On beating roulette: part 1

Beating roulette seems like a fool’s errand, much like beating the markets. After all, the roulette wheel is a physically random system isn’t it?

Being a great fool myself, I’ve thought about this rather a lot. I thought about it to the point where I thought I had a novel way to beat the roulette wheel, and invested a bit of effort and treasure into seeing if I could make it practical. As it turned out, my idea wasn’t so good. In the interests of inspiring some other fools out there, and because this was a fun project to think about for a while, I’m going to talk about some of the issues involved in designing a way to beat roulette. I have no intentions of following up on this project with another one, though I can think of two offhand which would do very well at beating roulette, if you have the engineering resources to dump into it. I probably won’t tell you about these, but anyone who reads this series ought to be able to figure them out on their own.

The way I see it, there are easier ways to make money: ones which don’t involve any risk of having your teeth kicked in by casino thugs, and which don’t involve substantial R&D costs. My idea was simple enough it had a chance of working in a period of time which would have been worth my efforts, which are otherwise better spent on other fool’s errands, like beating markets and chasing pretty girls.

History time: most say the roulette wheel was invented by Blaise Pascal in 1655 while he was attempting to create a perpetual motion machine. Ironic, as I was sort of looking at it as a perpetual money machine. Really though, Pascal invented the rotor part of the roulette wheel; the part which spins around at a deterministic speed. This isn’t the interesting random part of the device. Randomness is introduced by the scattering bumps and the pockets inside the rotor. Roulette was first played in its present form in the late 1700s in France. There are 36 numbers on the rotor of a roulette wheel, along with one or two zero pockets, which provide the house advantage. For the purposes of this discussion, the only important strategy is betting on a number. The payout on a number bet is 35:1, which makes it an exciting game. The house advantage on a single 0 wheel is 2.7%. For double zero wheels, 5.26%. This is a fairly small advantage to beat, so a forecasting algorithm doesn’t have to do very well to give you a reasonable probability of a profitable game.

There have been quite a few dumb ideas floated for beating roulette. The dumbest are pure bet sizing systems. If you go look at the literature, or just “gambling times,” there are a number of examples of people who thought they had a bet sizing system which would have allowed them to win. The problem, of course, is the house edge. Bet sizing systems are important, but they’re only important when you have an edge. Quite a few smart people thought they could use martingale betting to do this.
Another less dumb idea was looking for a dealers signature: the idea being that the croupier who sets the ball to spinning could bias the outcome. This makes a little more sense, but has an obvious downside: you need a roulette croupier confederate to make any money at it. As it turns out, it is not possible for the dealer to bias the outcome in any case.*

The simplest form of attack on Roulette which works is finding biased wheels. While roulette wheels are not supposed to have any bias, they often do, generally because they’re tilted, and so one of the quadrants is more likely than others. The bias could also be due to defects in the rotor or rotor pockets: modern wheels have very careful engineering to prevent this kind of bias: the pockets, for example, are generally machined from a solid piece of metal, so they can’t be dented or bent: a feature added by Huxley in the early 1980s. Several teams have managed to find biased wheels by keeping careful track of the outcomes; according to Wikipedia, the first was British engineer, Joseph Jagger in 1873. The problem with this sort of approach is, obviously, casinos don’t like it, and it’s something they have control over. An obvious countermeasure is to move the wheels around every few days; something they did to Jagger way back when. Since the wheels are physically indistinguishable, whatever statistics you gather on an individual wheel will be irrelevant in a few days. These days, wheels are actually networked, such that bias can be detected by the casino itself, and the wheel can be serviced when it begins to show the slightest bias. They also use wheels which are constructed in a way (shallow pockets, basically) which makes this sort of bias much less likely, even with strongly tilted wheels.

The first men to beat roulette using an actual forecasting algorithm were Ed Thorp and Claude Shannon in 1960. Ed Thorp was the guy who wrote beat the dealer and invented card counting in Blackjack. Claude Shannon was, well, he was freaking Claude Shannon: shoot yourself now if you don’t know who he was. How did the algorithm work? Key to their attack is the fact that you can place a bet well after the ball is in motion. Thorp and Shannon purchased a cheap wheel and did experiments (I did too; same wheel is still manufactured and available on ebay for a couple of bucks). You can see some of their early experiments on Thorp’s website, and read about it in a series of papers Thorp wrote for the Gambling Times, all linked below. To summarize: they found that one could predict the quadrant the ball would fall into with a large enough probability to make betting profitable by measuring the speed of the rotor (which is effectively constant for a given game: remember, it started out as Pascal’s perpetual motion machine), and the speed and decay rate of the orbiting ball. Essentially, they curve fit a spiral orbit of the ball around the track, and matched it against the orbit of the rotor. There is still plenty of randomness in the roulette wheel: even with a perfect measurement, the ball scatters off of the various randomizing bumpers and bounces around in the rotor pockets, but you could get enough of an edge that, with a careful betting system, you could consistently make money at it.

The velocity measurements were taken using microswitches built into a shoe, and the curve fitting done via analog computing. The engineering was fairly complex, involving various microswitches, analog computers, buzzers (which returned the quadrant) and radio transmitters (the electronics was not small enough to fit into one shoe). It also involved a fair amount of hand eye coordination; if you couldn’t accurately measure the velocity of the ball or the rotor, your forecast would be worthless. The ultimate required accuracy was about 10 milliseconds (or one ball diameter). This is within the bounds of human ability, though it requires concentration and practice, making it a two person project in general: one to take measurements, and one to place the bets. Ultimately, the Thorp-Shannon team (their wives were also involved) found that this wasn’t practical using the electronics they were using, which were unreliable and tended to do unpleasant things like shock the user, and so they moved on to other projects.

References:

Ed Thorp wrote a fascinating series on this subject for the Gambling Times. He’s generously made it available on his website here:
Paper 1
Paper 2
Paper 3
Paper 4

A paper on the analog computer they used to accomplish this feat (scanned jpg files unfortunately):
Page 1
Page 2
Page 3
Page 4
Page 5
Page 6

*Thorp papers on the croupier’s signature:
signature 1
signature 2

Thorp’s papers on the fallacy of roulette betting systems:
betting systems 1
betting systems 2
betting systems 3

Finally, the Shannon-Thorp team’s early experiments are documented and shown below:
the video of his early experiments
The documentation

The Thorp website is an amazing treasure of the mathematics of probabilistic systems, and anyone who cares about this sort of thing should read the whole damn thing. I did. Thorp himself appears to be made out of concentrated awesomeness, and has displaced Feynman as my intellectual hero:
http://www.edwardothorp.com