Mathematical Proof that Progressions will never Overcome a Negative Expectation Game

[johndoe]

Yes, johndoe, let's bring it into a reasonable real life perspective. We've all had fun with the infinite prospects; propose now a real life scenario where we have some finite amount of bankroll and the very wealthy house grants a no limit playing environment. Even this hardly seems real life, but just imagine it as a publicity stunt. Is the risk warranted? Is Thorp correct in this context?

As I've said and agreed completely all along, if there is a finite bankroll (or credit, or table limit, etc.), then of course the system does not work. I have no argument with this whatsoever.

The only reason an unlimited credit line is considered "unrealistic" is because casinos aren't that stupid. It would be a ridiculous mistake for them to make, but, unlike an infinite bank account, it's not inherently impossible.

Thorp gave exactly that qualification when he called using a martingale "reasonable".

As for the publicity stunt, as long as the casino is not extending credit, I think they'd greatly welcome this. I'd imagine that high-rollers regularly play with (effectively) no table limits regularly, as it is. And there are probably plenty of small players whose martingale fails due to a short bank long before they hit the table limit.


I think some of the allure of the martingale is that it would work quite well without credit or table limits, but people grossly underestimate the downside when those constraints exist.

 

[johndoe]

I think the missing element is the odds.
...
Martingale proponents say if you play long enough with an endless bankroll you can be guaranteed a 1 unit profit, even in a negative EV game. In a modestly negative game he may have an almost certain probability of winning 1 unit. However the probability of a win is never quite certain and a martingaler is ALWAYS RISKING MORE THAN IT'S WORTH WITH ANY BANKROLL to try for a 1 unit profit in a negative EV game.

I completely agree, with any finite bankroll. It's a horrible idea doomed to failure.

But if it's unlimited, the risk of losing over any reasonable period is so small that it might as well be zero. Let's see - one roll per minute, for simplicity we'll call it 50/50 odds on red. The risk is 1 in 2^(24*60) for a 24-hour session. That's 1 in 3x10^433. Pretty small. Adding up all the possible positions for all the particles in the universe you "only" get 5x10^591.

(Of course, the max bet size is similarly astronomical, but we didn't place a limit on it. This is still purely hypothetical.)

 

[aslan]

Did not find.

So my question is - If martingale will always succeed in the absence of limits,
is that some sort of paradox or violation of commonly held probability understanding? zg

Even without a house limit, your money could run out. There is no such thing as an unlimited bankroll.

 

[johndoe]

Even without a house limit, your money could run out. There is no such thing as an unlimited bankroll.

Right. An unlimited credit line is slightly more conceivable though - at least not completely impossible.

 

[zengrifter]

As I've said and agreed completely all along, if there is a finite bankroll (or credit, or table limit, etc.), then of course the system does not work. I have no argument with this whatsoever.

The only reason an unlimited credit line is considered "unrealistic" is because casinos aren't that stupid. It would be a ridiculous mistake for them to make, but, unlike an infinite bank account, it's not inherently impossible.

Thorp gave exactly that qualification when he called using a martingale "reasonable".

So, what do you think about this >>

If martingale will always succeed in the absence of limits,
is that some sort of paradox or violation of commonly held probability understanding? zg

In other words, given a -EV game, does it only come down to the infinite event where one loses every hand?
That is the reckoning?

But if so, could that ever happen in infinity with a +EV trial? zg

 

[k_c]

I completely agree, with any finite bankroll. It's a horrible idea doomed to failure.

But if it's unlimited, the risk of losing over any reasonable period is so small that it might as well be zero. Let's see - one roll per minute, for simplicity we'll call it 50/50 odds on red. The risk is 1 in 2^(24*60) for a 24-hour session. That's 1 in 3x10^433. Pretty small. Adding up all the possible positions for all the particles in the universe you "only" get 5x10^591.

(Of course, the max bet size is similarly astronomical, but we didn't place a limit on it. This is still purely hypothetical.)

All I'm saying is that with any bankroll the odds of success do not justify playing.

It's similar to an insurance bet in blackjack. Whenever probability of drawing a ten is less than 1/3 then insurance is negative EV and is not justifiable if you want to make a profit in the long run. If probability of drawing a ten is greater than 1/3 then insurance is positive EV.

In the martingale (applies to any bankroll):
if odds of failure = 1/bankroll then in the long run you break even.
if odds of failure > 1/bankroll then in the long run you lose
if odds of failure < 1/bankroll then in the long run you win

The martingale player is laying odds of bankroll to 1 that he will succeed.

Martingale proponents will only say that since they have been given an unlimited bankroll that they cannot possibly go broke. All this is is a delaying tactic to hide the inevitable failure of a martingale in a negative EV game. If you had a 0 probability of winning and martingaled an unlimited bankroll you could say the same thing. You will lose every trial, double the bet, lose again....keep losing and with an unlimited bankroll you will never go broke.

