Having read all about the Wizard of Odds I decided to email him to suggest he might like the article I wrote on the Doubling System in Roulette. He very kindly took a look and suggested he could not approve of the article and gave his reasons. Knowing that he is a very busy man I wondered whether to email back my response to his interpretation of the text and decided I would. Understandably he preferred not to discuss the matter any further and suggested I might put the info in this forum so that others might get a chance to read it and comment. That seemed a very good and helpful idea. I do seriously respect the Wizard of Odds, he is amazingly perceptive about probabilities and quite evidently a prolific writer. He is very good at maths and has genuine integrity when it comes to gambling for fun and not as a vice. So... in the best possible taste and to set the proverbial cat amongst the pigeons I am going to suggest that on the following point he is wrong.

I said: "Are you saying that the Martingale system does not work in theory?"

He said: "Yes, that is what I'm saying."

On this point, the Wizard of Odds is wrong! The Martingale Doubling Method cannot guarantee a win in reality. That is demonstrable and the article Doubling System in Roulette is very clear about that. But it does guarantee a win mathematically and so it does work in theory.

I would be very interested in any sensible (or not so sensible) comments on this article and the proposition that:

Toxic Drums

Quote:dwheatleyIn theory, communism works. In theory

Nice! The Wizard has noted the martingaling is good in the short term, I think he likes to focus on the long term to infinite aspects when calculating his numbers. If your goal was to win $100 and you are betting $5 at a time on red lets say, martingaling could work, but there is always that chance. The simple message here is no betting system can beat the house on an infinite scale of time, even with a huge bankroll the house limit can be reached quickly, if you are looking for easy money you should not be playing, just play to have fun!

This poster's particular simulation seems to involve doubling your bet always, regardless of whether you just lost the last bet. You double if you win or lose it seems. That's certainly a different approach, and certainly highlights what would *seem* to be true about his claim. How this conflicts with the Wizard's assertions is over my head.

Coincidentally, today for no particular reason I crunched out some number on a simple calculator (a lot of common ones will keep a function like "2X" alive without re-entering it). I took a look at what you would have if you started with one penny, and faced a losing streak you were trying to defeat with a Martingale. How quickly a large bankroll could get defeated with a streak a player could quite easily be facing is quite remarkable. Just starting with a penny!

Quote:WizardI think you need to [read] his article, which he linked to, to have a chance to understand his argument. As far as I can tell, he is saying that as long as you keep doubling up until a win, your bankroll with move upward. Of course I agree with that. However, his simulations don't indicate a maximum bet, so it isn't surprising that they trend upward. I'm not quite sure what our point of departure is. I'm already way over my quota for discussing betting systems for 2010, so asked Sam to post this here. If you have the inclination, have [a look] at it.

I think there is no point of departure really. I think it was a mild misunderstanding. I am a very hypothetical person (tree) and often depart on matters of practical reality. I originally wrote the software in the Loews Hotel (as was) in Monte Carlo because I was fascinated by the idea and wanted to see what the picture of the doubling method looked like.

In the article, SamSpruce both proves the math of the theory, as well as tells why it doesn't work.

It's a question of semantics.

Here's the short version:

When you consider only the math, the math cannot be altered. Math alone demonstrates that it works.

However, even as SamSpruce admits in the article, it only works if you have three things:

1 - Unlimited bankroll

2 - Unlimited betting limit

3 - Unlimited duration

Hmmm.... Those are the same obstacles mentioned in just about everything that has ever been written about the Martingale system.

Bottom line: He's right about one thing:

"...in theory." Yep. He's right about that!Quote:samspruceThe Martingale Method does work in theory!

No. Double on every loss. Return to one unit on a win.Quote:odiousgambitThis poster's particular simulation seems to involve doubling your bet always...

From his fourth paragraph:

Quote:The idea of the doubling system is that you have a basic stake of £1 and you play a round. If you win that's fine. If you lose you double your bet on the next round. If you lose again you double your bet again. You carry on doubling your stake until you win at which point you start with a £1 stake again.

