A Less Obvious Martingale Fallacy

[Dov]

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Infinity plus or minus any finite sum is still just infinity. So an infinite bankroll cannot be increased (or decreased) with the martingale system if you are betting finite sums.

If you have a finite bankroll, using the martingale system is demonstrably -EV in any game like roulette where each trial is -EV.

So either way -- finite bankroll or infinite bankroll -- you cannot turn a -EV game into a +EV game by martingaling it.

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Why do you say that you can use a finite bankroll over an infinite time period?

Surely, if the time period is infinitely long, then you can't increase it by another day.

Similarly you can't have an infinite number of trials either, because you would never be able to increment the number of completed trials by 1.

Why do you accept an increasing number of trials, but not an increasing bankroll?

[MMMMMM]

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Assuming the game is not biased, and the wager is close to 50-50, Baccarat or the pass line for example, there does exist a number of series where the probability of failure approaches a theoretical zero.

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Doesn't your so-called "theoretical zero" correspond to a so-called "theoretically infinite dollar amount (loss)" on the other side of the coin?

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Assume the game was fair, and I could play a progression that lasted for 1000 trials -- I would be quite comfortable playing for a lifetime.

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Well maybe you should be. However, for all the people who might do it along with you and be successful and happy, wouldn't there be a few who would bear a commensurate loss to your summed profits by hitting that nasty old extreme negative end of the tail? (actually it would be more money because of the house edge on roulette or baccarat or whatever).

[maurile]

I can't make sense of your post.

[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.

[maurile]

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Assuming the game is not biased, and the wager is close to 50-50, Baccarat or the pass line for example, there does exist a number of series where the probability of failure approaches a theoretical zero.

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Doesn't your so-called "theoretical zero" correspond to a so-called "theoretically infinite dollar amount (loss)" on the other side of the coin?

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Yes. Take roulette as an example. We'll bet on red, which pays 1-1. On any particular spin of the wheel, our chance of winning is 18/38 = 0.473684. Our chance of losing is thus 1 - 0.473684 = 0.526316.

I'll do the math for a 15-unit bankroll, and then I'll just post the results for successively higher bankrolls . . . you'll see for yourself that extrapolating up to an infinite bankroll does not produce a positive result.

If we have a 15-unit bankroll, we can fund four losing spins before we go broke in the martingale system. (1 + 2 + 3 + 4 = 15.) We are thus risking 15 units to win 1 unit. Our chance of winning = 1 - (0.526316^4) = 0.9232. We are thus 12-1 favorites to come out ahead. But laying 15-1 odds when we are only 12-1 favorites is a losing proposition. Thus we have a negative expectation with a 15-unit bankroll.

Here are the results for successively higher bankrolls:

3-unit bankroll (can fund 2 spins): we are laying 3-1 odds as a 2.61-1 favorite.
7-unit bankroll (can fund 3 spins): we are laying 7-1 odds as a 5.86-1 favorite.
15-unit bankroll (4 spins): we are laying 15-1 odds as a 12.03-1 favorite.
31-unit bankroll (5 spins): we are laying 31-1 odds as a 23.76-1 favorite.
63-unit bankroll (6 spins): we are laying 63-1 odds as a 46.05-1 favorite.
127-unit bankroll (7 spins): we are laying 127-1 odds as a 88.39-1 favorite.
255-unit bankroll (8 spins): we are laying 255-1 odds as a 168.8-1 favorite.
511-unit bankroll (9 spins): we are laying 511-1 odds as a 321.7-1 favorite.
1023-unit bankroll (10 spins): we are laying 1023-1 odds as a 612.1-1 favorite.
2047-unit bankroll (11 spins): we are laying 2047-1 odds as a 1164-1 favorite.
4095-unit bankroll (12 spins): we are laying 4095-1 odds as a 2212-1 favorite.
8191-unit bankroll (13 spins): we are laying 8191-1odds as a 4204-1 favorite.
16383-unit bankroll (14 spins): we are laying 16383-1 odds as a 7989-1 favorite.
32767-unit bankroll (15 spins): we are laying 32767-1 odds as a 15180-1 favorite.

And so on, up to a bankroll that approaches infinity, and can therefore fund a number of spins that approaches infinity. The trend of laying better odds than our chance of winning warrants never reverses itself: we are never taking the best of it.

As our bankroll approaches infinity, we approach becoming an infinity-to-one favorite to come out ahead. But the odds we are laying approach infinity-to-one even faster. The fact that it is theoretically possible for the roulette wheel to land on black N times in a row for any N means that, even in theory, the martingale never shows an expected profit.

[Dov]

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I can't make sense of your post.

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It seems to me like you accept the notion that we can have an infinite number of trials, each with its own result, as well as a cumulative result of all trials thus far conducted,

I was just wondering why you dismiss the applicability of this technique to a bankroll that can be considered to be large enough as to be equivallent to infinite.

If you like, then just always assume that more money will be available for the bankroll when you need to place a bet larger than the balance.

[maurile]

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It seems to me like you accept the notion that we can have an infinite number of trials, each with its own result, as well as a cumulative result of all trials thus far conducted.

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Yes.

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I was just wondering why you dismiss the applicability of this technique to a bankroll that can be considered to be large enough as to be equivallent to infinite.

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I never said you can't have an infinite bankroll. I just said -- correctly -- that when you add a finite sum to it, you are not increasing it. (Same with an infinite number of trials. If you add a trial to an infinite series of trials, you are not increasing the length of the series.)

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If you like, then just always assume that more money will be available for the bankroll when you need to place a bet larger than the balance.

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Yes, that's what an infinite bankroll is.

[Dov]

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Well, that argument meets to its complement, which is: by definition, you WILL lose, and lose many many many times in a row on occasion. And the longer you play, you will set new records for longer and longer losing streaks.

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Are you saying that eventually you will encounter a losing streak so long that it will be infinite?

If that's true, then you will also encounter a similar winning streak at some point.

I think that because we are assuming a game where you aren't drawing dead (so to speak) we dismissed this possibility by using an unlimited number of trials.

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The accumulated small wins you are thinking of are balanced by an extreme theoretical tail or far-out reach of the matrix of possible paths, on the negative side.

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The negative side is completely recovered with a single win. This is only possible because of the unlimited bankroll, but given that unlimited bankroll is a condition of this problem, I still don't see how our gambler can fail to bust the bank 1 unit at a time.

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The losses compound geometrically more and more, the rarer (longer) they are.

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This is why you need an unlimited bankroll. It overcomes this objection.

[maurile]

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If that's true, then you will also encounter a similar winning streak at some point.

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Possibly tomorrow. :fingerscrossed:

[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;-)