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Quantifying Probabilities for Gambling System Strategies

Probability and physics are helping make even roulette seem ultimately predictable.


The Perfect Bet: How Science and Math Are Taking the Luck Out of Gambling, is a new book by Adam Kucharski which explains how a mathematician who was studying craps and other dice games eventually led to the development of probability theory, how tiny computers were secretly used in the casino to successfully predict the outcome of the roulette game, and how bots playing poker are making rapid advances. What we thought wasn’t predictable is being turned upside down thanks to big advances in machine learning systems and big data. Math and science is helping us to quantify the probabilities of gambling games for the development of strategies.

As time passes, more parts of our daily life are being affected by algorithms, much of which we still don’t fully understand. In the age of information, trying to keep pace with our own creations are becoming quite a challenge. Kucharski tells us how we got to this point, and where we go from here in the following interview.

In the book you call gamblers the godfathers of probability theory, noting that it’s a newer area of mathematics than we might expect. Can you talk a little bit about how probability theory came out of gambling? 

One thing that I found remarkable about the history of math is that it’s only fairly recently that people looked into quantifying luck, so really for a long period of history, topics like geometry were the main study. There was a lot less interest in random events: it’s actually not until the 16th century that gamblers start to think of how likely things are and how that could be measured.

2220283848There was a gambler called Gerolamo Cardano: although a physician by profession, he had a pretty keen gambling habit. He was one of the first people to outline what’s known as the sample space. This is all of the possible outcomes you could get, say, if you’re rolling two dice together, there’s 36 ways they can land. And then of these 36 ways you can home in on the ones you’re interested in. This provided a framework for measuring these kinds of chance events.

That was of the first foundations of probability theory. From that point over the subsequent years, a number of other gambling strategy researchers built on those ideas, again often using bets and wagers to inspire the way they thought about these problems.

You recount several examples of scientists taking on certain gambling problems. Richard Feynman, for example, tells professional gambler Nick the Greek that it seems impossible for a gambling system to provide any advantage.   

Feynman was obviously famous for his curiosity. On his trips to Vegas, he wasn’t a big gambler, but he was interested in working out the odds. I think he started with craps, figuring that although it was pretty poor odds, he wouldn’t lose that much, and it might be a fun game. On his first roll he lost a load of money, so he decided to give up.

He was talking to one of the showgirls, who mentioned Nick the Greek. He was this famous professional gambler, and Feynman just couldn’t work how you could have the concept of a professional gambler because all the games are stacked against you in Vegas. On talking to him he realized what was actually happening was Nick the Greek wasn’t betting on the tables. He was making side bets with people around the table. He was almost playing off human flaws and human superstitions, because Nick the Greek had a very good understanding of the true odds.

If he made side bets at different odds he could kind of exploit that difference between the true outcome and what people perceive it to be. That’s a theme that continues throughout gambling: If you can get better information about what’s going to happen and you’re competing with people who don’t have much idea about how things are going to land or what the future might be, then that gives you a potentially quite lucrative edge.

There’s a sense that being at the whim of chance is somehow a very human position. Admitting things as totally unpredictable and leaving yourself up to fate is often part of the allure for the nonprofessional gambler. But then there’s a tension, because other people say maybe these are things that we can predict, and maybe this isn’t as much up to chance as we imagine it to be

It’s really been this almost tug of war between believing something is skill and believing something is luck, whether it’s in gambling or just in other industries. I think we have a tendency if we succeed at something to think it’s skill and if we fail at something to almost blame luck. We just say, “Wow, that’s chance, there is nothing I can do about it.”

The work of a lot of people who study these games is trying to think of a framework within which we can measure where we are in terms of skill and in terms of chance. Mathematician Henri Poincaré was one of the early people, in the early 1900s, interested in predictability. He said that when we have uncertain events, essentially it’s a question of ignorance.

He said that there are three levels of ignorance. Depending on how much information you have about the situation and what you could measure, things will appear increasingly random. Not necessarily because it’s truly a lucky event, but really it’s our perception that makes it appear unexpected.