Quote:odiousgambit...I naturally will bow to the Wizard's knowledge when he says it doesnt work even in theory.

Er... I think that was the misunderstanding. He missed the "theory" bit. He points out in a post here somewhere that he agrees, with no bankroll limit, the balance will continue to rise.

Quote:odiousgambitThis poster's particular simulation seems to involve doubling your bet always, regardless of whether you just lost the last bet...

Not at all odiousgambit. Just the regular old double when you lose. There are examples of constant single unit betting but just to illustrate the comparison.

Quote:dwheatleyIn theory, communism works. In theory

Very good point!

I've had FSA actuaries disagree with me on this. It really goes to show there is no such thing as infinity in real life, and arguments about the nature of infinity are kind of silly.

Math alone, without the limitations of bankroll or max bet or even duration, works.

After all, if you have a losing streak of infinity, with an unlimited bankroll, you can still make that infinity plus one bet.

Quote:samspruceQuote:odiousgambitThis poster's particular simulation seems to involve doubling your bet always, regardless of whether you just lost the last bet...

Not at all odiousgambit. Just the regular old double when you lose. There are examples of constant single unit betting but just to illustrate the comparison.

Well, I stand corrected, although to me that simulation was an interesting new wrinkle, assuming I understood what I was looking at.

But I note several of you will say it works in theory without apparently realizing you disagree with the Wizard on that....

Quote:jeremykayI agree that with unlimited resources (bankroll, time, etc) this system does work in theory. The fact that you need to put so many absurd conditions on it though just means it is that much worse in practicality! Besides that, martingale just isn't any fun... who would want to bet $5,000,000 to win $5 just because they already lost 20 times in a row. It's all about playing catchup.

Quite right jeremykay and there is a world of difference between theory and practice. For example:

Question: "Does the Martingale system work if there is no limit to the bankroll?"

Answer: "Yes."

Question: "Does it require an unlimited bankroll to work?"

Answer: "Yes."

Question: "Is it possible to have an unlimited bankroll?"

Answer: "No."

Conclusion: "Therefore it works in theory but not in practice."

Whether it works is not about whether it is fun or not. It is not about whether the imagined scenarios are absurd or not. It is only about trying to work out what is really going on beyond the limitations of the time and space we live in.

Without theory we wouldn't have computers to do all these lovely calculations!

Without theory we wouldn't have nuclear bombs.

Hmm - (walks off into distance scratching head.)

Quote:DJTeddyBear...if you have a losing streak of infinity, with an unlimited bankroll, you can still make that infinity plus one bet.

My point entirely! Any sequence of losses are totally wiped out by one (albeit theoretical) win.

Quote:DJTeddyBearWhen I said that I agreed that it works "In theory", I should have said, the MATH works in theory.

Math alone, without the limitations of bankroll or max bet or even duration, works.

After all, if you have a losing streak of infinity, with an unlimited bankroll, you can still make that infinity plus one bet.

And you can lose it too.:)

Read Contributions to the Founding of the Theory of Transfinite Numbers by Georg Cantor for a discussion of beyond infinity.

Wizard said: "Yes, that is what I'm saying."

SamSpruce said: "On this point, the Wizard of Odds is wrong!"

Wizard: "As far as I can tell, he [SamSpruce] is saying that as long as you keep doubling up until a win, your bankroll with move upward. Of course I agree with that." ... "I'm not quite sure what our point of departure is."

SamSpruce said "I think there is no point of departure really. I think it was a mild misunderstanding." ... "He [Wizard] missed the "theory" bit."

Wizard: "I won't even grant the Martingale works "in theory."

One thing that needs sorting out here:

Is there a difference between theory and practice?

The answer is a resounding "Yes."

In theory you will come out quits on average playing a 50/50 bet but the odds on coming out quits on any given sequence are very low. Most often you will either be up or down. Playing one sequence will tell you very little about the average case. So mathematicians developed probability theory by extending cases "theoretically" to the infinite. As the Wizard of Odds will know there is a huge subject and many complications in this area of study. But the difference between practice and theory is that theory will tell you more reliably about what is likely to happen than practise. Theory will tell you more about what is going on behind the scenes whereas practise is just one manifestation of the theory.

