Can anyone diprove this math?

Quote: Mosca

Right. And even still; he is proposing that one side be allowed to say when to stop, but not the other. I could reduce the proposal to, "If I am allowed to play until I win, and then stop, then I will always win." Well, yeah, if the other player is not allowed the same condition.

Who is the other player? The casino? Yes, it is exactly what makes this work - the casino cannot stop you while you are losing.

Quote: P90

The first bet is 1, the second bet is 2, the third is 4, so on till 128 as the last bet (1,2,4,8,16,32,64,128). It's [1..n]Sum(2^(n-1))=2^n-1.

Indeed. I stand corrected. Well, in that case what you called "effective house edge" is about 51%.
But, once again, I am not sure I can make any sense of this definition. It is not really a "house edge" so much just your expected total loss.

Quote: weaselman

But, once again, I am not sure I can make any sense of this definition. It is not really a "house edge" so much just your expected total loss.

"House edge" is just a legacy generic term that was never properly defined. Using EV instead works better in math, as it has only one interpretation.

Isn't using the term "infinity" when talking about the real world sort of the rhetorical equivalent of dividing by zero? Don't you automatically force yourself into "paradoxes" and contradictions by doing so?

Just asking. (Though, I suppose if you're a mathy person, such discussions might seem interesting even if they lead nowhere, sort of like English majors discussing the notion of free will as expressed in Milton's "Paradise Lost".)

Quote: P90

"House edge" is just a legacy generic term that was never properly defined. Using EV instead works better in math, as it has only one interpretation.

Never was defined? Everywhere I know, it's been always used as EV/(total wager) and without any ambiguity.

Quote: weaselman

I wanted to ask why you think it grew exponentially, but a more urgent question is how do plot the growth (and oscillation) of infinity? :)

You are not plotting an oscillation of "infinity," though you are plotting infinitely many points that oscillate around the line Y = E(X). Come on, you're smarter than this. The downside is clearly exponential, the upside is clearly linear. The lim_inf is an exponential function, the lim_sup is a linear function. The center of gravity is well defined for a distribution and needs no further comment.

The problem is that a progression neither wins nor loses with infinite bankroll. Such statements are meaningless.

If you complete one such infinite martingale and simply plot: {(N,W(N)), N >= 1} where W(N) is the amount won at hand N, then you have a set of points in the plane that are bounded on top by a linear function, on bottom by an exponential function (but nothing less than an exponential asymptotically), and has center of gravity a line {(N,E(N)), N >= 1} where E(N) is the expected value at trial N. The line y = E(X) is also the least squares best fit to the aforesaid plot.

This is really the end of the story and explains absolutely everything about an infinite martingale, if you'll take the time to think it through.

-Ms. D.

Quote: weaselman

Never was defined? Everywhere I know, it's been always used as EV/(total wager) and without any ambiguity.

With a lot of ambiguity. It can be calculated per base bet or per total wager (and Wizard's tables, for one, specify both). In multi-event bets, such as in craps, it can be calculated per resolution or per roll. That's four potential ways right off the bat (per base bet resolved, per wager resolved, per base bet per roll, per wager per roll).

And that's just for games with loss limited at explicit wager. For instance, in a game of 21 where you lose 3:2 on dealer blackjack, do you calculate it as betting 1 actual bet, or betting 1.5 actual bets and winning 0.5 back on loss? One could go on, but there is no point - it's enough to show that house edge is only unambiguous in the simplest of the games, with fixed loss and win amounts, fixed resolution time and method, and no pushes.

Quote: mkl654321

Isn't using the term "infinity" when talking about the real world sort of the rhetorical equivalent of dividing by zero? Don't you automatically force yourself into "paradoxes" and contradictions by doing so?

There are no contradictions in the real sense of the word, the set and number theories, both dealing with kinds of infinities are are talking about here, are both well-formulated and self-consistent, and do not include any contradictions (and are therefore incomplete, but that's a whole different topic).
All the "paradoxes" dealing with infinity are illusory, they arise from people trying to incorrectly apply their intuitions, "trained" on experiences with the finite entities, to infinity, and it just doesn't work.