One of the games that I think we expect to be least predictable is roulette. But as you point out, Claude Shannon, considered the father of information science, and Edward Thorp, who would later write one of the most popular books on card counting, made a strong case for being able to predict roulette spins.

For a long period of time, roulette has almost been like a case study for people interested in random events. Early statistics was honed by studying roulette tables because you had this process that was seen as very complicated to actually understand fully, but if you collected enough data then you could analyze it and try and look for patterns and see whether these tables are truly random.

Roulette strategy systemEdward Thorp, when he was a Ph.D. physics student, realized that actually beating a roulette table, especially if it’s perfectly maintained, isn’t really a question of statistics. It’s a physics problem. He compared the ball circulating a roulette table to a planet in orbit. In theory, if you’ve got the equations—which you do because it’s a physics system—then by collecting enough data you should be able to essentially solve those equations of motion and work out where the ball is going to land.

  The first wearable bit of tech was designed to be hidden under clothing so you could go into casinos and predict where the roulette ball will land.

Although in theory that could work, the difficulty is [that] in a casino, you actually need to take those measurements and perform those calculations to solve those equations of motion while you are there. So Thorpe then talks to Shannon, who was one of the pioneers of information theory and had all sorts of interesting contraptions and inventions in his basement. Thorpe and Shannon actually put together the world’s first wearable computer. The first wearable bit of tech was designed to be hidden under clothing so you could go into casinos and predict where the roulette ball will land.

Those early attempts were mainly let down by the technology. They had a method which potentially could be quite successful. But it was implementing it, for them at that time—that was the big challenge.

You mention, though, a much more successful attempt in 2004. 

This is the Ritz casino in London, where you have these newspaper reports of people who were initially said to have used a laser scanner to try and track the motion of the roulette ball. In the end, they walked away with just over a million pounds.

That’s an incredibly lucrative take, even for high-stakes casinos like that. This [attempt] reignited a lot of the interest in these stories because, although Thorp and subsequently some students at the University of California had focused on roulette tables, they’d always left out a bit of their methods. They’d never published all the equations.

  So there’s always this element of mystery and glamour around these processes. Actually, it’s only very recently that researchers in Hong Kong actually tested these methods properly and published a paper, Predicting the Outcome of Roulette. It actually said, that if we have this kind of roulette system it’s plausible that we can take into casinos and win. I think there’s been a number of other stories of gamblers trying out these methods in casinos, but it’s always been very secretive. It’s interesting how long it’s actually taken for some researchers to test this problem.

Toward the end of the book you talk about poker. If probability and physics are helping make even roulette seem ultimately predictable, poker seems a tougher nut to crack.

Exactly. Poker, on the face of it, seems like a perfect game for a mathematician because it’s just the probability that you get this card and someone else gets a card. Of course, anyone who’s played it realizes that it’s much more about reading your opponents and working out what they are going to do and what they think you’re going to do.

Early research into poker actually inspired a lot of the ideas of game theory: so, everything in A Beautiful Mind, about if players get together and try to optimize their strategy, they’ll come up with certain approaches. In more recent years, that link between science and gambling has continued, and actually a lot of the attempts at A.I. are focusing on these kinds of games.

Although historically we’ve seen games like chess beaten—and more recently, Go—these are what’s known as perfect information games. You have everything in front of you while you’re playing. So in theory at least there’s a set of moves that if you follow them exactly then you always get the optimal outcome.

If you’ve ever played tic-tac-toe: most people work it out pretty quickly. There’s just a set of fixed things that you can do, and that will always force a certain outcome. Whereas with poker, in poker you can’t do that because there’s an element of randomness. There’s hidden information in that you don’t know what your opponent’s cards are. You’ve got to adjust your strategy over time, and this is where things like bluffing and manipulating your opponents come into play.

I think for artificial intelligence that’s a really interesting challenge, because you’ve got this hidden information and risk-taking aspects. Arguably, that’s a lot closer to a lot of situations we actually face on a day-to-day basis. Whenever you go into a negotiation or you try to bargain for something, you’ve got information that you know, while they’ve got information that they know, and you have to adjust your strategy to account for the difference.