Also it is worth pointing out that there is no theoretical reality. Theories are based on certain asserted conditions. The theoretical case of no upper limit to a bet and no limit to the bankroll are imposed conditions for the sake of examining what happens in those conditions. There is no reality (that I know of) where there is no limit to the bankroll. So, given the theoretical condition that one more bet can always be made the Martingale system works. If the Wizard wishes to say that "his" theoretical condition is that there is a limit then "his" theoretical model predicts that the Martingale system will always eventually fail. So I guess the problem or disagreement perhaps resides in what the "theoretical" conditions are. I don't disagree with the Wizard that in reality there is no way that the Martingale system is a winner. I also expect that if one took a mathematical look at the investment required to secure a net gain over an infinite number of games that the returns would look something like the net rate of inflation that countries have been experiencing ever since economies have existed. I'm going to go away and do some more work on this.

Another interesting observation from the graphs is that if you run a sequence of plays with the Martingale system and then pick one play the odds are that it will be a net win. I will have to think some more on this too.

Quote:WizardI won't even grant the Martingale works "in theory." To put it another way, I don't concede it works, even with an infinite bankroll. Consider double-zero roulette. As the maximum bet the player can make gets bigger the house edge of the Martingale over a large enough sample size still stays right at 5.26%. If the limit approaches 5.26%, why would it not hold true at infinity?

I've had FSA actuaries disagree with me on this. It really goes to show there is no such thing as infinity in real life, and arguments about the nature of infinity are kind of silly.

I don't think you can assert things based on mathematical probabilities like "the limit approaches 5.26%" as evidence supporting your claim and then assert that that there is no such thing a infinity to dismiss another postulate. Your 5.26% is based on mathematical calculations using the concept of infinity. So does it exist or doesn't it?

If you're going to allow a number such as "infinity plus one", you have to allow for "infinity plus two". And sooner or later, maybe even as much later as infinity ^ infinity, you'll get out of that loosing streak.Quote:QFITQuote:DJTeddyBear...if you have a losing streak of infinity, with an unlimited bankroll, you can still make that infinity plus one bet.

And you can lose it too.:)

Quote:DJTeddyBearIf you're going to allow a number such as "infinity plus one", you have to allow for "infinity plus two". And sooner or later, maybe even as much later as infinity ^ infinity, you'll get out of that loosing streak.Quote:QFITQuote:DJTeddyBear...if you have a losing streak of infinity, with an unlimited bankroll, you can still make that infinity plus one bet.

And you can lose it too.:)

Why? I don't care if it is Aleph-infinity. The fact that it didn't work for the "first" infinite tries does not make it work for the next. "Extending" infinity because infinity was not enough is truly taking the concept from the ridiculous to the sublime, or vice-versa. As Napoleon said: "Du sublime au ridicule il n'y a qu'un pas."

Quote:samspruceI don't think you can assert things based on mathematical probabilities like "the limit approaches 5.26%" as evidence supporting your claim and then assert that that there is no such thing a infinity to dismiss another postulate. Your 5.26% is based on mathematical calculations using the concept of infinity. So does it exist or doesn't it?

Infinity is a concept. For example, it is perfectly correct to say that the sum for i=1 to infinity of (1/2)^i=1. I'm trying to ridicule infinity being invoked as a defense of the Martingale, because nobody has an infinite amount of money. To go further, there is not an infinite quantity of anything finite.

Resorting to infinity does not make time matter. The balls and cards still have no memory.

================

In Junior High we had a teacher who would teach us about the dangers of applying sloppy thinking to infinity. He would describe a stairway of length 4 and height 3. The stairway would be of length 7, because the steps would add up to 4, and the rises up to each step would add up to 3. (This is an idealized stairway, not a real one with lips).

If we let the number of steps get larger and larger so that they are too small to stand on, the length would still be 7. If we let the number of steps go to infinity, then the stairway would look like a straight line. However the length of a straight line is 5 by Pythagorean theorem = sqrt(3^2+4^2).

