The surge in activity caught the attention of lottery officials, who tried to stop the syndicate by shutting off the terminals they were using. As a result, the group members were able to buy up only 80 percent of the possible number combinations. It wasn’t enough to guarantee a win, but it was enough to put luck on their side; when the draw was announced, the syndicate had the winning numbers in their collection. Unfortunately, there were also two other winners, so the group had to share the jackpot. They still ended up with a profit of £310,000, however.
Simple brute force approaches like these do not require many calculations to pull off. The only real obstacle is buying enough tickets. It’s more a question of manpower than mathematics, and this reduces the exclusivity of the methods. Whereas roulette players have only to outwit the casino, lottery syndicates often have to compete with other teams attempting to win the same jackpot.
Despite the ongoing competition, some betting syndicates have managed to repeatedly—and legally—turn a profit. Their stories illustrate another difference from roulette betting. Rather than acting alone or in small teams below the official radar, many lottery syndicates have formed companies. They have investors, and they file tax returns. The contrast reflects a wider shift in the world of scientific gambling. What were once individual efforts have grown into an entire industry.
3
FROM LOS ALAMOS TO MONTE CARLO
BILL BENTER IS ONE OF THE WORLD’S MOST SUCCESSFUL GAMBLERS. Based in Hong Kong, his betting syndicate has bet—and won—millions of dollars on horse races over the years. But Benter’s gambling career didn’t begin with racing. It didn’t even begin with sports.
When he was a student, Benter came across a sign in an Atlantic City casino. “Professional card counters are prohibited from playing at our tables.” It wasn’t a particularly effective deterrent. After reading the sign, only one thought came to mind: card counting works. It was the late 1970s, and casinos had spent the previous decade or so clamping down on a tactic they saw as cheating. Much of the blame—or perhaps credit—for the casinos’ losses goes to Edward Thorp. In 1962, Thorp published Beat the Dealer, which described a winning strategy for blackjack.
Although Thorp has been called the father of card counting, the idea for a perfect blackjack strategy was actually born in a military barracks. Ten years before Thorp released his book, Private Roger Baldwin had been playing cards with fellow soldiers at the Aberdeen Proving Ground in Maryland. When one of the men suggested a game of blackjack, conversation turned to the rules of the game. They agreed on the basic format. Each player receives two cards, with the dealer getting one faceup and one facedown. Players then choose whether to hit, taking another card in the hope of getting a total bigger than the dealer’s, or stand, sticking with their current number. If taking a card sends a player’s total over twenty-one, they go “bust” and lose their stake.
Once all players have made their choice, it’s the dealer’s turn. One of the soldiers pointed out that in Las Vegas, the dealer must stand with a total of seventeen or higher. Baldwin was amazed. The dealer had to follow fixed rules? Whenever he’d played in private games, the dealer was free to do whatever he wanted. Baldwin, who had a master’s degree in mathematics, realized this could help him in a casino. If the dealer was subject to strict constraints, it should be possible to find a strategy that would maximize his chances of success.
Like all casino games, blackjack is designed to give the house an edge. Although the dealer and player both appear to have the same aim—drawing cards to get a total near to twenty-one—the dealer has the advantage because the player always goes first. If the player asks for one card too many, and overshoots the target, the dealer wins without doing anything.
Looking at some example blackjack hands, Baldwin noticed his odds improved if he took the value of the dealer’s faceup card into account when making decisions. If the dealer had a low card, there was a good chance the dealer would have to draw several cards, increasing the risk of the total going over twenty-one. With a six, for example, the dealer had a 40 percent chance of going bust. With a ten, that probability was halved. Baldwin could therefore get away with standing on a lower total if the dealer had a six, because it was likely that the rules would force the dealer to draw too many cards.
In theory, it would be simple for Baldwin to build these ideas into a perfect strategy. In practice, however, the vast number of potential blackjack hands made the task near impossible to do with pen and paper. To make things worse, a player’s choices in a casino weren’t just limited to hitting or standing. Players also had the option of doubling their stake, on condition that they would receive one more card to go with the two they already had. Or, if they had received a pair of cards showing the same number, they could “split” these into two separate hands.
Baldwin wouldn’t be able to do all the work by hand, so he asked Wilbert Cantey, a sergeant and fellow math graduate, if he could use the base’s calculator. Intrigued by Baldwin’s idea, Cantey agreed to help, as did James McDermott and Herbert Maisel, two other soldiers who worked in the analytics division.
While Thorp was working on his roulette predictions in Los Angeles, the four men spent their evenings working out the best way to beat the dealer. After several months of calculations, they arrived at what they thought was the optimal strategy. But their perfect system didn’t turn out to be, well, perfect. “In statistical terms, we still had a negative expectation,” Maisel later said. “Unless you got lucky, you’d still lose in the long run.” Even so, by the group’s calculations they had managed to reduce the casino’s edge to a mere 0.6 percent. In contrast, a player who simply copied the dealer’s rules—always standing on seventeen or higher—could expect to lose 6 percent of the time. The four men published their findings in 1956, in a paper titled ‘The Optimum Strategy in Blackjack.”