Quote: P90

With a lot of ambiguity. It can be calculated per base bet or per total wager

For games like roulette, that's not an issue.

Quote:

And that's just for games with loss limited at explicit wager. For instance, in a game of 21 where you lose 3:2 on dealer blackjack, do you calculate it as betting 1 actual bet, or betting 1.5 actual bets and winning 0.5 back on loss?

How much did you bet? If you bet 1 unit, then your wager is 1 unit. I don't see the ambiguity.

Anyway, consider a game like this. You bet $255 on red. The wheel is spun 8 times. If the ball falls on a red number at least once, you win $1, otherwise, you lose. Do you agree, that the house edge (traditionally defined) of such a game is about 0.198%?

Quote: Whiskeyjack

I've read all the information on betting systems and have come to the conclusion that to say there is no system available to beat the house is not true.

A simple Martingale or D'alembert MUST win against the house in the following scenario.

1. Unlimited bankroll.

2. Unlimited bets on the table.

Can you prove that this is incorrect?

You need a third condition:

3. The house has a limited bankroll.

But in that case, why bet? just buy the casino (or the parent company).

Quote:

The problem is that a progression neither wins nor loses with infinite bankroll. Such statements are meaningless.

Not if you define a "win" as the total amount lost is less than the total won after some (arbitrarily large) number of bets.
The notion of "loss" is indeed meaningless, if there is no apriori limit on the number of bets. That is why I am saying that you cannot lose.

Quote:

If you complete one such infinite martingale and simply plot: {(N,W(N)), N >= 1} where W(N) is the amount won at hand N,

But why do I need to care about hand N. I am only interested in the endpoints of the game - such K, where W(K) > 0.
Remember, we are not interested in the outcomes of individual bets, only the final result. You just keep doubling your bet until you win.

Quote: weaselman

For games like roulette, that's not an issue.

They become complicated when you apply betting systems.

Quote: weaselman

Anyway, consider a game like this. You bet $255 on red. The wheel is spun 8 times. If the ball falls on a red number at least once, you win $1, otherwise, you lose. Do you agree, that the house edge (traditionally defined) of such a game is about 0.198%?

It would seem so, if you apply that definition. However, it doesn't make much sense in such application, as it's normally used in games where the payout is even or higher. The vigorish on this bet would rather be 33.3%, as in 1-(170/255)/1.0.

Quote: P90

It would seem so, if you apply that definition. However, it doesn't make much sense in such application, as it's normally used in games where the payout is even or higher.

I am not seeing the difference. How does it matter if payout is higher or lower?

Quote: weaselman

I am not seeing the difference. How does it matter if payout is higher or lower?

Imagine a game where you bet $419, then I flip a coin, and whatever happens next, I take $1 and give you back $418.
Technically, by the same definition, this game only has a house edge of 0.24%.

But you have to agree it's meaningless, because what really happens is I just take your money, so for practical purposes it's a game with 100% loss (which also happens to be my vigorish).

Calculating HA as -EV*W is meaningful under a limited set of circumstances. For instance, a lottery ticket that costs $1 and pays $1M 5e-7 of the time has both its vigorish and house edge at 50%, and for all purposes can be said to return half. This simplicity rapidly diminishes with smaller and variable payouts and wagers.

Quote: weaselman

But why do I need to care about hand N. I am only interested in the endpoints of the game - such K, where W(K) > 0.

There are no endpoints.

Quote:

Remember, we are not interested in the outcomes of individual bets, only the final result.

There is no final result.

Quote:

You just keep doubling your bet until you win.

And with probability 1 you will win. But with probability 1, if you throw a dart at a dart board you will not hit a point with rational coordinates. That doesn't mean that such points and such possibilities don't exist.

--Ms. D.

Quote: DorothyGale

There are no endpoints.

Yes, there are

Quote:

There is no final result.

Yes, there is

Quote:

And with probability 1 you will win.

Exactly.

Quote:

But with probability 1, if you throw a dart at a dart board you will not hit a point with rational coordinates. That doesn't mean that such points and such possibilities don't exist.

Nothing to do with this. We are talking about discreet case. Rational numbers are a null-subset of the continuum. But in a countable set the only null-subset is the empty one.

