A Less Obvious Martingale Fallacy

[MMMMMM]

Here are some thoughts which have occurred to me over time regarding this betting system.

Many knowledgeable gamblers are familiar with the Martingale system of doubling up after each loss until a win, then restarting the series. On a game like roulette, betting red or black, this system results in a win of 1 unit each time a series is completed (example: bet 1 unit and lose, bet 2 units and lose, bet 4 units and win: which gives a net profit of 1 unit for the entire series).

Many understand that this system does not work in a casino where table limits exist, because eventually you will encounter a string of losses in which your next required bet in the series will exceeed the table limit.

Some good posters in other 2+2 forums have in times past asserted that it is only the table limits and bankroll limitations which prevent the Martingale from working. They believe that the system would in fact work if there were no table limits and if the gambler possessed an unlimited bankroll.

What they do not understand, is that even WITHOUT table limits, and WITH infinite bankrolls, this system still would not work in the long run.

It is easy to think that the system would work without such restrictions, because you get to keep playing until you win, at which point you have another unit of profit.

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

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

Counting your bankroll only at the times immediately after completion of a winning series is fallacious. If you were truly to play this game eternally, your chips would slowly go down, on average, due to the house edge. Yes, you would have some huge peaks and valleys, and you would eventually recoup bad losses after longer and longer series, but it would be the *average* pluses or minuses to your bankroll that would matter, not your arbitrarily chosen accounting points.

Let's say you could play this game until the universe itself winds down and comes to a complete grinding halt. Well, you would be expected to be down A LOT of money at that point.

You might have completed God-knows-how-many series by then, but this still would be true. This, of course, is because the house edge has been inexorably eating away at your bankroll despite your contrived betting pattern and your artificially chosen accounting points.

Another way of looking at it is to imagine a great matrix of all possible paths of outcomes. If the Red-or-Black wheel game were truly 50-50 odds (with no green zeros), then even with a doubling up Martingale system, the extremes of the matrix would balance the small accumulated wins, and you would break even in the long run. However, with the green zeros putting the advantage in the house favor, the matrix of possible paths is not weighted fairly. Hence some of your "get even" doubling up streaks will take longer than they would otherwise take with a 50-50 wheel, and some of your first bets of a new sequence will start with a loss whereas with a 50-50 wheel they would instead have started with a win.

Another way of looking at it is to imagine an immensely huge casino, so large you could never count all the gamblers in it, with each gambler seated at his or her own private roulette table and betting this system against the house.

At any given time after the first spin, the average bankroll of all the gamblers would be expected to be showing a loss--even though they are all playing the Martingale system.

Presuming a great computer system could instantaneously know the totals at each table and immediately sum the results, it would be ludicrous to take the accounting in any way other than simultaneously for all tables.

If you were the Pit Boss at the Cosmic Casino, and the Cosmic Casino Manager strolled by and asked you how the casino was doing today, it would be ludicrous for you to say, "Well, we have to wait until each player finishes their own individual series before I can tell you". If you tried it, the Cosmic Casino Manager would probably throw a fit and tell you that is ridiculous. Why, he might ask you, should we tally only when a player is winning? Would it make any sense to tally only when the players are losing? And, by the way, how did you get this job in the first place? Then, in a more kindly tone, he might say, "Look, it is more reasonable and far simpler to simply tally all the tables at once. Look, just press the "T" button here, for "Tally";-)).

Now suppose that Fast Eddy Felson were watching all of this from a catbird seat, as a special guest of the casino. Fast Eddie happens to believe in the Martingale as a winning system, as long as no limitations are present. He sees all the gamblers and gets a special offer from the Cosmic Casino Manager: Fast Eddie, you too can play this system against us. The only thing is, all tables are currently taken. But as our special esteemed guest, I will let you pick any table and I will have the patron occupying it removed. The only catch is you must start the series where he left off and take on his wins or losses up to this point.

Now, from his high vantage point, Fast Eddie can view acres and acres of tables, stretching out as far as the eye can see. The only thing is, he is so high up he can't see the chips and thus has no way of knowing which players are doing well or not. He also has no idea how long this game has been going on. Should Fast Eddie pick a table? Would you?

I hope all of this helps to illustrate why the Martingale System would not work, *even if* there were no real-world limitations on bet-sizing or on bankrolls.

All comments welcome.

[Dov]

I don't think this is true.

I think the infinite bankroll overcomes the house edge in the end.

This is because when you do complete a series you will have an EV of your old BR+1 betting unit.

You are still EV+, you just can't know the EV of a particular wager until the series is completed. You would have to divide your 1 betting unit of profit over all of the bets in the series.

You are correct in that during a losing series you will have experienced -EV bets. However, the metagame conditions allow this to actually be a +EV situation.

Taken from the perspective that there may be a more efficient use of your funds, I can understand that the Martingale would be -EV.

Under these conditions, though, I would expect that your BR would grow with the average # of trials in a series. If you are saying that the longer you play, the wider your variance will get, (which is true), it will still even out in the end when you do actually win, assuming that your winning chances are above 0.

As a matter of fact, I can't think of a situation where you have any chance to win that wouldn't guarantee that you do win except when you are a guaranteed loser. (like drawing dead)

I think you have somehow overlapped 2 concepts that don't, but I'm not completely sure where your error is or if I am the one who is mistaken. (I don't really think that I am, though, because by definition, you WILL win, and when you do, your BR will be larger than when it started.)

[[censored]]

Had lunch with Fast Eddy Felton, he is a cool cat.

[SheetWise]

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. 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. Because my confidence limits would be affected by my life expectancy, I would probably be happy, rich, and comfortable with a great many fewer. The casino would certainly have the advantage in the game you describe -- but they would probably want to take their lifetime (or quarterly earnings) into account as well.

SheetWise

[maurile]

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I think the infinite bankroll overcomes the house edge in the end.

This is because when you do complete a series you will have an EV of your old BR+1 betting unit.

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Your old bankroll was infinite. What's infinity plus one?

[Dov]

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Your old bankroll was infinite.

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So why were you playing in the first place?

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What's infinity plus one?

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According to your logic you could never lose anything either. What's infinity minus 1?

[maurile]

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Your old bankroll was infinite.

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So why were you playing in the first place?

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What's infinity plus one?

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According to your logic you could never lose anything either. What's infinity minus 1?

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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.

[MMMMMM]

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(I don't really think that I am, though, because by definition, you WILL win, and when you do, your BR will be larger than when it started.)

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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.

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.

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

Also, although this is tangential and more complicated, if you had an infinite number of gamblers playing this system, might you have at least one gambler who would never get to make a single winning bet? So he might never complete the first series?

[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

[MMMMMM]

[ QUOTE ]
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I think the infinite bankroll overcomes the house edge in the end.

This is because when you do complete a series you will have an EV of your old BR+1 betting unit.

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Your old bankroll was infinite. What's infinity plus one?

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Just for clarity's sake, I wish I had written "unlimited bankroll" instead of "infinite bankroll".

Not that your question isn't interesting on its own merits.