I was recently reminded by the introduction of Bitcoin gambling service SatoshiDice that the rest of the world doesn’t live side-by-side with casino-style gaming like I and the other 1.38 million residents of Las Vegas (and to a lesser extent the 2.7 million folks that live in Nevada) have grown so accustomed to. I’ve grown so accustomed to the presence of slot and video poker games in every 7-11, restaurant and bar that I honestly don’t know how non-Nevadans break a $100 bill any more. I’ve grown so accustomed to gaming that I almost forgot how easy it is for visitors to think they’ve found / created a winning system, a way to beat the casinos at their own game. Unfortunately the casinos’ “own game” is called “basic statistical math” and it’s not so easy to beat.
In the “Everybody’s Got a System” series I’ll explore some of the more common methodologies clever and not-so-clever folks have come up with for beating the system, explore whether or not they work and why. This week’s installment: The “Martingale” or “Double-Up” betting system.
The Martingale betting system is very simple: Pick a game with roughly a 50/50 win/loss ratio, place a bet. If you win, collect your winnings and place the same bet again. If you lose, double your previous bet each round until you win, then return to your initial bet. It seems foolproof because eventually you have to win and each time you win you increase your pot by the amount of your initial bet. It seems entirely too simple to fail, but there is always a flaw.
In this case, the flaw in the math is that your money is not unlimited. The closer to unlimited your funds are, the longer you can hold out, but eventually you will hit a losing streak that bankrupts you.
Instead of talking about “hands” of something like blackjack, let’s talk about what we’re really interested in: rounds of Martingale betting. We’ll define one round as however many hands it takes for you to eventually win and earn your initial investment. Let’s say you start with a $1 bet and have $1 Million in chips. We can do the math easy enough and find that on round 20 our total betting of $1,048,575 will exhaust our entire bank. This means that we can only endure 19 losses in a row in any given round and on our 20th loss we lose everything. How often will you encounter a round with 20 losses in a row? Well let’s look at an actual game. In blackjack, the house has, on average, about a 0.5% advantage. This means that 50.5% of the time you will lose, so you’ll encounter 20 straight losses on average once every 1/(0.505^20)=859,354.662… rounds. Even if we round up to a nice even 860,000 rounds that means that you’ll only earn $860,000 for every $1,048,575 you spend – about an 18% loss. This means that while you think every round is worth $1 it is in fact worth -($0.18) and the house, in the long run, still wins.
Now one thing that fascinates me about gambling and its associated psychology is how it tends to crop up in unexpected places. A lot of folks will tell you that there’s no such thing as a safe investment, and they’re probably right, so how do all of these big banks and brokerages always seem to turn a profit? Realistically what they’re doing isn’t much different than the Martingale system, making risky bet after risky bet until one of them pays off big enough to cover all their previous losses. But they’ve got to lose eventually, right? Absolutely. Unfortunately, when they do finally lose the now-infamous phrase “too big to fail” comes into play. Imagine in the previous example that when you finally hit round 20 you can count on some third party, like the government, to pay half of the cost of your final failure – $524,287.50 – so now you’ve earned $860,000 and failed to the tune of $524,287.50 for a net profit of $335,712.50 or about $0.39 per round. This is why those of us who get math are upset about bailouts: It’s not about whether or not the banks are actually ”too big to fail,” it’s about taxpayer dollars subsidizing a business model that is built to fail; if they’re really “too big to fail” then maybe they need to start actually trying not to fail.
But I digress – as excellently as many of these systems apply to economics of various scales, these articles are about the systems themselves and as we’ve already shown, Martingale just doesn’t work. For that reason, most casinos don’t really seem to mind Martingale betters – they’re in it for the long run anyway and most of them don’t have our theoretical million dollar bankroll. Walk up to a table game with $1,000 and you’ll be bankrupt after 10 losses, or roughly 927 rounds. A few folks will beat the odds and win big, go on to tell others about their unbeatable system, but that’s what the casinos want anyway – they’re intentionally designed to occasionally produce big winners to be a vocal minority, drawing others in to lose even more. Casinos are playing the long game, backed by scale, time and solid statistical math. They always win because the math always works the same way.
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