Quote: P90

Imagine a game where you bet $419, then I flip a coin, and whatever happens next, I take $1 and give you back $418.
Technically, by the same definition, this game only has a house edge of 0.24%.

Well said. Consider myself convinced that there needs to be some kind of a positive payout for "house edge" to have any meaning.The question of whether it needs to be even money or higher is still open though ...

Quote: weaselman

...

Thank you for opening my eyes with your insightful comments -- you really know your stuff -- amazing.

--Ms. D. Out.

Quote: weaselman

Well said. Consider myself convinced that there needs to be some kind of a positive payout for "house edge" to have any meaning.The question of whether it needs to be even money or higher is still open though ...

I would say it is a meaningful metric when payout is at least even money. When it's less, you have to weight the winnings for the metric to be meaningful. Vigorish becomes a better metric in that situation.

Quote: P90

I would say it is a meaningful metric when payout is at least even money. When it's less, you have to weight the winnings for the metric to be meaningful.

I guess, what confuses me here is why you want to weigh the winnings when it is less then even money, but not when it is more than even money. Why the asymmetry?

Quote: weaselman

I guess, what confuses me here is why you want to weigh the winnings when it is less then even money, but not when it is more than even money. Why the asymmetry?

It simply tends to be the same or very similar result when winnings are above even money, so it doesn't make a difference how it is calculated. The difference is only pronounced when underpaying on a small-payout bet.

Quote: weaselman

There are no contradictions in the real sense of the word, the set and number theories, both dealing with kinds of infinities are are talking about here, are both well-formulated and self-consistent, and do not include any contradictions (and are therefore incomplete, but that's a whole different topic).
All the "paradoxes" dealing with infinity are illusory, they arise from people trying to incorrectly apply their intuitions, "trained" on experiences with the finite entities, to infinity, and it just doesn't work.

If infinity could be defined and dealt with in the context of real-world experience, then there wouldn't be several dozen mutually contradictory posts on this thread. Postulating "infinite" ANYTHING as a condition of something that happens on Planet Earth will just lead to GIGO. This is one of those concepts that only has validity in Math World (which can still be a fun time had by all, which is why many tenured university professors go there and never come back).

Quote: mkl654321

If infinity could be defined and dealt with in the context of real-world experience, then there wouldn't be several dozen mutually contradictory posts on this thread.

Yes, of course. This problem is however not unique to the infinity. Try discussing motion at very high speeds with reasonably smart people, even those, that have heard of relativity, but haven't really mastered it through everyday application. Or quantum mechanics. There are literally mind bungling things there, such as quantum tunneling, teleportation, delayed choice etc., which are all very real, and take place in the real world around us all the time.

Quote:

Postulating "infinite" ANYTHING as a condition of something that happens on Planet Earth will just lead to GIGO. This is one of those concepts that only has validity in Math World (which can still be a fun time had by all, which is why many tenured university professors go there and never come back).

Well ... not ANYTHING ... but when talking about concepts that are inherently finite, then sure.

For example, Whiskeyjack poses the question as "unlimited" (ie, infinite), then defends it by saying the the longest streak he could get in 2,500,000 trials is 26....

But

2,500,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,00

0,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,00

0,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,00

0,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,00

0,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,00

0,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,00

0,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000

isn't even a measurable fraction of infinity. Such a fraction, regardless of how many zeros you wish, is undefinably small. Imagine a page of zeros circling the earth, it is still an indefinably small fraction of infinity.

Quote: Mosca

isn't even a measurable fraction of infinity. Such a fraction, regardless of how many zeros you wish, is indefinably small. Imagine a page of zeros circling the earth, it is still an indefinably small fraction of infinity.

Certainly, one has the arithmetic of infinities, a la Cantor, in which case Aleph_naught - (any integer) = Aleph_naught. The statement you made is hyperbole meant to make a point. The point is, I assume, that infinity is larger than any integer. Well, yes, the smallest "infinity" (Aleph_naught) is the cardinality of the integers, so by definition it is larger than any integer. Relationships between integers and Aleph_naught are usually expressed in terms of "limits" and "asymptotic analysis."

-Ms. D.

