A Less Obvious Martingale Fallacy

[PairTheBoard]

I think the key to understanding it is to realize that the Casino's viewpoint that I've talked about several times is just as valid as the Martingaler's viewpoint.

Best Wishes

PairTheBoard

[Grivan]

Dov you are assuming the gambler can end whenever he chooses. This is not fair if you are saying he must play for an infinite amount of time. A more reasonable way to look at it would be to allow time to run, but have it stop at some random point in the future (the casino and the gambler having no affect on when it stops). Once it stops look at who is ahead and by how much. Now do this trial a ton of times and you will clearly see that the casino is the winner in this circumstance.

Back to the scenario where it never ends. The gambler cannot choose to stop when he is ahead, he must always keep going until there is a point where he is behind again. On average the gambler will always be behind assuming the trial has run long enough.

[drudman]

[ QUOTE ]
[ QUOTE ]
At this point, is do we have consensus that OP was wrong?

[/ QUOTE ]

Only among those without the math background to understand that he is basically right.

PairTheBoard

[/ QUOTE ]

Well, that's me, but I'll play Devil's Advocate.

1) Each completed series adds +1 to Martingaler.
2) Each series completes.
3) Martingaler is guaranteed to be +x, where x=number of series completed.

Which of the above is false, and why?

[MMMMMM]

[ QUOTE ]
1) Each completed series adds +1 to Martingaler.
2) Each series completes.
3) Martingaler is guaranteed to be +x, where x=number of series completed.

[/ QUOTE ]

Well for one thing, each series hasn't completed yet (you are looking into the future, not the past). So how do you KNOW that each series completes?

[PairTheBoard]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
At this point, is do we have consensus that OP was wrong?

[/ QUOTE ]

Only among those without the math background to understand that he is basically right.

PairTheBoard

[/ QUOTE ]

Well, that's me, but I'll play Devil's Advocate.

1) Each completed series adds +1 to Martingaler.
2) Each series completes.
3) Martingaler is guaranteed to be +x, where x=number of series completed.

Which of the above is false, and why?

[/ QUOTE ]

I think it's true that the event, Martingale Series Completes, occurs infinitely many times with probabilty 1.

But when the Casino goes ahead of the player by a record amount let's call that the completion of the Casino's Anti-Maringale Series.

Then it's also true that the event, Anti-Martingale Series Completes, occurs infinitely many times with probabilty 1.

So just saying that the player's series or the Casino's series completes infinitely many times doesn't tell us who is really winning because they both happen. We need a more precise way of seeing who is really winning. Grivan has the right idea when he says,

Grivan --
"A more reasonable way to look at it would be to allow time to run, but have it stop at some random point in the future (the casino and the gambler having no affect on when it stops). Once it stops look at who is ahead and by how much. Now do this trial a ton of times and you will clearly see that the casino is the winner in this circumstance."

The "random point in the future" needs to be nailed down a little better, but Grivan's basic point is exactly right.

PairTheBoard

[drudman]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
At this point, is do we have consensus that OP was wrong?

[/ QUOTE ]

Only among those without the math background to understand that he is basically right.

PairTheBoard

[/ QUOTE ]

Well, that's me, but I'll play Devil's Advocate.

1) Each completed series adds +1 to Martingaler.
2) Each series completes.
3) Martingaler is guaranteed to be +x, where x=number of series completed.

Which of the above is false, and why?

[/ QUOTE ]

I think it's true that the event, Martingale Series Completes, occurs infinitely many times with probabilty 1.

But when the Casino goes ahead of the player by a record amount let's call that the completion of the Casino's Anti-Maringale Series.

Then it's also true that the event, Anti-Martingale Series Completes, occurs infinitely many times with probabilty 1.

So just saying that the player's series or the Casino's series completes infinitely many times doesn't tell us who is really winning because they both happen. We need a more precise way of seeing who is really winning. Grivan has the right idea when he says,

Grivan --
"A more reasonable way to look at it would be to allow time to run, but have it stop at some random point in the future (the casino and the gambler having no affect on when it stops). Once it stops look at who is ahead and by how much. Now do this trial a ton of times and you will clearly see that the casino is the winner in this circumstance."

The "random point in the future" needs to be nailed down a little better, but Grivan's basic point is exactly right.

PairTheBoard

[/ QUOTE ]

I disagree. It's obvious that the casino will be a large winner if the betting must randomly stop at some point. But that wasn't what OP said. OP said that it is a fallacy that with an infinite bankroll Martingaling is successful. That is wrong, because the player chooses when to stop, not the casino.

As for the anti-Martingaler series objection, it seems pretty weak. It doesn't even make sense. There is no chain of bets that ends with the casino winning a unit back from the player. Obviously the casino will win some ridiculous bets, but the player will complete his series by definition, and all of those ridiculous record bets will become immaterial. The player will be +1 unit.

