A Less Obvious Martingale Fallacy
 

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.

Re: A Less Obvious Martingale Fallacy
 

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.)

Re: A Less Obvious Martingale Fallacy
 

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

Re: A Less Obvious Martingale Fallacy
 

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

Re: A Less Obvious Martingale Fallacy
 

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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?

Re: A Less Obvious Martingale Fallacy
 

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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?

Re: A Less Obvious Martingale Fallacy
 

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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.

Re: A Less Obvious Martingale Fallacy
 

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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?

Re: A Less Obvious Martingale Fallacy
 

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

Re: A Less Obvious Martingale Fallacy
 

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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.

Re: A Less Obvious Martingale Fallacy
 

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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?

Re: A Less Obvious Martingale Fallacy
 

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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).

Re: A Less Obvious Martingale Fallacy
 

I can't make sense of your post.

Re: A Less Obvious Martingale Fallacy
 

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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.

Re: A Less Obvious Martingale Fallacy
 

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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.

Re: A Less Obvious Martingale Fallacy
 

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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.

Re: A Less Obvious Martingale Fallacy
 

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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.

Re: A Less Obvious Martingale Fallacy
 

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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.

Re: A Less Obvious Martingale Fallacy
 

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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:

Re: A Less Obvious Martingale Fallacy
 

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;-)

Re: A Less Obvious Martingale Fallacy
 

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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?

Re: A Less Obvious Martingale Fallacy
 

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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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No, because I'm going to use my lifespan as a definition of infinity [img]/images/graemlins/wink.gif[/img] . For arguments sake, if infinity is a lifetime -- and you play 24 hours a day until death -- you can find a number of progressions with a finite bankroll that will give you any confidence level you want, and I do believe in a theoretical zero. While a long sequence of losses is required to lose, it isn't to win. I'm saying neither one will happen.

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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?

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I'm saying no. That's why nobody will play this game. I'm not suggesting the house edge ever goes away, it's negative expectation -- but if the player can control the wagering to this degree, that expectation will never be realized.

Re: A Less Obvious Martingale Fallacy
 

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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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You have to be careful to distinguish between (a) an extremely large but still finite bankroll, and (b) a literally infinite bankroll.

For any extremely large but finite bankroll, we can do the math directly and see that we are still -EV (in roulette or any other game where each trial is -EV).

If you jump to a literally infinite bankroll, then the casino also has to have a literally infinite bankroll (to cover our biggest possible wagers). So the answer to why you won't eventually bust the casino one unit at a time is that you can't bust an infinite bankroll one unit at a time. You can keep subtracting one from the casino's 'roll all you want, but you will never bust it.

Re: A Less Obvious Martingale Fallacy
 

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No, because I'm going to use my lifespan as a definition of infinity

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This doesn't really work, though.

As an example, 5 years ago it would have been extremely rare for any professional poker player to play more than 125,000 hands in a year.

It is not uncommon for us to be able to play 4, 5, or even 6 times that number of hands now. Does that mean that the lifetime of a pro poker player has been lengthened?

Why put an artificial limit on this?

Re: A Less Obvious Martingale Fallacy
 

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If you jump to a literally infinite bankroll, then the casino also has to have a literally infinite bankroll (to cover our biggest possible wagers). So the answer to why you won't eventually bust the casino one unit at a time is that you can't bust an infinite bankroll one unit at a time. You can keep subtracting one from the casino's 'roll all you want, but you will never bust it.

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Fair enough, but if the game is actually +EV for the casino, then there should be a finite bankroll capable of defeating the unlimited Martingale bankroll, no?

If not, then you need to conclude that the Martingale played with an unlimited bankroll actually does overcome the house edge, regardless of its size.

I know this is true because you will eventually run into a series that overwhelms the finite side.

Re: A Less Obvious Martingale Fallacy
 

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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

Re: A Less Obvious Martingale Fallacy
 

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Fair enough, but if the game is actually +EV for the casino, then there should be a finite bankroll capable of defeating the unlimited Martingale bankroll, no?

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Yes, any finite bankroll will defeat the Martingale. A finite bankroll on the part of the casino means a finite betting limit -- so the player will be unable to keep doubling his bets past a certain point.

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If not, then you need to conclude that the Martingale played with an unlimited bankroll actually does overcome the house edge, regardless of its size.

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With literally infinite bankrolls, neither party has an edge. If either side has a finite bankroll, then the house will have an edge (assuming a game like roulette where the house has an edge on each individual trial).

Re: A Less Obvious Martingale Fallacy
 

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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]

Re: A Less Obvious Martingale Fallacy
 

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Yes, any finite bankroll will defeat the Martingale. A finite bankroll on the part of the casino means a finite betting limit -- so the player will be unable to keep doubling his bets past a certain point.