Quote: DorothyGale

The statement you made is hyperbole meant to make a point. The point is, I assume, that infinity is larger than any integer.
-Ms. D.

Yes, that is true; I don't have the mathematical language to express it that way. There is a secondary point, though, that the premise can be disproven just by thinking about it in the colloquial understanding of its framework, if you let go of the concept of 2,500,000 as being a large number of trials, or if you understand that you've rigged the solution by the way you've structured the problem. Either way it's wrong.

Well of course table limits are in place to limit the ability to Martingale meaningfully.

From a practical sense, what kind of bet is 26 losses in a row. In a martingale, $5 -> $10 -> $20 -> $40 -> $80 -> $160 -> $320 -> $640 -> $1,280 -> $2,560 -> $5,120 -> $10,240 -> $20,480 -> $40,960 -> $81,920 -> $163,840 -> $327,680 -> $655,360.

That's 18 losses in a row for a loss of $1,310,715.

Odds of that happening is about 9.6 in a million. Not likely at all but not insignificant. Will you as a player run into this situation in your life time? There's a 50/50 probability of this happening in about 72,000 spins. If you play the martingale exclusively even for 4 hours a week for 20 years, with a spin every 2 minutes, you'll see 120 spins / week x 52 weeks = 6,240 spins x 20 years = 124,480 spins, and you'll likely run into the situation of losing over $1 million -- all for a win of $5.

Quote: Whiskeyjack

In the following scenario.
1. Unlimited bankroll.
2. Unlimited bets on the table.

Unlimited bankroll is something you might have,
No table limits is something a casino is just not ever going to allow you to have.
They are not fools.

>So indeed you can see that both Martingale and D'Almbert were right in their thinking
Whoa there. Martingale was the OWNER of a casino who ran around urging customers to Double Up to Catch Up. He was indeed right in his thinking. He was thinking in his capacity as the Casino Owner.

>If you wish to beat the game you need to figure out how to beat the table limits.
Get the Casino Manager drunk? How are you going to beat table limits. They exist to protect the casino.

Quote: boymimbo

That's 18 losses in a row for a loss of $1,310,715...and you'll likely run into the situation of losing over $1 million -- all for a win of $5.

With no guarantee that after that 18th loss there won't be a 19th loss, but what the heck, if you indeed win and get that five dollars... how much have you tipped the waitress that night?

True martingalers don't tip.

If I was a casino, I'd raise the table limits by an order of magnitude for anyone playing a Martingale strategy.

It's a good way to get the entire bankroll for very little risk.

Think about the last spin. They can have $2560 on the line... but the casino is only really risking a total loss at the table of $5.... there's a nice over round for them.

Quote: P90

True martingalers don't tip.

But an infinite Martingaler with an infinite bankroll could tip an infinite amount and still have the same infinite bankroll after he did so.

(This whole discussion is infinitely silly.)

Quote: weaselman

Yes, of course. This problem is however not unique to the infinity. Try discussing motion at very high speeds with reasonably smart people, even those, that have heard of relativity, but haven't really mastered it through everyday application. Or quantum mechanics. There are literally mind bungling things there, such as quantum tunneling, teleportation, delayed choice etc., which are all very real, and take place in the real world around us all the time.
Well ... not ANYTHING ... but when talking about concepts that are inherently finite, then sure.

I guess what bothers me is that you can't really talk about "winning" and "losing" if infinity+X=infinity for any value of "X", positive or negative. So an infinite Martingale would work exactly as well with a 100% house advantage as under any other condition. The question of whether an "infinite Martingale" would "work" is therefore moot.

Quote: P90

True martingalers don't tip.

absolutely true.
I can even say true gamblers don't tip.

Quote: mkl654321

I guess what bothers me is that you can't really talk about "winning" and "losing" if infinity+X=infinity for any value of "X", positive or negative.

The amount you win (or lose) is finite. If you consider the total result of all your bets without adding it to the bankroll, the "win" (positive result") can be defined without ambiguity.
The utility of such a win would, of course, be questionable (since it does not increase your bankroll), but that's a different story. If you have infinite amount of money, the utility of pretty much anything money can buy becomes questionable to you.