A Less Obvious Martingale Fallacy

[SheetWise]

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Why put an artificial limit on this?

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Because there are limits. Using infinity is an artificial abandonment of constraint -- and why you can't define the expected outcome of the game.

My calculation would require a definition of zero (impossibility), and I have seen people define it differently for different purposes. But once that limit is defined, we can apply it to a lifetime of play and achieve that confidence level.

I don't know what to do with infinity [img]/images/graemlins/confused.gif[/img]

[PairTheBoard]

Correct.

I posted this same idea in the Probabilty Forum for the case where bet sizes remain constant but players with finite bankroll bust out due to variance. The difference here is that average bet sizes increase over time and the Cosmic Casino makes EVEN MORE MONEY on average over time because of it. It makes no difference whether bet sizes increase at random or due to long losing streaks.

PairTheBoard

[Dov]

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It makes no difference whether bet sizes increase at random or due to long losing streaks.

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If Cosmic Casino has to repay the entire amount lost to them plus 1 unit when the gambler finally wins, then how are they making money here?

[PairTheBoard]

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It makes no difference whether bet sizes increase at random or due to long losing streaks.

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If Cosmic Casino has to repay the entire amount lost to them plus 1 unit when the gambler finally wins, then how are they making money here?

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What you will see as you look over all the tables at the Cosmic Casino is a lot of players who are ahead a little bit, and a few players who are stuck huge amounts. The identities of the players who are ahead and who are stuck keep changing. But as time goes on it becomes more and more likely that the Casino will be ahead overall. What matters is who's rack the money is in at any point in time. Overall, the liklihood is that the Casino will have more money won and its racks than the players. As far as an individual player goes, if you're going to count yourself as being ahead after a win, you also have to count yourself as being behind when on a horrendous losing streak. You may spend more Time being ahead, but if you weight your time spent ahead or behind by the Amount you are ahead or behind, you will spend more Weighted Time being behind.

PairTheBoard

[Dov]

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What matters is who's rack the money is in at any point in time.

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I still don't see why if each individual player comes out 1 unit ahead at the end of a series, and you extend them out to infinity, you think the casino will win in the end.

If this were true, then why put a table limit at all? Why not let people keep doubling the bets?

Are you saying that it would be too risky for a real casino to allow someone to do this?

Wouldn't that contradict what was said earlier about them making more money when more money is wagered?

There must be a reason for table limits. Obviously EV isn't everything.

[maurile]

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Are you saying that it would be too risky for a real casino to allow someone to do this?

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Yes. Binion's used to be famous for being willing to accept huge bets. It let a guy bet $1 million on craps a couple times, I think. (Back when $1 million meant something.)

But there comes a point where it's definitely too risky. If I bet a few billion or whatever on red at Harrah's, I will own the casino if I win. The board of directors at Harrah's doesn't want to take that risk.

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Wouldn't that contradict what was said earlier about them making more money when more money is wagered?

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No.

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There must be a reason for table limits. Obviously EV isn't everything.

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EV isn't everything. There's variance, too.

[PairTheBoard]

Dov --
"I still don't see why if each individual player comes out 1 unit ahead at the end of a series, and you extend them out to infinity, you think the casino will win in the end."

I'm not looking at what happens "in the end". We never get to the end. All we can look at is what happens over time as we look at longer and longer measures of time played. Yes, there are times you will be ahead. In fact, most of the time you will be ahead. But there is no point in time where you will be ahead forever. There will always be times in the future when you are behind again. During those periods when you are behind it makes no sense to say, "It doesn't matter that I'm behind now because I know I'm going to be ahead again in the future." If it did make sense then the opposite could be said when you are ahead, "It doesn't matter that I'm ahead now because I know I will be behind again in the future." What does matter is how much time you spend ahead or behind AND BY HOW MUCH. If you weight the amount of time spent ahead and behind by the AMOUNT you are ahead or behind and look at future times T then you can say the following. As T gets larger and larger, the Weighted Time you are behind will exceed the Weighted Time you are ahead by an average amount determined by the House Edge.

PairTheBoard

[Dov]

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What is less obvious is that it is STILL all one long game.

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I think this is where the problem is.

It IS still one long game, but we adjust the parameters artificially according to short term results.

The Martingale with an infinite bankroll (the way I think you intended it) actually breaks the long run into many short run games which all have a +EV of 1 unit spread across as many wagers as it takes to win 1 game.

As soon as a win is achieved, the system 'resets' itself as if it were actually starting over again for the first time.

What is the difference between bets in the Martingale system when it is the 1st bet of a new series or the next bet following a win?

For all intents and purposes, it is NOT one long game.

[MMMMMM]

Hi Dov,

I can see how what you are saying appears to make sense, but I think that is an illusion.

Time for sleep now, so later;-)

[felson]

There's no way I am betting my money according to Martingale.

F.E.F.