Martingale proponent will say, "ahhh but if I have any chance at all of winning I must eventually win at least once." However, your chance of winning 1 unit is never justifiable in a negative EV game if you want to make a profit.

 

[Coach R]

If I had an unlimited bankroll, I wouldn't worry about odds, percentages, true counts or anything that made my brain hurt

 

[aslan]

Right. An unlimited credit line is slightly more conceivable though - at least not completely impossible.

So-called unlimited credit lines exist in name only.

 

[zengrifter]

In the martingale (applies to any bankroll):
if odds of failure = 1/bankroll then in the long run you break even.
if odds of failure > 1/bankroll then in the long run you lose
if odds of failure < 1/bankroll then in the long run you win

keep losing and with an unlimited bankroll you will never go broke.

I guess that settles it? zg

 

[k_c]

So-called unlimited credit lines exist in name only.

The black-red roulette martingale with unlimited credit that johndoe suggested probably wouldn't fail within a person's lifetime. The danger to the casino would be if the person died in the middle of a long losing streak - they would have no way to collect what was owed to them. :laugh:

 

[aslan]

The black-red roulette martingale with unlimited credit that johndoe suggested probably wouldn't fail within a person's lifetime. The danger to the casino would be if the person died in the middle of a long losing streak - they would have no way to collect what was owed to them. :laugh:

Holy cash cow, Batman! Infinite credit dilemma!!! View attachment 7114

 

[zengrifter]

The black-red roulette martingale with unlimited credit that johndoe suggested probably wouldn't fail within a person's lifetime.

Still hedging?? It would never fail. zg

 

[k_c]

Still hedging?? It would never fail. zg

No hedging - a person's lifetime is a very short time when compared to eternity. In a -EV game a martingale provides a very good chance of being ahead in its initial stages before it eventually fails. Howeever no matter how large bankroll is there is that nagging, if miniscule, chance that a person may never even see a single win in their lifetime, the chance that they will experience some 1 unit wins but are in the midst of a long losing streak when they die and owe much nore than they have won, and the certainty that no bankroll will not be enough if progression is played out to its conclusion.

Martingale players, do not fear though. There is a place where common sense can be suspended, non sequiturs can prevail, and reality can be replaced by fiction zenzone.

Addtionally we have a testimonial by a satisfied martingale user:

"No matter the odds me swears by me martingale." -Plopeye the Sailor

 

[QFIT]

Right. An unlimited credit line is slightly more conceivable though - at least not completely impossible.

There is no such thing as an unlimited credit line.

 

[zengrifter]

No hedging - a person's lifetime is a very short time when compared to eternity. In a -EV game a martingale provides a very good chance of being ahead in its initial stages before it eventually fails. Howeever no matter how large bankroll is there is that nagging, if miniscule, chance that a person may never even see a single win in their lifetime,

There is no such thing as an unlimited credit line.

We removed the stop of lifetime many posts ago.
Time is unlimited as well.
Therefore, a fully unbounded martingale or other negative progression,
as noted by Thorp et al will always win, yes?

Early in this thread we acknowledged the elementary proposition that a negative progression MUST ALWAYS lose in a -EV environment, given typical or even atypical parameters.

So we said okay of course and noted that an unbounded progression would ALMOST SURELY prevail, however. We did not expect anything more than a disdainful acknowledgement of that elementary truth. But curiously, THAT is where the heated debate began, right down to attempted math-proofs and repeated ridicule of such proposed non-reality scenario.

Further, subtle hedges were continually injected like "just to win 1u", and "not guaranteed in any lifetime", etc.

So my trailing question(s) is, for the last time -
Is an unbounded martingale almost surely to win in a -EV environment?
And if so, does that constitute a paradox or a suspension of normal statistical logic and/or probability? zg

 

[Machinist]

There is no such thing as an unlimited credit line.

Try telling our wonderful el presidente that.....:laugh::laugh: Lets not forget "all" and i mean ALL the ijiots on the Hill..... Left, right , under , above, and hidden.......



Machinist

 

[johndoe]

There is no such thing as an unlimited credit line.

I never said there was. I just said it wasn't completely impossible, unlike an "infinite bankroll" that people get so hung up about on this entirely hypothetical case.

Though Machinist certainly has an excellent point.

 

[zengrifter]

The black-red roulette martingale with unlimited credit that johndoe suggested probably wouldn't fail within a person's lifetime. The danger to the casino would be if the person died in the middle of a long losing streak - they would have no way to collect what was owed to them. :laugh:

It was a promo event for publicity - not real money.
The casino industry went along in order to establish that martingale
betting will always win if you got the money and the balls to use it. zg

 

[21gunsalute]

It was a promo event for publicity - not real money.
The casino industry went along in order to establish that martingale
betting will always win if you got the money and the balls to use it. zg

Unless the casino says you can't play anymore during a long losing streak.

 

[zengrifter]

Unless the casino says you can't play anymore during a long losing streak.

Casinos are known to change the rules in the middle of a promo.
But in this case it is to their benefit to lose, so they won't stop the game. zg