The player isn't even really betting on roulette. He's betting that he won't have an infinitely long losing streak. It's a lock, Jerry!

[MMMMMM]

[ QUOTE ]
I disagree. It's obvious that the casino will be a large winner if the betting must randomly stop at some point. But that wasn't what OP said. OP said that it is a fallacy that with an infinite bankroll Martingaling is successful. That is wrong, because the player chooses when to stop, not the casino.


[/ QUOTE ]

Actually, I said this:

"If you intend to keep repeating the system, you cannot logically "take accounting" at only those times when you fancy to take accounting."

But even if you do NOT intend to play forever, or repeatedly, the system does not work. Yes, it will almost surely work for you if you do it just once or a few times. But for every time you try it, there exists the possibility of indefinitely prolonged losing streaks. Though unlikely, the penalty of a horrid streak may be so severe as to obviate all gains from prior runs.

You cannot be sure that will not occur. And if you and a countless number of gamblers were all playing the same system, from the same bottomless bankroll, on countless tables simultaneously, you guys would be expected to be showing a net loss all along the way. In other words, even thoughg you and many of the gamblers might be winning as you complete as series, therew would exist your horrendously unlucky counterparts who would be losing more than you guys are winning.

And you have no way of knowing in advance if you are going to be one of the ones for whom the luck seems "normal", or "good"--or absolutely unbelievably, never-endingly horrible. Just as you have no way of knowing if the next streak you embark on will get worse....and worse...and worse...

For all the small wins "earned" by completing the series, there exists a theoretical counterpart or series which is losing that much and more. And you can't know in advance what your next lot will be.

Yes, it seems like a lock to win a small amount (perhaps again and again). But theoretically speaking, it isn't.

[PairTheBoard]

drudman --
"As for the anti-Martingaler series objection, it seems pretty weak. It doesn't even make sense. There is no chain of bets that ends with the casino winning a unit back from the player. Obviously the casino will win some ridiculous bets, but the player will complete his series by definition, and all of those ridiculous record bets will become immaterial. The player will be +1 unit."

It's not weak. After the end of every Martingale series that puts the player ahead by amount A, the Casino is looking for a series of bets LLLLLLL...L consisting of n straight losses for the player where 2^n > A. That is a series of bets that ends with the Casino taking one or more chips from the player's pocket. The fact that they play on from that point is no more valid than the fact that they play on after the end of a Martingale Series. What's weak is your refusal to look at it from the Casino's perspective.

PairTheBoard

[Dov]

I thnk the part of this that's so hard for us non-math people is the fact that there is only 1 series that ends with a L, and that is not considered to be a complete series.

Losing forever is the only way that the Martingaler loses. Every other sequence results in a gain. And he simply continues until he wins.

I understand the point about having multiple people playing at the same time and I think it is a good one.

The problem with the accounting is this, as I see it:

The casino's edge is ok because they are offering the game. This is the 'price' they are charging you for the privelege (option) to play.

The Martingaler is not required to play. However, we have stipulated in our problem that he must continue playing forever.

Now, since the Martingaler plays as he sees fit, he only takes account of when he completes a series. This is because he knows that the rest of it is just waiting.

The casino takes account after every bet. They have no way of knowing if he will, in fact, place another bet or not.

I'm starting to think that they may both be right. Still, it seems that the casino isn't taking everything into account.

They should be able to understand that if the Martingaler continues to play, and never quits when he loses, then they are almost certain not to win.

This is not against the rules of the game, as someone else was saying that it would be taking future bets into account. I think this is imperative.

Otherwise, it would be kind of like saying that as long as your preflop game is perfect, your postflop skills don't matter because you already made your money in the beginning. If something happens later in the hand to change the EV of your situation, and you don't recognize it, you will lose, even though you were originally the favorite.

I think that this is what happens here. The Martingaler is not playing the same game as the casino.

The casino plays one bet and wins every time. The Martingaler plays one series and wins every time. They are actually playing different games.

They both win.

How did I do?

[drudman]

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
At this point, is do we have consensus that OP was wrong?

[/ QUOTE ]

Only among those without the math background to understand that he is basically right.

PairTheBoard

[/ QUOTE ]

Well, that's me, but I'll play Devil's Advocate.

1) Each completed series adds +1 to Martingaler.
2) Each series completes.
3) Martingaler is guaranteed to be +x, where x=number of series completed.

Which of the above is false, and why?

[/ QUOTE ]

Pairtheboard,

Like I say above, I'm playing Devil's Advocate. Which is false, 1 or 2? If both are true, then 3 is true. You say 3 is false, so please make your objection.

Bet 1. Win. +1

Bet 1. Lose. -1
Bet 2. Win. +1

Bet 1. Lose. -1
Bet 2. Lose. -3
Bet 4. Win. +1

...and so on. Are you trying to say that at some point:

Bet 1. Lose. -1
Bet 2. Lose. -3
...
Bet x. Win. <not +1>