The Perfect Bet

by Adam Kucharski

  After Thorp put together his winning blackjack system, he turned his attention to the problem of such bankroll management. Given a particular edge over the casino, what was the optimal amount to bet? He found the answer in a formula known as the Kelly criterion. The formula is named after John Kelly, a gunslinging Texan physicist who worked with Claude Shannon in the 1950s. Kelly argued that, in the long run, you should wager a percentage of your bankroll equal to your expected profit divided by the amount you’ll receive if you win.

  For the coin toss above, the Kelly criterion would be the expected payoff ($0.50) divided by the potential winnings ($2.00). This works out to 0.25, which means you should bet a quarter of your available bankroll. In theory, wagering this amount will ensure good profits while limiting the risk of chipping away at your funds. A similar calculation can be performed for horse racing. Betting teams know the probability that a horse will win according to their model. Thanks to the tote board, they can also see what chance the public thinks it has. If the public thinks a victory is less likely than the model suggests, there might be money to be made.

  Despite its success in blackjack, there are some drawbacks to the Kelly criterion, especially at the racetrack. First, the calculation assumes you know the true probability of an event. Although you know the chance of a coin coming up heads, things are less clear in horse racing: a model just gives an estimate of the chance a horse will win. If a team overestimates a horse’s chances, following the Kelly criterion will cause them to bet too much, increasing their risk of going bust. Consistently overestimate by twofold—for instance, by thinking a horse has a 50 percent chance of victory when in reality it has a 25 percent chance—and it will eventually lead to certain bankruptcy. For this reason, syndicates generally bet less than the Kelly criterion would encourage, often wagering only a half or third of the suggested amount. This reduces the risk of having a “rough ride” and losing a large chunk—or worse, all—of their wealth.

  Wagering a smaller amount can also help teams overcome one of the quirks of the betting market in Hong Kong. If you think betting on a certain horse will have a big expected payoff, the Kelly criterion will tell you to put a lot of money on it. In the extreme case, when you are certain of a result, you should stake everything you have. Yet in pari-mutuel betting, this is not necessarily a good idea. A horse’s odds depend on the amount wagered, so the more people bet on it, the less money you’ll make if it wins.

  Even one large bet can shift the whole market. For instance, you might compare your model’s predictions and the current odds and notice that you can expect a 20 percent return if you bet on a certain horse. Put down $1.00 and it won’t change the overall odds much, so you’ll still expect to bag $0.20 if the horse wins. If you have deep pockets, you might choose to bet more than $1.00. The Kelly criterion will certainly be telling you to do so. But if you make a $100.00 bet, it might lower the odds a little. So, your actual profit will be only 19 percent. Still, you’ve made $19.00.

  You might decide to go bigger and bet $1,000. This could shift the odds quite a bit. If a few thousand dollars have already been staked on that horse, it might knock your expected profit down to 10 percent, which means a payoff of $100. Eventually, there comes a point at which putting more money on a horse actually reduces your profits. If the expected return for a $2,000 bet is only 4 percent, you’d be better off wagering a lower amount.

  The potential for bets to move the market isn’t the only problem you’d have to deal with. All the calculations above assume that you are the last person to bet, and so know the public odds. In reality, devising an optimal strategy isn’t that straightforward. At the track, there is a lag on the tote board, sometimes of up to 30 seconds, which means more bets might come in after you’ve picked your horse.

  The total betting pool at Happy Valley might be $300,000 when a team place their bets, but it will probably grow by at least another $100,000 by the time the race starts. Syndicates need to adjust for this influx of cash when deciding how to bet; otherwise, a strategy that initially looks like it will generate a big return could end up producing a mediocre profit. They can’t assume that the extra money will be placed on random horses either. In the past decade or so, scientific betting has become more popular, and there are now several syndicates operating in Hong Kong that use models to predict races. These teams are likely to be the ones behind any last-minute betting. “The late money tends to be smart money,” Bill Benter said. Teams therefore have to assume the worst: others will also bet on the favorable horse, so any potential profits will have to be divided among more people.

  UNTIL SYNDICATES STARTED USING scientific approaches in Hong Kong, successful racetrack betting strategies were few and far between. The techniques are now so effective—and the wins so consistent—that teams like Benter’s don’t celebrate when their predictions come good. Much of the reason for Benter’s early success was the unique setup available to gamblers in Hong Kong. At Happy Valley, gamblers don’t have to bet in person at the track. They can call in their selections by phone. This was one of the main reasons Benter and Woods chose Hong Kong. It removed an additional complication and meant they could concentrate on updating their computer predictions rather than worrying about how to place bets in time. Combined with the good availability of data and active betting market, it made Hong Kong the ideal place to implement their strategy.

  Gradually, others noticed the appeal of Hong Kong, too. As a result, it is now extremely difficult for betting teams to make money at the city’s racetracks. With competition increasing in Hong Kong, the ideas first introduced by Bolton and Chapman are spreading to other regions, including the United States. Over the past decade, scientific betting has become a major part of US horse racing. It’s been estimated that teams using computer predictions bet around $2 billion a year at American racetracks, almost 20 percent of the total amount wagered. This sum is all the more impressive when you consider that computer teams cannot bet at several of the large racetracks.