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This is not a finite bankroll for the casino. It is a finite bankroll for the player. The assumption was that the player is allowed to wager any amount he likes and chooses to wager 1 unit after a win and double the previous wager after a loss.

A finite bankroll for the casino would be a limited amount of capital that they could pay winning wagers with. (ie: an individual player wins)

Re: A Less Obvious Martingale Fallacy
 

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This is not a finite bankroll for the casino. It is a finite bankroll for the player.

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Same thing. The betting limit depends on the lesser of the two bankrolls. So if the bank's bankroll is finite, then effectively so is the player's. He can't double up his bets anymore once he gets to the point where the casino can't cover his bet.

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The assumption was that the player is allowed to wager any amount he likes and chooses to wager 1 unit after a win and double the previous wager after a loss.

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This requires an infinite bankroll for the casino as well. How can the player wager an infinite sum if the casino only has $100?

Re: A Less Obvious Martingale Fallacy
 

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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.

Re: A Less Obvious Martingale Fallacy
 

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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.

Re: A Less Obvious Martingale Fallacy
 

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

Re: A Less Obvious Martingale Fallacy
 

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

F.E.F.

Re: A Less Obvious Martingale Fallacy
 

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

F.E.F.

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Do you ever say anything that is even remotely useful or insightful on this board? Seriously, you must be the number 1 rank troll on 2+2.

Lawrence

Re: Making Unlimited Money With Flip-A-Coin
 

Suppose I repeatedly flip a coin for $1 each time; no doubling strategy. If it's a fair coin, then the swings in my bankroll, both up and down, will grow without bound. That is, for any X, there will come a time when I am $X ahead and a time when I am $X behind. So if I have an unlimited bankroll and an unlimited amount of time, I can make a million dollars. I simply wait until I am a million dollars ahead, which will eventually happen, and then I quit, never to play the game again. But the expected value of the number of flips I must make before winning a million dollars is infinity. And I must *never* play the game again after reaching my goal. I'll leave it as food for thought what, if anything, this has to do with the so-called martingale betting system.

Re: Making Unlimited Money With Flip-A-Coin
 

Jason1990 --
"But the expected value of the number of flips I must make before winning a million dollars is infinity."

Actually, this expected value would be a finite number. This is easy to see. After only about 100 Trillion flips, being up a million would only amount to being about 1/10 of a standard deviation over the mean of 50 Trillion heads. After a Billion*Billion flips it's almost like a coin flip for you to be ahead by a million dollars.

With a fair coin you will of course spend just as much time on average being behind a million dollars as ahead. This is similiar in principle to what I'm saying would happen with the martingale backed by unlimited funds.

PairTheBoard

Re: Making Unlimited Money With Flip-A-Coin
 

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Jason1990 --
"But the expected value of the number of flips I must make before winning a million dollars is infinity."

Actually, this expected value would be a finite number. This is easy to see.

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What you are seeing is an illusion. The expected number of flips it takes to win *one* dollar is infinite.

Re: A Less Obvious Martingale Fallacy
 

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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."

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Isn't this exactly how we decide whether or not to continue playing in a poker game?

We figure our Sklansky bucks based on the situation and go from there, right?

If you know that you are going to be ahead in the future, and that the win will overtake the loss, then this is a +EV situation.

I don't see why this is any different than say entering a pot from LP with a small PP hoping to spike a set and recover the PF 'mistake' through implied odds on the flop. You have taken a -EV individual situation and given it a +EV spin by using implied odds.

The Martingale seems to have implied odds that will cover all of your losses + 1 unit.

I don't see how you can get around this. I understand you to be saying that when you finally do win, you should be ahead more than 1 unit if your expectation was positive. I also understand that you want to measure each individual bet. I don't think this is entirely accurate, though.

I posted a 44 hand from EP a few weeks ago where each individual action (bet, call fold) was correct, but each STREET had errors on it. (flop, turn, river). Maybe what we should be calculating is the effective odds of the Martingale. You are using pot odds on every bet.

BTW, I know that this is not a viable gambling technique, but it is interesting to consider.

Meh, I have to go now. Be back later.

Re: Making Unlimited Money With Flip-A-Coin
 

[ QUOTE ]
[ QUOTE ]
Jason1990 --
"But the expected value of the number of flips I must make before winning a million dollars is infinity."

Actually, this expected value would be a finite number. This is easy to see.

[/ QUOTE ]
What you are seeing is an illusion. The expected number of flips it takes to win *one* dollar is infinite.

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ok. I think you are working with a definition of the word "win" which you have in mind but are not stating. What I took "win" to mean was to "be ahead" by that amount. Certainly the expected number of flips you would make before finding yourself ahead by one dollar at that point in time is not infinite. If you mean to say that being ahead by a dollar doesn't mean you've won a dollar because you keep playing forever then ok. In that case you never "win" anything.

PairTheBoard