A Less Obvious Martingale Fallacy

[PairTheBoard]

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Take accounting the way it's always done. At the end of every roullette spin. With just one Martingaler it's hard to see how to measure all the ups and downs. But with a million Martingalers each getting one spin at a time, you can easily measure the net flow of funds after every set of million spins - one for each martingaler - to see the funds flowing out of the Martingaler's collective pockets and into the Casino Coffers. That's the original point of MMMMMM and it can be easily simulated for proof.

PairTheBoard

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A million martingalers is not enough. Even with a million, the casino's cashflow will continue to oscillate positive and negative and you will still have the issue of when to take accounting. With a million martingalers, there is a positive probability on each spin that all million will win. When that happens for the n-th time (and it will happen infinitely often if the game is played forever), then the casino will be down at least a million * n dollars.

By the way, as I said, there will be infinitely many spins in which all million martingalers win simultaneously. But I doubt you would ever see one of those in any simulation. The expected number of spins before you see that event is about 10^325,000.

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This is true. Although I don't think the question of when to take an accounting should be an issue. Take an accounting after every spin.

With ever increasing bet sizes over time the 1,000,000 Martingaler Results get to look more like a single Martingaler's Results with Huge Singleton Bets swamping the smoothing out effect of the numerous Martingalers. You might keep the results smoother by increasing the pool of Martingalers at play by a certain percent after each spin. Maybe this would help in looking for some kind of rigourous statement for what happens as N goes to infinity.

If bet sizes were fixed we could say that after N trials there is a certain probabilty a gambler will be ahead playing against a certain house edge. It would be nice if we could formulate a similiar statement for a properly expanding pool of Martingalers.


PairTheBoard

[drudman]

The sim is flawed because you can't take accounting at any finite point, as per the OP. Martingaling works because a win is inevitable. In an infinite series, you will see all of the wins, and they will cancel out all of the losses and then some. The "then some" increases one by one into infinity.

[jason1990]

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In an infinite series, you will see all of the wins, and they will cancel out all of the losses and then some.

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Or you could say,

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In an infinite series, you will see all of the losses, and they will cancel out all of the wins and then some.

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Imagine you're flipping a coin for $1 a flip. It may not be as obvious here, but there will definitely come a time when you are $1 ahead. If you call that time the end of a "sequence," then you can watch your money grow $1 at a time and consider all your losses to be "funny money."

But that's just one perspective. The casino is free to regard the first time you're down $1 as the end of their "sequence." If they do that, then they're watching your money shrink $1 at a time and they consider all your wins to be "funny money."

[MMMMMM]

haha, you said what I said, but in a third as many words or less;-)

[MMMMMM]

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The sim is flawed because you can't take accounting at any finite point, as per the OP. Martingaling works because a win is inevitable. In an infinite series, you will see all of the wins, and they will cancel out all of the losses and then some. The "then some" increases one by one into infinity.

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It is a strange form of accounting which counts only the peaks and not the valleys. More especially so because the valleys gradually get ever deeper and deeper.

Picture a wave oscillator chart, fluctuating above and below a fixed line of zero. The first win puts the wave 1 above the zero line. A dip will put it below.

As time goes on, you would observe some higher highs and lower lows. This is virtually inevitable, and this trend would continue. But the lower lows would slowly get greater and greater.

From the casino's perspecive, those lower lows (for the gambler) are mathematically expected.

So let's say one way of accounting would be: score the higher highs and lower lows ONLY. They both should occur. Then average the higher highs and lower lows, because you really can't fairly count ONLY the highs, any more than the casino can fairly count ONLY the lows.

I'm not saying the above is the best way of scoring. It's not. But it is better than your completely lopsided way of scoring by counting only the highs which is 100% player-centric. And it would demonstrate that the player on average is getting further and further behind.

Look at it this way too: If you can count only the highs, the casino can count only the lows. So by that method, whichever side has the Cosmic Accountant in its employ is the side that is winning. Or better yet, give them both an Accountant trained in this manner of accounting, so they can both be winning! Does that really make any sense at all?

[SheetWise]

MMMMMM-

You have to remeber that the Martingaler never has any losses. Losing hands are simply an investment that gets you one step closer to a win. I would carry them on the balance sheet as an asset -- perhaps as a prepaid expense, or maybe an account receivable. [img]/images/graemlins/wink.gif[/img]

[Komodo]

</font><blockquote><font class="small">Svar till:</font><hr />
The sim is flawed because you can't take accounting at any finite point, as per the OP. Martingaling works because a win is inevitable. In an infinite series, you will see all of the wins, and they will cancel out all of the losses and then some. The "then some" increases one by one into infinity.

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That is exactly here where many peoples thinking is flawed. The win isnt inevitable. In an infinite serie, the probability will be closer and closer to 100%, but never 100. The assumption that every martingaler will be able to complete his serie is matchematically impossible. It will be 99,99999999999999999999999999 etc etc but never 100%.

[mosta]

is your claim that EV is &lt; 0? skimming the thread I didn't notice any mathematical statements from you. I'd LOVE to see you try to _prove_ this (ie, not with analogies and word pictures). do you do any proofs on near 200 score IQ tests?? do you know what a proof is??

[MMMMMM]

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is your claim that EV is &lt; 0? skimming the thread I didn't notice any mathematical statements from you. I'd LOVE to see you try to _prove_ this (ie, not with analogies and word pictures). do you do any proofs on near 200 score IQ tests?? do you know what a proof is??

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Are you addressing me?

Of course the gambler's EV is less than zero on a roulette wheel, using the Martingale or any other system.

EV = total volume bet * advantage (or disadvantage). In this case the advantage for the gambler is negative on every spin (positive for the house), so no matter how many spins he might make (&gt;1) his EV is &lt;0. The house edge on a roulette wheel is fixed somewhere around 5.25% as I recall. So his EV would be -5.25% * total amounts bet.

Also, I never claimed an IQ near 200.

I do sense a hostile tone in your post. Why?

[MMMMMM]

It is also a mathematical fact that NO betting system can overcome the house advantage when utilized against an independent-trials-based game with a built-in house advantage on every trial. The house extracts an EV of the house edge times the volume bet.

I will not attempt to PROVE it, but any mathematician will tell you that that is so.

It should also be obvious that it is so: because the EV for the gambler is negative on every single roll.

Adding a string of negative numbers cannot produce a positive number. Not a formal proof, but pretty basic, and clearly true.