The problem is in sloppy thinking. Even if the steps get so small that you can't see them, they do not actually go to a straight line. You are doing something similar with time. That's why the mathematicians spent so much time devising formal definitions of infinity and of limits.

Cheers,

Alan Shank

Suppose you have an infinite bankroll and no table limit, and are trying to win 1 betting unit. While your probability of winning tends to 1 over however many bets it might take, technically... your probability of winning is not 1. It only tends to 1; some function describing your probability of winning in terms of the # of bets it takes has a limit that converges to 1, but it's not actually 1. That is, you might lose the next bet. Every time.

So, in theory, you may not even win that 1 betting unit. Therefore... Martingale doesn't work, even in theory

And of course when placed in any confines of gambling, even with a billion dollar bankroll, the martingale is a proven loser given enough trials. Expanding the bounds of 'theory' to include infinity really shouldn't happen in a gambling-based scenario. It may be convenient, not to mention interesting for a debate, but its still wrong.

You all make valid points.

I retract my 'infinity plus one' comment and agree that Martingale doesn't work, even in theory.

When you double your bet, you are effectively doubling the size of the finite because you can express that doubling as a number. And you can keep doubling as much as you want until you get the one units that you want as you can always express the units as a number and the number of losses as a unit. You will never reach infinity in this case, you you need to think, is there a point where you can no longer multiply a finite number by two, and for me the answer is no.

Ok, you're still reading. So, the Martingale doesn't work because there is a house edge. But it does work if there is a player advantage. So, let's say someone builds a 100,000 number roulette wheel with no 0 or 00. In this 100,000 number roulette wheel there are 50,001 blacks and 49,999 reds. Both bets pay even money. You're telling me that the Martingale would work if you're betting on black, because there is a pleyer advantage, but it wouldn't work if you're betting on red, because there is a house advantage of .0001 (I don't know the exact number of zeros, but you get my point)

So in my mind, that is extremely hard to wrap my head around. I guess I just don't know why 1,000 or more reds can't come up in a row because you have the slight advantage, but 1,000 or more blacks can come in a row when you have the slight disadvantage.

I hope this either made sense, or you stopped reading. And no, I don't play roulette, because the game is boring and has a high house edge.

1. If I remember correctly, the math of infinite series rests on the idea of convergence since we are not going to get to infinity but can only approach it. Therefore a more correct statement is for a fraction x/n, the limit of x/n will approach infinity as n approaches zero. I think that if n gets to zero, then the value of the fraction is undefined.

2. What about 0/0? I think the answer is zero.

I don't think so.Quote:matilda2. What about 0/0? I think the answer is zero.

Elementary algebra says x/x=1

Heck, even 4th grade math says anything divided by itself is 1.

But was there an except 'zero' in there?

I did? When?Quote:matildaOh Mr Bear: First you stated that anything divided by zero is infinite.

Quote:DJTeddyBearI don't think so.Quote:matilda2. What about 0/0? I think the answer is zero.

Elementary algebra says x/x=1

Heck, even 4th grade math says anything divided by itself is 1.

But was there an except 'zero' in there?

0/0 is an "imaginary" number. Anything divided by 0 is an "imaginary" number.

I don't remember much about them except that I never understood the point of the imaginary numbers, except maybe that there must always be an answer in math. Maybe Wiz or someone else smarter than me in the area can add to this.

x/0 is "infinity", but only in the asymptotic sense as this only arises from limits in calculus.

Does infinity exist? It does not exist in the same sense that the real numbers exist (although, philosophically, we could argue numbers don't "exist" either, they only exist "by accident"). Infinity is only a concept, but the concept exists and is reasonably easy to define. For example, you can have an infinite number of something, just like you can have four of something. But infinity is not in the set of real numbers.

So... I take the fence on this one.

Quote:AZDuffmanQuote:DJTeddyBearI don't think so.Quote:matilda2. What about 0/0? I think the answer is zero.

Elementary algebra says x/x=1

Heck, even 4th grade math says anything divided by itself is 1.

But was there an except 'zero' in there?