  Betting teams are also targeting events in other countries. Like Swedish harness racing, in which horses pull drivers around the track on two-wheeled carts. Imagine a modern version of a Roman chariot race, without the swords and capes. The techniques are growing in popularity at racetracks in Australia and South Africa, too. An idea that began as a piece of academic research has turned into a truly global industry.

  It is worth mentioning that it’s not cheap to set up a scientific betting syndicate. To gather the necessary technology and expertise—not to mention hone the prediction method and place the bets—costs most teams at least $1 million. Because betting strategies are expensive to run, teams in the United States often seek out racetracks that offer favorable gambling conditions. Several tracks have noticed the bump in profits that comes with the syndicates’ huge bets and now encourage computer-based approaches. They even strike deals with betting teams, handing out rebates if the syndicates place large volumes of bets.

  These difficulties mean that, although Bolton and Chapman enjoyed the problem-solving aspect of racetrack predictions, they have never really been that interested in gambling careers. Aware of the cost and logistics involved in implementing their strategy, they were happy to remain in academia. “We would joke that we could do it,” Bolton said. “Every so often we’d hear how much money was being made and how large these operations had got, but it wasn’t for us.”

  The success of scientific betting in horse racing is all the more remarkable because historically there has been a limit to how much gamblers can predict outcomes. The problem is not limited to horse racing. Whether betting on sports or politics, it has often been difficult to get ahold of the necessary information and to create reliable models. Even if gamblers did manage to come up with a decent prediction, the strategies could be tricky to implement. But at the start of the twenty-first century, that all changed.

  4

  PUNDITS WITH PHDS

  WHEN A NEW BLACKJACK SYSTEM HIT BRITAIN IN
2006, WORD of its success traveled quietly but quickly. No disguises were required, or card counting, or even visits to casinos. Admittedly, the profit margin was on the sort of scale that would buy pints rather than penthouses, but the system worked. All it required was a computer, a good chunk of spare time, and willingness to do something dull in return for beer money. Students loved it.

  The strategy emerged as a result of the new Gambling Act, passed by the government a few months earlier. It meant that UK-based companies could now provide online casino games as well as traditional sports betting. In the rush for new customers, firms started offering signup bonuses. Bet £100 and get an extra £50 free—that sort of thing. At first glance, such a bonus doesn’t seem to help much with blackjack. In an online game, it’s much easier for casinos to ensure that cards are dealt randomly, making card counting impossible. If you use the Four Horsemen’s optimal blackjack strategy instead, taking the dealer’s card into account when making your decision, you can expect to lose money over time. But the signup bonuses tipped things back in the players’ favor. People realized that the bonuses would in effect subsidize any losses. Playing the ideal strategy, players would probably lose some of the £100—but not much—and once they’d bet the required total, they would get the bonus. They would usually have to bet this, too, before it could be withdrawn; fortunately, they could simply repeat the previous approach to limit their losses.

  During 2006, gamblers hopped from website to website, sitting through hundreds of blackjack hands to build a collection of bonus money. It didn’t take long for betting companies to clamp down on what they called “bonus abuse” and exclude games like blackjack from their signup offers. Although there is nothing illegal about setting up a single account to obtain a bonus—indeed, that’s the point of a signup bonus—some gamblers pushed the advantage too far. The first conviction for bonus abuse came in the spring of 2012, when Londoner Andrei Osipau was jailed for three years for using fraudulent passports and identity cards to open multiple betting accounts. For those who operated within the law in 2006, profits were far more modest than the £80,000 Osipau was reported to have made. Still, the fact that these bonuses could be exploited illustrates three crucial advantages that gamblers have gained in recent years.

  First, the explosion of online betting has meant a far wider range of games and gambling options. In real-life casinos, new games are generally good news for gamblers. According to professional gambler Richard Munchkin, casinos rarely understand how much of an advantage they are offering when they introduce new games. The blackjack loophole that appeared in 2006 showed that the same is true in online gambling. And when the Internet is involved, news of a successful strategy travels much, much faster. The second advantage is the ease with which gamblers can implement a potentially profitable system. Rather than having to dodge casino security or visit bookmakers, they can simply place bets online. Whether through websites or instant messaging, access is quicker and easier than ever before. Finally, the Internet has made it much easier to get ahold of the vital ingredient for many successful betting recipes. From roulette to horse racing, the limited availability of data has dictated where and how people gamble. But these limitations are fading away. As a result, people are targeting a whole host of new games.

  EVERY AUTUMN, RECRUITMENT TEAMS descend on the world’s best mathematics departments. Most are from the usual crowd: oil firms wanting fluid-dynamics researchers or banks trying to find specialists in probability theory. But in recent years, another type of firm has started to appear at the career events hosted by British universities. Instead of discussing business or finance, they focus on sports such as soccer. Their career presentations are rather like watching a very technical prematch analysis. Formulae and data tables—which most companies hide from prospective applicants—fill the talks. The events have more in common with a lecture than a job pitch.

