Why the Martingale doesn't work

[Bjorn]

This is a random post, inspired by random annoyance with random people on the Internet, which is random.  Feel free to ignore it, criticize it, or use it to prove to people why they should never go to Vegas.

The Martingale system is a scheme which purports to tell you how to always win while gambling.  The premise is the following: you find a game which in which there is a bet which pays off at 1 to 1 (in other words, if you win, you get back your stake and profit equal to your stake, doubling your money).  Just to fill in details, we'll assume that you're going to play American roulette (which has 38 numbers, 18 red, 18 black, as well as 0 and 00), and bet on red, which pays off at 1-1.  Now, bet $1 on red.  If you win, this is a dollar profit for you.  If you lose, you now bet $2 on red.  If you win this second bet, then you've made a dollar profit again -- you win $2 on top of your current stake, but you lost $1 on your first bet.  If you lose, you double your bet again, and repeat.  The idea is that you will win eventually, and by doubling your bet each time, you will be able to pay off all the money you lost getting up to the point where you win.  

This sounds good in theory, but it hinges very critically on the assumption that you can always double your bet.  This isn't the case; you only walk in with so much money in your wallet, and the house will generally have maximum and minimum bets, which changes the game critically.

Let's pretend you enter the casino with $300 to bet.  Now you must win within eight bets.  On the eight bet, you will be putting up $128, and will have staked a total of $255; if you lose at this point, you would now be required to bet $256, for a total of $511.  Now, how much can you expect to win on average?  The chance of losing on any given bet is 18/38.  So the worst case scenario is when you lose eight times in a row ((18/38)^8), and when that happens, you'll lose $255.  On the other hand, the rest of the time, you'll win $1.  So we get:

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$1*(1-(18/38)^8) - $255*(18/38)^8 = -$0.51

In other words, on average, you're going to lose half a dollar every time you play.

Okay, so obviously, we started off with a bankroll that was too small.  The question that naturally follows is: how big a bankroll do we need to guarantee that, on average, we'll win?  So let's generalize the equation above.   Let's let p be the chance that we lose on any single bet, and n be the maximum number of bets we can make.  Then we can write down our expected return per betting cycle, and try to make it positive:

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$1*(1-p^n) - $(2^n-1)*p^n > 0
1 - p^n - (2p)^n + p^n > 0
1 > (2p)^n

Since n is greater than 1, this only happens if 2p < 1, or in other words, p < 0.5.  Or, to put it another way, you have to be more likely to win than to lose.  Otherwise, it just doesn't matter how much money you started off with, you'll end up losing money.

Intuitively, the idea is that the casino is cheating.  The odds on red winning are 1-1, which implies that it will come up red 50% of the time.  But because of 0 (and in American roulette, 00), red comes up less than half the time.  This mismatch of posted odds and true probabilities means that you don't win enough money when you do win to balance out the amount of money it costs when you actually do lose.  In the example above, to break even you'd need to win 255 cycles for every time you lost eight bets in a row; but the odds are such that you'll actually only win  509 times for every two times you lose eight bets in a row.  So over the long haul, you end up losing money.

This is just a basic reality of gambling: to win in the long haul, you need the effective odds (the probablity of winning conditioned by the pay-off for winning) to be in your favour.  And if this is the case, then the house is losing money.  So: if you gamble, you lose.

[Ranma_007]

I'd have to disagree with the last sentence. A skilled gambler can make money in Vegas.

They just have to play the right game. It's been shown that people who count cards in blackjack and adjust basic strategy for that can boost their win rate over 50%. Sure, it might be 50.6% or something like that, but over the long run, they will eventually come out ahead.

And a skilled poker player *coughmyselfcough* can use their skills to beat the bad players and win money, even if the house is taking a 5% rake every hand. I've seen it and experienced it.

It all depends on the type of game. :)

[Anastasia]

This always felt like common sense to me. Why would a casino run a game against the house where the player could expect a net profit in the long run? This would be financial suicide unless it took a forbidding amount of skill to attempt.

What's your opinion of games that don't play directly against the house, like poker? (I.e., using rake as is normal in Texas High/low.)

[Anastasia]

I'd have to disagree with the last sentence. A skilled gambler can make money in Vegas.

They just have to play the right game. It's been shown that people who count cards in blackjack and adjust basic strategy for that can boost their win rate over 50%. Sure, it might be 50.6% or something like that, but over the long run, they will eventually come out ahead.

I'm rusty on Vegas lore, but can't you get tossed for counting cards?

[Bjorn]

I'm rusty on Vegas lore, but can't you get tossed for counting cards?

Generally, you can.  Also, casinos can and do take various actions to minimize the potential gains of card counting (replacing the deck as frequently as possible, for example).

Strictly speaking, my statement applies to pure games of chance.  Games like poker, which have a skill element, are a different story, and it is in fact possible to win consistently with them.  This is exactly why you don't play against the house; they make their money off table fees or percentages and the like.

It's also worth noting that this statment only really applies in fixed-odds betting.  In parimutuel betting, for example, a skilled bettor can consistently win, in theory.  Of course, these winnings come at the expense of the unskilled bettors, so in theory, if every bettor is skilled, then you actually lose money winning.  It's a bit odd.

[Slacker]

Only if they catch you.

Honestly, there's nothing they can do to keep you from counting in your skull.

Oh, sure, if you start to win a bit TOO much, they can ask you to move on.  Or, you know, go to town on your fragile, fragile bones with a hammer.

Old school is the best school, after all.

[Jon]

I'm rusty on Vegas lore, but can't you get tossed for counting cards?

There's nothing illegal about doing a bit of arithmetic in your head. But if they suspect you, they'll take measures. Like sending someone to talk to you to break your concentration, or shuffling more often. And in Nevada, they can ban you from the casino at any time, for any reason. And casinos share databases of suspected counters with other casinos.

Of course, if you're trying to use a device to assist you with your counting, then they can also get you arrested.

[Jason_Miao]

Only if they catch you.
Honestly, there's nothing they can do to keep you from counting in your skull.

Excuse me, sir.  Would you mind coming with us?

We noticed that you keep varying your bet size.  Now, we can't actually prove that you count cards.  But that's okay.  This is a private business, not a public forum.  So we're in our rights to ban you whenever we feel like it.  Such as now.

Not only that, I'd advise against going into other casinos to gamble.  You see, we're going to send your picture everywhere else.  So why waste their and your time?

Please enjoy the gift shop.

[Dracos]

This is a violation of my arthematic rights!

Dracos