The Eudaemonic Pie

by Thomas A Bass

  Another Quixote in search of a system for beating roulette was the English mathematician Karl Pearson. Pearson invented the field of statistics. To him we owe such ideas as the normal curve, standard deviation, and the correlation coefficient—which, unfortunately, he used to “prove” that Jews are inferior to northern Europeans. Pearson took two weeks of data from the permanences, or daily run of numbers recorded in Le Monaco, and analyzed them for statistical fluctuations.

  Publishing his findings in an article entitled “Science and Monte Carlo,” Pearson wrote: “If Monte Carlo roulette had gone on since the beginning of geological time on this earth, we should not have expected such an occurrence as this fortnight’s play to have occurred once on the supposition that the game is one of chance.… Monte Carlo roulette, if judged by returns which are published apparently with the sanction of the Société, is, if the laws of chance rule, from the standpoint of exact science the most prodigious miracle in the nineteenth century.” Pearson’s data had been faked by journalists preferring to keep their tally in the casino bar. But the scandal gave him the opportunity to call for closing the casinos and using their resources to endow “a laboratory of orthodox probability” designed to further Pearson’s social Darwinism.

  Feodor Dostoyevsky, given to epileptic seizures that left him in a state of near-idiocy for days on end, employed a roulette system that operated more in the realm of psychology than of statistics; it had to do with continence and emotional equanimity. “I do know the secret,” he wrote to his sister-in-law after a gambling session at Wiesbaden, “and it is extremely stupid and simple: it consists in controlling one’s-self the whole time, and never getting excited at any phase of the game. That is all; in that way one can’t possibly lose and must win.”

  The difficulty with his system, as Dostoyevsky realized, is that “I have an evil and exaggeratedly passionate nature. In all things I go to the uttermost extreme; my life long I have never been acquainted with moderation.” Sigmund Freud thought the system had other problems, of a sexual nature. “The ‘vice’ of masturbation is replaced by the addiction to gambling; and the emphasis laid upon the passionate activity of the hands betrays this derivation,” said Freud in an essay titled “Dostoevsky and Parricide.” “The irresistible nature of the temptation, the solemn resolutions, which are nevertheless invariably broken, never to do it again, the stupefying pleasure and the bad conscience which tells the subject that he is ruining himself (committing suicide)—all these elements remain unaltered in the process of substitution.”

  One of the more inventive roulette schemes is described by Alexander Woollcott in his story “Rien ne va Plus.” With the success of Louis Blanc’s operation at Monte Carlo, the casino owners in Nice and elsewhere on the coast tried to put a spoke in his wheel by broadcasting exaggerated reports on the suicide rate at Monaco. It was getting hard to swim there, they said, with so many dead bodies washed up on the beaches. Dining one night with friends on a terrace in Monte Carlo, the narrator of Woollcott’s story is “eating a soufflé and talking about suicide.” Earlier in the day he had watched a well-dressed young man lose all his money in the salles privées of the casino, and now there are reports that the man has been found dead on the beach with a bloodied shirt and a gun in his hand. To avoid publicity, agents from the casino tuck ten thousand francs into the corpse’s dinner jacket pocket, “so that the victim would seem to have ended it all from Weltschmerz.” But as soon as the agents are out of sight, the corpse jumps up. With tomato sauce still smeared on his shirt front, he races to the casino and uses their ten thousand francs to win a hundred thousand more.

  Perhaps the most successful of all systemiers was Marcel Duchamp. Having already launched his artistic career into the outer reaches of Dada and surrealism, Duchamp in 1924 perfected a system for playing roulette in which “one neither wins nor loses.” Forming a company to exploit his scheme at Monte Carlo, he designed an issue of thirty stock certificates for sale at five hundred francs apiece. These bear on their face a Monte Carlo roulette layout and wheel with a superimposed photograph of Duchamp taken by his friend Man Ray. The photo shows a satyrlike Duchamp with horns and a beard made out of shaving cream. Signed by Rrose Sélavy (eros c’est la vie), Chairman of the Board, these pieces of paper are now worth far more than five hundred francs apiece. Although Duchamp managed at the time to sell only two of them, this capital financed a month-long-trip to the roulette tables at Monte Carlo, where, to his great satisfaction, Duchamp broke even.

  Given the impossibility of devising a winning mathematical system, the only feasible approach to beating roulette lies in physical prediction. This requires a device intelligent enough to comprehend its laws of motion and swift enough to calculate its outcome while the game is in play. It requires, in other words, a computer. Analog computers could be built to these specifications in the 1960s. Digital computers followed in the 1970s. Whether the idea preceded the technology, or vice versa, microcomputers and roulette prediction were lovers attracted to each other at first sight.

  It was Edward Thorp who pioneered the use of analog computers to beat roulette. As early as 1962, in the first edition of Beat the Dealer, he declared himself in possession of “a method for beating roulette wheels whether or not they are defective!” In the gung-ho prose of a scientist trying for suspense, Thorp added cryptically: “I played roulette on a regulation roulette wheel in the basement lab of a world-famous scientist. We used the method and steadily averaged 44 per cent profit. In an hour’s run, betting no more than $25 per number, we won a fictional $8,000!”

  As to why he was writing a book about his success, rather than retiring to Cap d’Antibes, Thorp explained that “there are certain electronic problems which have so far kept the method from being used on a large scale in the casinos.”

  Seven years after mentioning it in Beat the Dealer, Thorp was mathematically more explicit, although still vague about the practical details, when he published his article in the journal of the International Statistical Institute. This is where he discusses briefly, in remarks directed toward readers with a background in probability theory, his development of the Newtonian and quantum methods for physical prediction. His collaborator, the “world-famous scientist,” remains unnamed.

  Only recently, in a series of articles written for the magazine Gambling Times, has Thorp been more forthcoming about the nature of his roulette project. The name of his partner also appears in print for the first time—Claude Shannon. While still a graduate student, Shannon had worked out the equations for comprehending switching electrical networks. The general expression of his findings, which became known as information theory, is now applied to switching networks as diverse as telephone exchanges, the computer, and the human brain. Shannon’s “basic idea,” as he put it, “is that information can be treated very much like a physical quantity such as mass or energy.”

  Shannon was in his early forties, a professor at MIT and a recognized luminary in applied mathematics, when Edward Thorp in December 1960 had the temerity to knock on his office door. Thorp had just finished his doctorate in mathematics at UCLA and taken his first job as an instructor at MIT. His dissertation was titled “Compact Linear Operators in Normal Spaces,” but what really interested Thorp was the theory of gambling. Writing programs to simulate play in blackjack, he had a field day on the computers at MIT while devising both the basic and more sophisticated versions of his card-counting strategy.

  About to announce his findings at a meeting of the American Mathematical Society, Thorp thought he should rush his talk into print and save his ideas from being pirated. He targeted the Proceedings of the National Academy of Sciences, a prestigious journal that publishes articles only on the recommendation of its members. The sole mathematician and National Academy member at MIT was Claude Shannon.

  “I was able to arrange a short appointment one chilly December afternoon,” said Thorp. “But the secretary warned me that Shannon was only going to be in for a few minutes
, not to expect more, and that he didn’t spend time on subjects (or people) that didn’t interest him (enlightened self-interest, I thought to myself).

  “Feeling both awed and lucky, I arrived at Shannon’s office for my appointment. He was a thinnish, alert man of middle height and build, somewhat sharp featured. His eyes had a genial crinkle and the brows suggested his puckish incisive humor. I told the blackjack story briefly and showed him my paper.”

  Shannon quizzed Thorp for possible errors in his analysis. Finding none, he told him to condense the paper and change its title from “Fortune’s Formula: The Game of Blackjack” to something more academically neutral. (It was published as “A Favorable Strategy for Twenty-One.”) Shannon then asked if he was working on other problems in gambling.

  “I decided to spill my other big secret,” said Thorp, “and told him about roulette. Several exciting hours later, as the winter sky turned dusky, we finally broke off with plans to meet again on the roulette project.”

  Shannon lived in a large wood-frame house on one of the Mystic Lakes north of Cambridge. Down in the basement he and Thorp went to work in a laboratory that Thorp described as “a gadgeteer’s paradise. It had perhaps a hundred thousand dollars worth of electronic, electrical and mechanical items. There were hundreds of categories, like motors, transistors, switches, pulleys, tools, condensers, transformers, and so on.” The two men ordered a regulation roulette wheel from Reno and set it up on Shannon’s billiard table. Around the wheel they stationed a strobe light, a clock, a movie camera, and the switches needed to coordinate the strobe and clock while filming the ball in motion. Their research—which agreed with that later conducted by Eudaemonic Enterprises—concluded that roulette is a highly predictable game.

  Thorp and Shannon then set to work to build a computer. They came up with a transistorized analog device the size of a cigarette pack. It received data via four push buttons, which were compressed on successive revolutions of the rotor and ball in front of a fixed point. Because theirs was an analog computer, representing variables by means of voltages, Thorp and Shannon were limited in the complexity of their program, and they either ignored or were ignorant of many factors needed for physical prediction on any but tilted roulette wheels.

  Claude and Betty Shannon, Edward and Vivian Thorp, and their computer checked into the Riviera Hotel on the Las Vegas Strip in 1962. Thorp was already well known in Nevada, having made a big splash the previous year publicizing his card-counting strategy. Two professional gamblers had staked him $10,000 to prove his system, and he had parlayed that sum, during his spring vacation from teaching at MIT, into $21,000—a tidy profit of more than 100 percent. Having thus launched a million card counters as a plague on their tables, Thorp was not a welcome sight to the casino owners of Nevada. Later he was barred from play and took to wearing disguises; he grew a beard, donned wraparound sunglasses, and traveled always in the company of friends. But at this stage of his notoriety he still had access to most of the clubs, and none of them suspected that he was out to beat them at a game other than blackjack.

  The Thorps and Shannons spent a week at the Riviera. They took in some shows, lounged around the pool, played diversionary blackjack, and did their best to beat roulette. For security reasons they had developed a two-person system, with a radio transmitter built into their “cigarette pack” computer. The radio informed the bettor of the winning octant by means of a do-re-mi scale whose tempo was calibrated to the ball-wheel configuration it was meant to be mimicking. These radio signals were picked up by a hearing aid and “a little bitty loudspeaker with flesh-colored wire attached to it, which we shoved into our ear canal. The trouble,” said Thorp, “is that the wires kept breaking. So we shopped around and got steel wires about the size of a hair, but even these were fairly fragile.

  “Sometimes I was the bettor, and sometimes I was the data taker. We traded off. But it would take us a while to get wired up in our hotel room and get in there. It was a real hassle putting it all together. It’s a long, tedious project, even though it’s conceptually very simple.”

  “Difficulty with read-out devices” is Thorp’s modest confession in print as to why he and Shannon gave up on their roulette computer after three or four sessions. During a recent conversation in his office at the University of California at Irvine, Thorp was more graphic in describing a general snafu of broken wires, gibbering beeps, shocks, and other electronic failures. He and Shannon tried sporadically over a number of years to debug their system, until they finally abandoned it as a bright idea whose implementation had eluded them.

  Their computer ended up in Shannon’s basement, “where it’s gathering dust,” said Thorp. “If I were doing the whole thing over today, I’d use digital technology, a microprocessor. It’s really the way to go. You don’t have to use linear approximations or the other kinds of approximations that analog computers like. You could solve the equations and put in the right curves.” He stopped and sat blinking behind his desk, as if frightened by a moment of misplaced enthusiasm. “But I’d never go back to it,” he said quickly. “It would require a huge amount of work.”

  It was the latest advance in computer technology—from analog to digital microcircuitry—that provided the breakthrough chance for roulette prediction. Understanding their different modes of operation explains why, in this instance, digital computers are superior to analog. Analog computers, so named because they work through electrical analogs, represent variables by means of voltages. Numbers are correlated, like those on a speedometer, to continuous gradations in physical quantity.

  “To make an analog roulette computer,” as Norman explained it, “you build a circuit which mimics electronically what’s happening physically to the ball and rotor. You model these forces by means of a voltage that decreases with time, but you make the model operate ten times faster than the game in play.

  “It’s simple enough to program an analog roulette computer. But if you want to change your algorithm—the equation you’re using to predict what’s going to happen—you have to rewire the computer, because the program in an analog device is the circuit itself.”

  Rather than operating by means of electrical analogs, digital computers, as their name implies, function in the realm of digits or numbers. This allows them nearly infinite storage capacity, great logical facility, and easy programmability. Instead of having to solder in new transistors and rewire circuits, the modification of a program in a digital computer requires nothing more than the addition of a number.

  “We opted for digital over analog because it was more flexible,” Norman said. “It allowed us a wider range of predictive models without having to mess around with changing the circuitry for every model. It’s also more general purpose. Not only can a digital computer make the predictions, it can also send out signals to transmitters, communicate with toe clickers, and generate solenoid buzzes.”

  Although technologically superior, there was one difficulty entailed in opting for a digital over an analog computer. It meant that the equations of motion involved in predicting roulette, rather than being approximated by continuous gradations of electrical analogs, had to be solved. Digital computers tolerate no approximations. They operate solely in the black and white world of numbers ordered into equations with solutions. Where Thorp had experimented with an empirical system based on what he thought were standard values taken from a normal roulette wheel, Eudaemonic Enterprises was striving after a system at once more universal and specific. They hoped to solve the equations governing every kind of behavior in the roulette cosmos. And at the same time they wanted algorithms flexible enough to account for the minute differences found with each specific wheel. Neither they nor their microprocessor were prepared to tolerate any behavior left unexplained.

  7

  Strange Attractors

  The squirming facts exceed the squamous mind, if one may say so. And yet relation appears…

  Wallace Stevens

  “Connoisseur of Chaos�
�

  In the spring of 1978 Eudaemonic Enterprises, nearing its second anniversary as a company, announced that the Pie was about to be served. There would be more than enough to go around, but anyone hoping to be first in line for a taste of roulette richness was advised to show up in Santa Cruz immediately. From up and down the coast Eudaemons arrived and threw themselves into round-the-clock sessions building computer hardware, practicing on the eye-toe coordination machine, designing costumes, discussing gambling theory, and attending classes, complete with a twenty-five-page manual and homework, on how to beat roulette by computer. This frenzy of soldering, sewing, clicking, and betting was directed toward an imminent junket to South Lake Tahoe and Reno. Doyne’s winning session that winter at the Golden Gate had already proved the efficacy of the system. Designed to be free of broken wires and other snafus, the computer was now ready, they thought, for its first major raid on the casinos.

  With people camped in the house and out in the yard in sleeping bags, 707 Riverside looked like an ashram devoted to studying the tao of physics. Friends provided background music on the piano, while Ralph Abraham dropped by for an occasional chat on gambling theory and casino disguise.

  “Ralph and I spent a lot of time,” said Doyne, “talking about what it takes to convince the casinos you’re real.” They also discussed problems raised by Richard Epstein in his book The Theory of Gambling and Statistical Logic, which Ralph was teaching that spring in a course on the mathematics of gambling. Given a bank of x number of dollars, what percentage of it do you want to wager at each play of the game? Is it advantageous to bet on one or more than one number at a time? For a more detailed examination, these questions were referred to Alan Lewis, a specialist in statistical mechanics.