0/0 is an "imaginary" number. Anything divided by 0 is an "imaginary" number.

I believe "imaginary" numbers are those involving the square root of minus one (i).

This is just terminology, of course.

Cheers,

Alan Shank

Quote:goatcabinQuote:AZDuffmanQuote:DJTeddyBearI don't think so.Quote:matilda2. What about 0/0? I think the answer is zero.

Elementary algebra says x/x=1

Heck, even 4th grade math says anything divided by itself is 1.

But was there an except 'zero' in there?

0/0 is an "imaginary" number. Anything divided by 0 is an "imaginary" number.

I believe "imaginary" numbers are those involving the square root of minus one (i).

This is just terminology, of course.

Cheers,

Alan Shank

I think you are right-like I said I barely passed the course. But I think there was also some "answer" for 0/0 which was imaginary. Though again, I barely passed the course, seeing no use for Algebra at the time.

Consider the equation 8/2 = X, then solving; 8 = 2X and then X = 4

Using the same logic consider 8/0 = X. then solving; 8 = 0X. In this case there is no solution to the problem. No solution exists. There is no number multiplied by 0 that equals 8, including infinity. Therefore the result of dividing a number by zero cannot be equal to infinity.

Now consider 0/0 = X. then 0 = 0X. This equation is solved by all possible values for X, but there is no single answer that can be determined.

If it doesn't work in practice, then the theory is deficient, incomplete or just plain wrong. When you see an equation like F=ma, theory, you can test it in the real world and see that, indeed, theory corresponds to practice. If in practice F<>ma, then the theory would be wrong.

Ok. Some theories are approximations or can only describe part of a phenomenon. Newton's theory of universal gravitation, for example, only works with two bodies. But you will not ever find a betting system that overcomes the house advantage by itself. Any such system would only work with an offsetting player advantage, such as counting cards, setting dice, or cheating.

As for the number 0, it has properties different from those of other numbers. A math teacher once explaned that dividing by nothing simply cannot be done, therefore any division by 0, even 0/0, is not a real mathematical expression.

It helps to refer to 0 as "nothing." Thus ten times nothing is obviously still nothing, and ten divided by nothing doesn't make sense.

If the probability of winning is anything less than one, than it is possible then to have a series of losses despite the probability. On a finite bankroll, Martingaling does not work because you will eventually run into a series of losses that will bankrupt your finite bankroll.

That said, one could argue that because the bet size is 2^(n-1) where n is the number of losses in a row, n is << 2*^n and therefore your bet will hit infinity before the number of losses does. I'll buy that.

Quote:matildaUsing the same logic consider 8/0 = X. then solving; 8 = 0X. In this case there is no solution to the problem. No solution exists. There is no number multiplied by 0 that equals 8, including infinity. Therefore the result of dividing a number by zero cannot be equal to infinity.

Who says infinity times 0 isn't 8? The statement is completely vacuous. Infinity isn't a number and can't be operated on within the set of real numbers.

You can't use algebra as contra-proof to x/0 being infinity. It can only be examined using calculus, but even so, only holds asymptotically. In a sense, I agree that x/0 is not anything, but in another more important sense, x/0 is indeed infinity.

Quote:AZDuffmanQuote:DJTeddyBearI don't think so.Quote:matilda2. What about 0/0? I think the answer is zero.

Elementary algebra says x/x=1

Heck, even 4th grade math says anything divided by itself is 1.

But was there an except 'zero' in there?

0/0 is an "imaginary" number. Anything divided by 0 is an "imaginary" number.

I don't remember much about them except that I never understood the point of the imaginary numbers, except maybe that there must always be an answer in math. Maybe Wiz or someone else smarter than me in the area can add to this.

Roughly speaking an imaginary number is a number who's square root is a negative number. e.g. -2x-2=4 and 2x2=4. There is no "real number" which is the square root of -4. It's a little more complicated because the "imaginary unit" is defined as the square root of -1 so any real number multiplied by the square root of -1 is an imaginary number (except 0 and perhaps infinity for the people that think infinity is a number :o)