So much of our strategizing depends on accounting for having incomplete information. It reminds me of Bill Benter’s horse-racing models. He models hundreds of variables to predict how the horses will run, but he also cautions about mistaking correlation for causation, especially when the correlations seem wildly counterintuitive. He’s saying don’t try to explain the models, as long as they work. That’s what often happens with machine learning: we put a machine to work on some problem, feed it massive amounts of data, and it returns with these correlations that we never would have expected. They’re totally counterintuitive, but we kind of just have to take them because they’re revealing something.

I think the approach that these scientific bettors use has been really interesting. One of the questions that I found the most unexpected answer to was when I asked, “What criteria do you used to make your strategy?” And with Bill Benter and these horse-racing syndicates, they’re really just interested in which horses will win. They don’t want to explain why it will win.They just want a model where if you put in enough information you will get a reliable prediction.

I think that goes against a common notion that somebody who’s good at gambling is almost an expert and has a lot of knowledge of the narrative of the sport, whereas actually, a lot of these scientific teams treat it much more like an experiment. As long as it gives a good result, they don’t mind how they get there.

And similarly with these bots, because you’ve got so much complexity in terms of how they learn—they have billions and billions of games against each other. It’s very difficult to understand why bots might choose a certain strategy. This even goes back to the early days of machine learning, Alan Turing’s question of can machines surprise us? Can they come up with something unexpected? Machine learning is increasingly showing that they can, because they can just learn so far beyond what their creators are capable of.

In many cases, these poker bots are turning up with strategies that humans would never have thought to attempt, because they’ve simply been able to crunch through that many games, and they’ve refined their strategies. It’s a really interesting development we’re seeing in terms of what the minimal amount of information or strategy is that you need to be successful. We really try to identify questions that maybe people wouldn’t traditionally ask.

That brings us back to this notion that chance is both something that can’t be explained away any further, and yet there’s something deeply human about the desire to create a story to explain why things happen. Computers are now showing us strategies and explanations we never could have arrived at on our own; as you say, they’re outpacing their creators. What are some of the ramifications of that process?

One of the things that really surprised me in writing the book is how quickly these developments are happening. Even the Go victories this year: I think lots of people didn’t expect it to happen that suddenly. And likewise with poker: last year some researchers found the optimal solution for a two-player limit game. Now you got a lot of bots taking on these no-limits stakes games—where you can go all-in, which you often see in tournaments—and they’re faring incredibly well.

In many cases, these poker bots are turning up with strategies that humans would never have thought to attempt.

The developments are happening a lot faster than we expected and they’re going beyond what their creators are capable of. I think it is a really exciting but also potentially problematic line, because it’s much harder to unpack what’s going on when you’ve got a creation which is thinking much further beyond what you can do.

I think another aspect which is also quite interesting is some of the more simple algorithms that are being developed. Along with the poker bots which spend a huge amount of time learning, you have these very high-speed algorithms in gambling and finance, which are really stripped down to a few lines of code. In that sense, they’re not very intelligent at all. But if you put a lot of these things together at very short time scales—again, that’s something that humans can’t compete with. They’re acting so much faster than we can process information; you’ve got this hidden ecosystem being developed where things are just operating much faster than we can handle.

This goes beyond simply teaching bots to play poker or Watson winning at Jeopardy! There are wider ramifications.

Yes. And I think the increasing availability of data and our ability to process it and create machines that could learn on their own, in many ways, it’s challenging some of those early notions about learning machines. Even some of the criticisms and limitations that Alan Turing put forward when they were first coming up with these ideas, they’re now being potentially surpassed by new approaches to how machines could learn.

You have these poker bots, instead of learning to play repeatedly, they’re developing incredibly human traits. Some of these bots, people just treat them like humans: they refer to them in human terms because they bluff and they deceive and they feign aggression. Historically, we think of these behaviors as innate to our species, but we’re seeing now that potentially these are traits you could have with artificial intelligence. To some extent it’s blurring the boundaries between what we think is human and what’s actually something that can be learned by a machine.