It happened that Thorp had already booked a trip to Las Vegas when the paper came out. It was meant to be a relaxing holiday with his wife: a few days of dinner tables rather than blackjack tables. But just before they left, a UCLA professor told Thorp about the soldiers’ research. Ever curious, Thorp wrote down the strategy and took it along on his trip.
When Thorp tried the strategy in a casino one evening, slowly reading from a crib sheet while he sat at the table, his fellow gamblers thought he was crazy. Thorp was drawing cards when he should stick and turning down cards when he should take them. He doubled his bet after receiving weak cards. He even chose to split his paltry pair of eights when the dealer had a much stronger hand. What on earth was he thinking?
Despite Thorp’s apparently reckless strategy, he didn’t run out of chips. One by one, the other players left the table with empty pockets, but Thorp remained. Eventually, having lost eight of his ten dollars, Thorp called it a night. But the little excursion had convinced him that the soldiers’ strategy worked better than any other. It also made him wonder how it could be improved.
To simplify calculations, Baldwin had assumed that cards were dealt randomly, with each of the fifty-two cards in the deck having an equal chance of appearing. But blackjack isn’t really so random. Unlike roulette, in which each spin is—or at least should be—independent of the last, blackjack has a form of memory: over time, the dealer gradually works through the deck.
Thorp was convinced that if he could record which cards had previously been dealt, it would help him anticipate what might come up next. And because he already had a strategy that in theory pretty much broke even, having information about whether the next card would be high or low was enough to tip the game in his favor. He soon found that even a tactic as simple as keeping track of the number of tens in the deck could turn a profit. By counting cards, Thorp gradually turned the research of the four Aberdeen soldiers—later dubbed the “Four Horsemen of Aberdeen”—into a winning strategy.
Although Thorp made money from blackjack, it wasn’t the main reason he made all those trips to Vegas. He saw it more as an academic obligat
ion. When he’d first mentioned the existence of a winning strategy, the response wasn’t exactly positive. People ridiculed the idea, just as gamblers had done during his first attempt. After all, Thorp’s research challenged the widely held assumption that blackjack couldn’t be defeated. Beat the Dealer was Thorp’s way of proving that his theory was right.
THAT SIGN IN ATLANTIC CITY always stuck with Bill Benter, so when he heard about Thorp’s book during a year studying abroad at Bristol University, he headed to the local library to get a copy. He had never seen anything so remarkable. “It showed that nothing was invulnerable,” he said. “Old maxims about the house always having the edge were no longer true.” Upon his return to the United States, Benter decided to take some time away from study. Switching his university campus in Cleveland, Ohio, for the casinos of Las Vegas, he set to work putting Thorp’s system into action. The decision was to prove extremely lucrative: in his early twenties, Benter was making about $80,000 a year from blackjack.
During this time, Benter met an Australian who was also making a tidy sum from card counting. Whereas Benter had gone straight from lecture theaters to casino floors, Alan Woods had started training as an actuary after leaving college. In 1973, his firm was commissioned by the Australian government to calculate the house edge on games in the country’s first legal casino. The project introduced Woods to the idea of profitable blackjack systems, and over the next few years he spent his weekends beating casinos around the globe. By the time he met Benter, Woods was a full-time blackjack player. But things were becoming harder for successful gamblers like them.
In the years since Thorp had published his strategy, casinos had become better at spotting card counters. One of the biggest problems with counting—aside from the mental focus required—is that you have to see a lot of cards before you have enough information to make predictions about the rest of the deck. During this time, you have little choice but to use Baldwin’s optimal system and bet small amounts to limit your losses. And when you eventually decide that the upcoming cards are likely to be favorable, you need to dramatically increase your stakes to make the most of your advantage. This gives a clear signal to any casino staff on the hunt for card counters. “It’s easy to learn how to count cards,” as one blackjack professional put it. “It’s hard to learn how to get away with it.”
Keeping a mental note of card values isn’t illegal in Nevada (or anywhere else, for that matter), but that didn’t mean Thorp and his strategy were welcome in Las Vegas. Because casinos are private property, they can ban anyone they please. To evade security, Thorp started to wear disguises on his visits. With casinos on the lookout for big changes in betting patterns, gamblers started to search for a better way to play blackjack. Rather than count cards until things looked promising, what if it were possible to predict the order of the entire deck?
MOST MATHEMATICIANS IN THE early twentieth century had read Poincaré’s work on probability, but it seemed that hardly anyone truly understood it. One of a few who did was Émile Borel, another mathematician based at the University of Paris. Borel was particularly interested in an analogy Poincaré had used to describe how random interactions—like paint in water—eventually settled down to equilibrium.
Poincaré had compared the situation to the process of card shuffling. If you know the initial order of a deck of cards, randomly swapping a few cards around won’t completely mess up the order. Your knowledge about the original deck will therefore still be useful. As more and more shuffles are made, however, this knowledge becomes less and less relevant. Much like paint and water mixing over time, the cards gradually become uniformly distributed, with each card being equally likely to appear at any point in the deck.
