), then Ken G's claim is actually true - every strategy within this class has expected winnings of zero.
Correct.
(Relax either of the bounds, and it stops being true.)
Incorrect. All you can ever do is maximize the chance of a small winning by accepting a remotely tiny chance of losing an enormous amount. This is always true, regardless of whether or not there are bounds placed on the bets. But more importantly, in practical reality, there is simply no such thing as a "betting strategy" that gives a nonzero expectation value in a fair casino with no house odds (which, of course, do not exist in reality). Thus, every single person who ever thought they had a strategy to "beat the house" is just another sucker, the house loves them. This is because, in practical reality, there is always a limit on how much money a player actually has to bet, and no single player is ever allowed to dominate the action. So from the house's point of view, there are no strategies at all, there is just X money being bet, and the house odds times X being won by the house. The house never cares about what is going on in the minds of people involved in betting that X, they couldn't care less even if they all use the same misguided "strategy." The reason the house doesn't care is that as long as new players keep walking in the door, there's no concern about what stage of their "strategy" any individual player happens to be in at the time.
Last edited by Ken G; 2017-Dec-24 at 09:59 PM.

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It doesn't have to be extreme. I can give myself a ⅔ chance of winning by starting at \$1000 and stopping when I'm \$500 ahead.

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Yes, indeed that's more or less just what I do when I gamble, which is not often and not for long. I like the idea that I will usually beat the casino, but I'm under no illusion that this bothers them or that I will end up ahead in the long run.

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