The Quants: How a New Breed of Math Whizzes Conquered Wall Street and Nearly Destroyed It

by Scott Patterson

  Thorp believed it was possible to beat roulette even without help from flaws in the wheel. Indeed, the absence of defects made it easier, since the ball would be traveling along a predictable path, like a planet in orbit. The key: because croupiers take bets after the ball is set in motion, it is theoretically possible to determine the position and velocity of the ball and rotor, and to predict approximately which pocket the ball will fall into.

  The human eye, of course, can’t accomplish such a feat. Thorp dreamed of a wearable computer that could track the motion of ball and wheel and spit out a prediction of where it would land. He believed he could create a machine that would statistically forecast the seemingly random motion of a roulette wheel: an observer would don the computer and feed in information about the speed of the wheel; a bettor, some distance away, would receive information via a radio link.

  Thorp purchased a cheap half-scale wheel and filmed it in action, timing the motion with a stopwatch that measured in splits of hundredths of a second. Thorp soon realized that his cheap wheel was too riddled with flaws to develop a predictive system. Disappointed, he tabled the idea as he worked to finish graduate school. But it gnawed at him, and he continued to fiddle with experiments.

  One evening, his in-laws visited him and his wife, Vivian, for dinner. They were surprised when Thorp didn’t greet them at the door and wondered what he was up to. They found him in the kitchen rolling marbles down a V-shaped trough and marking how far the marbles spun across the kitchen floor before stopping. Thorp explained that he was simulating the path of an orbiting roulette ball. Surprisingly, they didn’t think their daughter had married a lunatic.

  The Thorps made their first visit to Las Vegas in 1958, after Thorp had finished his degree and begun teaching. The frugal professor had heard that the rooms were cheap, and he was still toying with the idea of beating roulette. The smoothness of the wheels in Las Vegas convinced Thorp that he could predict the outcome. Now he just needed a solid, regulation-size wheel and suitable laboratory equipment.

  Thorp had also decided to try out a blackjack strategy he’d recently come across. The strategy was from a ten-page article in the Journal of the American Statistical Association by U.S. Army mathematician Roger Baldwin and three of his colleagues—James McDermott, Herbert Maisel, and Wilbert Cantey—who’d been working at the Aberdeen Proving Ground, a military facility in Maryland. Among blackjack aficionados, Baldwin’s group came to be known as the “Four Horsemen,” although no one in the group actually tested the strategy in Las Vegas. Over the course of eighteen months, the Four Horsemen punched a massive amount of data into desktop calculators, plotting the probabilities involved in thousands of different hands of blackjack.

  Ever the scientist, Thorp decided to give Baldwin’s strategy a whirl in Las Vegas. While the test proved inconclusive (he lost a grand total of $8.50), he remained convinced the strategy could be improved. He contacted Baldwin and requested the data behind the strategy. It arrived in the spring of 1959, just before Thorp moved from UCLA to the Massachusetts Institute of Technology.

  At MIT, Thorp found a hotbed of intellectual creativity that was quietly revolutionizing modern society. The job he stepped into, the coveted position of C. L. E. Moore Instructor, had previously been held by John Nash, the math prodigy who eventually won the Nobel Prize in economics in 1994 for his work on game theory, a mathematical approach to how people compete and cooperate. (Nash later became known as the subject of A Beautiful Mind, the book and movie about the competing forces of his genius and mental illness.)

  That first summer in Cambridge, Thorp crunched the numbers on blackjack, slowly evolving what would become a historic breakthrough in the game. He fed reams of unwieldy data into a computer, seeking hidden patterns that he could exploit for a profit. By the fall, he’d discovered the rudimentary elements of a blackjack system that could beat the dealer.

  Eager to publish his results, he decided on a prestigious industry journal, The Proceedings of the National Academy of Sciences. The trouble: the journal accepted papers only from members of the academy. So he sought out the only mathematics member of the academy at MIT, Dr. Claude Elwood Shannon, one of the most brilliant, and eccentric, minds on the planet.

  On a November afternoon in 1960, Ed Thorp walked briskly across MIT’s leaf-strewn campus. A cold wind whistled off the Charles River. The freshly minted mathematics professor shuddered, and his nerves jangled at the very thought of sitting down face-to-face with Claude Shannon.

  Few figures at MIT were more intimidating. Shannon was the brains behind two of the twentieth century’s greatest intellectual advances. The first was the application of the binary number system to electronic circuits, which laid the groundwork for the birth of the computer. Shannon’s great breakthrough had been to take a two-symbol logic in which problems are resolved by the manipulation of two numbers, 1 and 0, and apply it to a circuit in which a 1 is represented by a switch that is turned on and a 0 by a switch that is turned off. Sequences of on and off switches—essentially strings of 1s and 0s—could represent nearly any kind of information.

  Shannon was also a founding father of information theory: how to encode information and transmit it from point A to point B. Crucially, and controversially, Shannon asserted at the start that while messages “frequently have meaning … [such] semantic aspects of communication are irrelevant to the engineering problem.” In other words, information, as a technical matter, is completely devoid of meaning and context. Instead, it is purely statistical, and therefore encodable.

  This was highly counterintuitive. Most scientists prior to Shannon had assumed that the fundamental element of communication was meaning, and nothing but meaning. Shannon changed all that.

  Thorp didn’t want to talk to Shannon about the binary code or information theory, however. He wanted to talk about blackjack. He was still on edge as he stepped into Shannon’s office. Shannon’s secretary had warned him that the busy professor had only a few minutes to spare.

  Thorp spat out his blackjack results as quickly as he could and showed Shannon his paper. Shannon was impressed and said that Thorp had made a significant theoretical breakthrough. He agreed to submit the paper, which was called “A Winning Strategy for Blackjack.” But he had one suggestion.

  “I think you might want to change the title.”

  “Okay,” Thorp said, confused. “Why?”

  “The Academy can be a bit stodgy. And this title has a bit too much of a whiff of the casino. How about ‘A Favorable Strategy for Twenty-One’? That should be boring enough to pass the smell test.”

  Thorp agreed, and his few minutes were up. As he stood, Shannon asked, “Are you working on anything else in the gambling area?”

  Thorp paused. He’d kept his roulette research largely secret, and he hadn’t worked on it for months. But maybe Shannon would find it interesting.

  “I’ve been conducting some studies of the game of roulette,” he said, “and have had some … interesting results.”

  “Really?” Shannon said, his eyes lighting up. He gestured for Thorp to sit down again. “Continue.”

  Several hours later, Thorp left Shannon’s office into the darkening November night.

  Thorp started paying regular visits to Shannon’s home later that November as the two scientists set to work on the roulette problem. Shannon called his home “Entropy House,” a nod to a core concept in information theory, borrowed from the second law of thermodynamics. The law of entropy essentially means everything in the universe will eventually turn into a homogenous, undifferentiated goop. In information theory, Shannon used entropy as a way to discover order within the apparent chaos of strings of seemingly random numbers.

  Shannon’s three-story wooden house overlooked the Mystic Lakes, several miles northwest of Cambridge. One look indoors told Thorp why Shannon likened it to a theory about the inexorable slide of the universe into utter randomness. It was a disorderly “gadgeteer’s paradise,” as Thorp later describ
ed it, packed with electronic and mechanical contraptions. Shannon was obsessed with automatons, machines that mimic human behavior, and he was especially fond of creating mechanical juggling dolls and coin tossers. He was a notorious unicyclist and impressed visitors by navigating a long tightrope stretched across his yard. One visitor was astounded by Shannon’s daughter, who could ride a unicycle and skip rope at the same time. Shannon for a time was obsessed with trying to calculate how small one could make a unicycle and still ride it.

  Science fiction writer Arthur C. Clarke visited Shannon’s house a number of times. A device Shannon called the “ultimate machine” left him unnerved. “Nothing could be simpler,” Clarke later wrote. “It is merely a small wooden casket, the size and shape of a cigar box, with a single switch on one face. When you throw the switch, there is an angry, purposeful buzzing. The lid slowly rises, and from beneath it emerges a hand. The hand reaches down, turns the switch off and retreats into the box. With the finality of a closing coffin, the lid snaps shut, the buzzing ceases and peace reigns once more. The psychological effect, if you do not know what to expect, is devastating. There is something unspeakably sinister about a machine that does nothing—absolutely nothing—except switch itself off.”

  Thorp and Shannon ordered a regulation roulette wheel from Reno for $1,500 and put it on a dusty slate billiard table. To parse its motion, they clocked it to the hypnotic pulse of a flashing strobe light. To time the ball, they would depress a switch each time it made one revolution around the wheel. The switch also triggered the strobe, marking where the ball stood at the moment the switch was hit. This let Thorp and Shannon gauge how well they were timing the ball, since it showed them how early or late they were in hitting the switch.

  The results were ingenious, and perhaps doomed to fail. After much trial and error, Thorp and Shannon calculated a method to predict, with favorable odds, which octant of the roulette wheel the ball would tumble into. The wheel contained eight octants—six octants with five pockets each and two with four, making up the thirty-eight pockets on the wheel. If they could predict the octant, that tipped the odds sharply in their favor. If they bet on all four or five numbers in the predicted octant and their method proved accurate, winning would be guaranteed. It would be cheating, of course, and if they were caught, there was a predictably high chance that large, thick-necked casino bouncers with hairy knuckles would exact a price. But that was a concern for another day.

  Thorp and Shannon designed a computer the size of a cigarette pack and embedded it in a pair of shoes. It had two switches: one switch turned on the computer, and the other timed the spinning of the rotor (one toe click when the wheel started and another when it made a single revolution). The computer calculated the results and transmitted which octant to bet on in eight tones to another person wearing a primitive sort of headphone in one ear. In all probability it was the world’s first wearable computer.

  However, technical problems doomed the project. The headphone wires often broke. One time Thorp, who generally wore the headphone and placed the bets, noticed a woman staring at him with horror. He promptly headed for the bathroom. In a mirror he saw the speaker jutting from his ear like an alien insect.

  Though Shannon didn’t lead Thorp to riches at the roulette wheel, the professor did make a key contribution to his younger colleague’s blackjack strategy. While Thorp had devised a winning approach to blackjack, a key unanswered question remained: how much should a bettor wager if he doesn’t want to risk financial ruin? Shannon told Thorp that the answer could be found in a 1956 paper by John Kelly Jr., a physics researcher at Bell Laboratories in Murray Hill, New Jersey. The paper described how much a gambler with inside information about the winner of a series of baseball games between two equally matched teams should wager if there is a certain amount of noise (and hence a possibility that the information could be faulty) in the channel communicating that information.

  Thorp realized he could use Kelly’s betting system to optimally regulate how much he wagered on various scenarios in blackjack. In simplest terms, when his odds of winning rose, he tossed more chips on the table. When his odds got worse, he backed off.

  A good way to size up Kelly’s system is by comparison with another gambling strategy: doubling down. Say you bet $10 on a hand of blackjack and you lose. If you bet $20 on the next hand and win, you’re up again. But you could lose that, of course. Bet $40, win, and you’re back ahead. Doubling down, also known as martingale betting, has been a time-honored strategy practiced by gambling legends such as Casanova. But there’s an obvious flaw in the strategy: gambler’s ruin. Eventually the martingale gambler will run out of money. The odds of this happening, if the gambler keeps playing, are 100 percent.

  Kelly, however, limited the amount from a player’s billfold that could be placed on any bet. The only time a player would go all in would be when the odds of winning are 100 percent, a very rare event that almost never happens in a casino—although Thorp would discover such opportunities on Wall Street several years later.

  The mathematics of Kelly told him exactly how much to add or subtract, based on the amount in his billfold, in order to achieve the maximum gains. The formula described, in the words of Kelly, how a gambler could “cause his money to grow exponentially,” while at the same time avoiding the curse of gambler’s ruin.

  In January 1961, Thorp presented his blackjack paper to the American Mathematical Society. Since the AMS wasn’t as conservative as the National Academy, which had already received the paper, Thorp provocatively titled it “Fortune’s Formula: A Winning Strategy for Blackjack.” A reporter for the Associated Press picked up on the paper and wrote a story about a brilliant math professor who’d cracked blackjack. The story appeared in newspapers nationwide. Suddenly Ed Thorp was famous.

  The article also caught the eye of a number of enterprising gamblers always on the make for a new system. Thorp fielded a flood of requests about the nature of his system, as well as offers to back him. One of the most generous came from a New York businessman who promised to pony up $100,000. Thorp was eager to test his theory, but he didn’t think he needed that much cash. He decided to accept $10,000 and promptly headed for Reno.

  The same day Thorp beat the dealer in that ratty Reno casino at five in the morning, he awoke in the afternoon eager to continue his experiment. After a hearty meal, he met with one of his financial backers, known as the mysterious “Mr. X” in the book he would later write detailing his system, Beat the Dealer. Later that day, a “Mr. Y” arrived.

  Mr. X was, in fact, a New York businessman with connections to organized crime. His name was Emmanuel “Manny” Kimmel, a short, white-haired racketeer with his fingers in everything from numbers games in Newark, New Jersey, to East Coast horse tracks. He was also part owner of a company called Kinney Parking, which owned sixty-four parking lots in New York City. A 1965 FBI memo on Kimmel said he was “a lifetime associate of several internationally known hoodlums.” Mr. Y was Eddie Hand, a car-shipping magnate and Kimmel’s regular high-stakes gambling pal.

  After Hand arrived, they went to Harold’s Club, a famous casino located in an enormous building in the center of downtown Reno. It was a significant step up from the second-rate casino Thorp had played in the night before, and it would represent an even more rigorous test of his system.

  They sat down at the $500-maximum tables, the highest amount possible. Within fifteen minutes they’d won $500, playing hands ranging from $25 to $250.

  The dealer hit a concealed button with her foot. Thorp watched as the casino’s owner, Harold Smith, marched toward them across the casino’s floor.

  “Good evening, gentlemen,” Smith said, all smiles and glad-handing. Thorp wasn’t fooled for a second. He’s out to stop me.

  After a few more hands, the deck had about fifteen cards left. Typically, dealers play out a deck until only a few cards are left. One way to trip up card counters is to shuffle the deck early.

  “Shuffle,” Smith s
aid to the dealer. With the newly shuffled deck, Thorp and Kimmel kept winning, since the tens strategy can start paying off after only four cards are dealt, though the odds remain relatively slim, mandating careful bets. As the next deck was about halfway through, Smith nodded at the dealer.

  “Shuffle.”

  Thorp’s system still kept picking up favorable odds after several hands. The dealer started shuffling after dealing only two hands. While the system still worked, the repeated shuffling significantly curbed favorable opportunities. Thorp and Kimmel finally left, but they’d already pocketed several thousand dollars.

  The combination of Thorp’s winning blackjack model and Kelly’s optimal betting system was powerful. Thorp and Kimmel continued to beat the dealer, despite a number of hurdles thrown their way. After several days, they had more than doubled their initial $10,000 stake.

  Soon after Thorp announced his results in Washington, D.C., he was watching a TV program about gambling. A reporter asked a casino owner whether gambling ever paid off.

  “When a lamb goes to the slaughter, the lamb might kill the butcher,” the owner said. “But we always bet on the butcher.”

  Thorp smiled. He knew that he’d beaten the butcher. As he would later write: “The day of the lamb had come.”

  After his first excursion to Vegas, Thorp began work on Beat the Dealer. Published in 1962, the book quickly became a New York Times bestseller—and struck terror into the heart of casino bigwigs everywhere.

  Thorp continued to rack up gains at blackjack tables on several return trips to Las Vegas. Dealers were on the lookout for the gambling professor. He began wearing disguises, well aware of stories about card counters getting hauled into side alleys or casino basements for brutal beatings.

  One day in 1964, when he was playing at a baccarat table in Las Vegas, he was offered a cup of coffee with cream and sugar. He took a few sips, then started feeling odd.