Inspired by Poincaré’s work, Borel found a way to calculate how quickly the cards would converge to this uniform distribution. His research is still used today when calculating the “mixing time” of a random process, whether card shuffles or chemical interactions. The work also helped blackjack players tackle a growing problem.
To make life difficult for card counters, casinos had started using multiple decks—sometimes combining as many as six—and shuffling the cards before all of them had been dealt. Because this made it harder to keep count, casinos hoped it would help stamp out any player advantage. They didn’t realize that the changes also made it harder to shuffle the cards effectively.
During the 1970s, casinos often used a “dovetail shuffle” to mix up the cards. To perform the shuffle, the pack is split in two, and then the two the halves are riffled together. If the cards are perfectly riffled, with cards from each half alternating as they fall, no information is lost: the original order can be recovered by simply looking at every other card. Even if cards fall randomly from each half, however, some information remains.
FIGURE 3.1. A dovetail shuffle. (Credit: Todd Klassy)
Suppose you have a deck of thirteen cards. If you perform a dovetail shuffle, the cards might end up swapping around as follows:
The shuffled deck is far from random. Instead, there are two clear sequences of rising numbers (shown in boldface and plain text above). Actually, several card tricks rely on this fact: if a card is placed into an ordered deck and the deck is shuffled once or twice, the extra card will usually stand out because it won’t fit into a rising sequence.
For a fifty-two-card deck, mathematicians have shown that a dealer should shuffle the cards at least half a dozen times so as not to leave any detectable patterns. However, Benter found that casinos would rarely bother to be so diligent. Some dealers would shuffle the deck two or three times; others seemed to think that just once was enough.
In the early 1980s, players began to use hidden computers to keep track of the deck. They would enter information by pressing a switch, and the computer would vibrate when a favorable situation cropped up. Keeping track of the shuffles meant it didn’t matter whether casinos used several decks. It also helped players avoid giving a clear signal to casino security. If the computer indicated that good cards were likely to come up on the next hand, players didn’t have to increase their bets substantially to profit. Unfortunately for gamblers, the advantage no longer exists: computer-aided betting has been illegal in American casinos since 1986.
Even without the clampdown on technology, there was another problem for players like Woods and Benter. Much like Thorp, they had gradually found themselves banned from most casinos around the globe. “Once you become well known,” Benter said, “it’s a very small world.” With casinos refusing to let them play, the pair eventually decided to abandon blackjack. Rather than leave the industry, however, they instead planned to take on a much grander game.
WEDNESDAY NIGHTS AT THE Happy Valley racecourse are busy. Seriously busy. Tucked behind the skyscrapers of Hong Kong Island on a patch of what used to be swampland, over thirty thousand spectators crowd into its stands. Cheers rise above the sound of engines and car horns from the nearby Wan Chai district. The jostling and the noise are signs of how much money is at stake. Gambling is a big part of life at Happy Valley: an average of $145 million was bet during each race day in 2012. To put that in perspective, in that same year the Kentucky Derby set a new American record for betting on a horse race. The total was $133 million.
Happy Valley is managed by the Hong Kong Jockey Club, which also runs the Saturday races at Sha Tin racecourse across the bay in Kowloon. The Jockey Club is a nonprofit organization and has a reputation for running a good operation: gamblers are confident that the races are honest.
Betting in Hong Kong operates on a so-called pari-mutuel system. Rather than gamblers placing a bet with a bookmaker at fixed odds, gamblers’ money goes into a pool, with the odds depending on how much has already been wagered on each horse. As an example, suppose there are two horses racing. A total of $200 has been bet on the first, and $300 on the second. Adding these together gives the total betting pool. The race organizers begin by subtracting a fee: in Hong Kong it’s 19 percent, which, if the total was
$500, would leave $405 in the pot. Then they calculate each horse’s odds—the amount you’d get back if you wagered $1 on it—by taking the total available winnings ($405) and dividing it by the amount bet on that horse, as shown in Table 3.1.
TABLE 3.1. An example tote board.
Invented by Parisian businessman Joseph Oller, who also founded the Moulin Rouge cabaret club, pari-mutuel betting requires constant calculations and recalculations to produce correct odds. Since 1913, these calculations have been easier thanks to the invention of the “automatic totalizator,” commonly known as the “tote board.” Its Australian inventor, George Julius, had originally wanted to build a vote-counting machine, but his government had no interest in his design. Undeterred, Julius tweaked the mechanism to calculate betting odds instead and sold it to a racetrack in New Zealand.
In a pari-mutuel system, spectators are effectively betting against each other. The race organizers take the same cut regardless of which horse wins. The odds therefore depend only on which horse the bettors think will do well. Of course, people have all sorts of different methods for picking a successful horse. They might go for one that’s been putting in some impressive performances. Perhaps it’s won a few races recently or looked confident in practice. It might run well in certain weather. Or have a respected jockey. Maybe it’s currently a good weight or a good age.
If enough people bet, we might expect the pari-mutuel odds to settle down to a “fair” value, which reflects the horse’s true chances of winning. In other words, the betting market is efficient, bringing together all the scattered bits of information about each horse until there’s nothing left to give anyone an advantage. We might expect that to happen, but it doesn’t.
The Perfect Bet Page 5