  Many of the approaches are familiar to mathematicians. But although researchers might use the techniques to study ice sheets or epidemics, these firms have found a very different application for the methods. They are using scientific methods to take on the bookmakers. And they are winning.

  Modern soccer predictions began with what would otherwise have been a throwaway exam question. During the 1990s, Stuart Coles was a lecturer at Lancaster University, a few miles from the sweeping fells of England’s Lake District. Coles specialized in extreme value theory, which deals with the sort of severe, rare events that are like nothing ever seen before. Pioneered by Ronald Fisher in the 1930s, extreme value theory is used to predict the worst-of-the-worst-case scenarios, from floods and earthquakes to wildfires and insurance losses. In short, it is the science of the very unlikely.

  Coles’s research spanned everything from storm surges to severe pollution. At the prompting of Mark Dixon, another researcher in the department, Coles also started to think about soccer. Dixon had become interested in the topic after looking at a statistics exam given to final-year students at Lancaster. One of the questions involved predicting the results of a hypothetical soccer match, but Dixon spotted a flaw: the method was too simple to be useful in real life. It was an interesting problem, though, and if the ideas were extended—and applied to actual soccer leagues—it might lead to an effective betting strategy.

  It took a couple of years for Dixon and Coles to develop the new method and get it ready for publication. The work eventually appeared in the Journal of Applied Statistics in 1997. With the research finished, Coles went back to his other projects. Little did he realize how important the soccer paper would turn out to be. “It was one of those things that at the time seemed inconsequential,” he said, “but looking back it had a massive impact on my life.”

  TO PREDICT HORSE RACES in Hong Kong, scientific betting teams assess the quality of each horse and then compare these different quality measurements to work out the probable result. It’s tricky to do the same in soccer. Although it might be possible to weigh up each team’s qualities and calculate which team is likely to be successful over an entire season, it is much harder to work out who is likely to win in a given match. A team that plays well against one set of opposition can look sluggish against another. Or one shot might go in while another bounces off the woodwork. Then, you have the players. Sometimes a talismanic performance will lift a whole team; sometimes a team will carry along weak players. This tangle of on-pitch activity means that things are much messier from a statistical point of view. During the 1970s, a few researchers had even come to the conclusion that a single soccer match was so dominated by chance that prediction was hopeless.

  By choosing to study soccer matches, Dixon and Coles were clearly walking into difficult territory. There was one thing on their side, however. In the United Kingdom, betting odds were generally fixed several days ahead of the match. Unlike the hectic last-minute betting at Hong Kong’s racetracks, anyone analyzing soccer matches would have plenty of time to come up with a prediction and compare it to the bookmakers’ odds. Even better, there were plenty of potential wagers available. Thanks to a well-established soccer betting market in the United Kingdom, there are all sorts of things to bet on, from half-time score to the number of corner kicks.

  Dixon and Coles chose to start with the big question: Which team was going to win? Rather than trying to predict the final result directly, they decided to estimate the number of goals that would be scored before the final whistle. To keep things simple, the pair assumed that each team would score goals at some fixed rate over the course of a game and that the probability of scoring at each point in time was independent of what had already happened in the match.

  Events that obey such rules are said to follow a “Poisson process.” Named after physicist Siméon Poisson, the process crops up in many walks of life. Researchers have used the Poisson process to model telephone calls to a switchboard, radioactive decay, and even neuron activity. If you assume something follows a Poisson process, you are assuming that events occur at a fixed rate. The
world has no memory; each time period is independent of the others. If a match is goalless at half time, it won’t make a second-half goal more likely.

  Having chosen to model a soccer game as a Poisson process—and hence assuming that goals are scored at a consistent rate over the course of the match—Dixon and Coles still needed to know what the scoring rate should be. The number of goals in a match would probably vary depending on who is playing. How many goals should they expect each team to score?

  Early in their 1997 paper, Dixon and Coles set out the things you need to do if you want to build a model of a soccer league. First, you need to somehow measure each team’s ability. One option is to use some sort of ranking system. Perhaps you could hand teams a certain number of points after each match and then add up the total points earned over a given time period. Most soccer leagues hand out three points for a win, one for a draw, and nothing for a loss, for example. Representing each team’s ability with a single number might show which team is doing well, but it’s not always possible to convert rankings into good predictions. A 2009 study by Christoph Leitner and colleagues at Vienna University of Economics and Business provided a good illustration of the problem; they came up with forecasts for the Euro 2008 soccer tournament using rankings published by the sport’s governing body Fédération Internationale de Football Association (FIFA) and found that the bookmakers’ predictions turned out to be far more accurate. To make money betting on soccer, it seems that you need more than one measurement for each team.

  Dixon and Coles suggested splitting ability into two factors: attack and defense. Attacking ability reflected a team’s aptitude at scoring goals; defensive weakness indicated how poor they were at stopping them. Given a home team with a certain attacking ability and an away team with a certain defensive weakness, Dixon and Coles assumed that the expected number of goals scored by the home team was the product of three factors: