The Eudaemonic Pie

by Thomas A Bass

  The system did indeed work, with startling accuracy. Chips seemed to get sucked off the layout and deposited in front of Doyne with one winning bet after another. They accumulated into mounds that attracted the usual crowd of pit bosses and players sniffing the scent of Lady Luck. With a look of ferocious, almost idiotic concentration on his face, Doyne ignored everyone around him and concentrated solely on the microswitches in his shoes and the predictions being tattooed onto his stomach.

  Within half an hour the usual problems set in—short circuits caused by wires pulling loose and sweat. “It could have been disastrous,” said Browne, who was watching by then from behind the slot machines. “Doyne was doing fine, betting dime chips, when one of solenoid buzzers jammed and started heating up. He rushed off to the bathroom, fixed the solenoid, and returned. But not much later, it happened again. He had barely sat down to play when he had to jump up and exclaim, ‘Boy, have I got the shits!’

  “This happened several more times, with the solenoid becoming progressively hotter, and Doyne running to the bathroom faster and faster, saying, ‘I’ve got the shits today! Boy, do I have the shits!’ On the last trip to the toilet the pit boss followed him and sat in a neighboring stall. The croupiers must have thought he was crazy, and I guess he was. Doyne was jumping around like a gnat on a skillet. But I’m sure they never dreamt of the poison that brought on craziness like this.”

  Doyne cashed in his chips and quit after four and a half hours in the casino. Betting dime stakes, in several hundred trials, the profit was small. But what mattered was the computer’s advantage over the house, which he and Browne figured conservatively at 25 percent.

  “I was elated,” said Doyne. “We had proved we could go into a casino, set parameters, play on an unknown wheel, and beat the house by at least a twenty-five-percent advantage, which is a very hefty margin. Now all we needed to do was raise the stakes.”

  6

  The Invention of the Wheel

  The main thing is the play itself. I swear that greed for money has nothing to do with it, although heaven knows I am sorely in need of money.

  Feodor Dostoyevsky

  Although the paternity of the computer, the law of probability, and the game of roulette are all ascribed to Pascal, I can report—after extensive research into the matter—that while the first two offspring have an undeniable claim to his name, the third is illegitimate. There are, however, good reasons for implicating Pascal in the act of conception. This is a case of mistaken identity, but one made with good cause.

  After inventing the mechanical adding machine, Pascal secured a monopoly on its exploitation and supervised the manufacture out of wood, ivory, ebony, and copper of more than fifty Pascalines. Among his contemporaries, his fame rested primarily on this invention (even the great Descartes applied for a demonstration). But Pascal increasingly turned his thoughts to more abstract speculation in gaming and theology, and on his way to an irascible sainthood at the Jansenist convent of Port-Royal, he paused briefly to invent the mathematics of probability.

  In 1654 a friend had written him to ask if Pascal could solve the problème des parties, or problem of points. When players end a card game before its natural conclusion, how should they divvy up the pot? Pascal in turn posed the question to Pierre de Fermat, noted mathematician and jurist in Toulouse, and together, in a series of written exchanges, they worked out the mathematical basis for the theory of probability. In solving the problem of points, the stakes should be divided, said Pascal, according to the concept of mathematical expectation, that is, according to each player’s probability of winning the game.

  His contemporaries marveled that chance could have laws. Catching the tone of their stupefaction, Pascal said of his accomplishment: “Thus, bringing together the rigor of scientific demonstration and the uncertainty of chance, and reconciling those things which are in appearance contrary to each other, this art can derive its name from both and justly assume the astounding title of the Mathematics of Chance.”

  Gamblers and popular historians further ascribe to Pascal the invention of roulette. (The word comes from the French for “small wheel.”) In early versions of the game, the Greeks spun shields on sword points, and the Roman emperor Augustus had a rotating chariot wheel installed in the gaming room of his palace. While these devices employed a wheel and a stationary pointer, the modern game of roulette utilizes a more complicated mechanism, with rotor and ball revolving in counterdistinction to each other.

  The story about Pascal’s inventing roulette most likely stems from a toothache that seized him one night at the Port-Royal monastery in the spring of 1657. According to the account of his sister, Pascal leapt out of bed and turned his thoughts to a particular mathematical problem. After several nights of pacing his room, he simultaneously cured his toothache and made his greatest contribution to pure mathematics. Pascal was searching for the formula to a curve known as the cycloid, or “Helen of geometry,” because of the fascination it had exercised over the minds of scientists from Nicolas of Cusa to Galileo and Descartes. Pascal defined this curve as the line described by a nail on the rim of a rolling wheel as it rises from the ground and falls again to meet it. The problem “considers the rolling of wheels,” he said, “and is for this reason called roulette.“

  Whether Pascal actually experimented with rolling wheels remains a matter of conjecture, although he did in fact solve the equations necessary to describe the cycloid. Gottfried Leibnitz, on reading Pascal’s treatise on the subject, Histoire de la roulette, ten years after its author’s death, would generalize Pascal’s equations into the integral calculus, which allows for the study of continuously changing quantities and remains to this day the great tool of modern mathematics. “Nothing astonished me so much,” said Leibnitz about the History of Roulette, “as the fact that Pascal seemed to have his eyes obscured by some evil fate; for I saw at a glance that the theorem was a most general one for any kind of a curve whatever.”

  Roulette—not the curve of the cycloid, but the game of chance we know today—has assumed its own important, if checkered, place in the history of mathematics and physics. Scientists interested in studying its laws of motion have included James and Daniel Bernoulli, Laplace, Poisson, Poincaré, Claude Shannon, and Edward Thorp, many of whom, on lining up at the baize to examine more closely the wheel’s physical and statistical attributes, have discovered that the game in play often has less to do with the stately procession of planets and more with the greed of men.

  Roulette makes its first official appearance in 1765 when Gabriel de Sartine, a police lieutenant who thought he saw in it a gambling device immune to cheating, introduced it into the casinos of Paris. During the Revolution of 1789 royalist émigrés fled with their roulette wheels to Bath and other British resorts. By the early nineteenth century the game had spread to Continental health spas at Wiesbaden, Bad Homburg, Baden-Baden, Saxon-les-Bains, and Spa itself.

  After impoverishing the hapless émigrés and minor nobility of Germany, the casinos were closed by the Prussian government in 1872. One smart operator simply packed up his concession and transported it from Bad Homburg to the principality of Monaco. At the time of Louis Blanc’s arrival, Monte Carlo, the capital of this independent nation, was described in a contemporary account as consisting of “two or three streets upon precipitous rocks; eight hundred wretches dying of hunger; a tumble-down castle, and a battalion of French troops.” That the royal family and citizens of Monaco are considerably better off today is due to the fact that Blanc cut them in for 10 percent of the action, which soon became substantial indeed. With roulette as its main attraction, Monte Carlo ruled supreme as the world center of gambling up until the rise of Las Vegas after World War II.

  Roulette first reached the United States, like many exports from France, via New Orleans. It was played extensively in our original casinos: the paddle-wheel steamers that plied the Mississippi with loads of cotton bales and confidence men. Roulette had come ashore to be indulge
d in privately or illegally in Saratoga, New York City, New Orleans, Chicago, and Denver by the time Las Vegas took the fateful step of legalizing it and other games of chance in 1931. The Fremont Street casinos offered, at best, a brawling sumptuousness, and it took another fifteen years before the entrepreneurial talent of a New York gangster named Benjamin “Bugsy” Siegel would turn Las Vegas into the gambling mecca of the world.

  In 1946, just across the town line in a blighted stretch of greasewood known as the Strip, Bugsy Siegel opened the Flamingo Club, the first of Nevada’s famous pleasure domes. Bugsy was gunned down by the mob within a year, but his idea had already taken hold. Siegel, in a stroke of genius, had seen that a completely encapsulated fortress of delight, designed around a theme of historic excess, such as that of Arabian sheiks, Spanish grandees, or Roman emperors, would become the ideal destination for late-twentieth-century tourism. Packaged into segments of one, three, five, seven, or more days, the nobility of leisure could be made available to everyone. Suitably elevated by the grandeur of Bugsy’s ambiance and décor, the tourist could adopt other attributes of the nobility—their love of play, largeness of gesture, and disdain for monetary loss. What better environment, thought Siegel, for fleecing people at the tables?

  Within a decade a dozen more pleasure domes had sprung up on the Strip. Today Las Vegas offers more than a hundred. France legalized gambling two years after Nevada, and roulette quickly appeared in casinos scattered along the coasts at Deauville, Biarritz, Nice, and Le Touquet. It could also be found at Estoril in Portugal, in Rome, Venice, San Remo, and Salzburg. When Britain legalized gambling in 1960, roulette was reintroduced as a strong component in what quickly became the country’s leading industry. (Britons spend more than $5 billion annually on gambling, while Americans wager in excess of $100 billion a year.) Roulette today can be played throughout the Caribbean and South America, in Marrakech, Macao, Quintandinha, Constantsa, and even Mbabane, Swaziland, where a combination casino—hot springs offers the rare opportunity to see South African racists rubbing shoulders with Swazi tribesmen.

  For all its regularity, the game of roulette in play produces an outcome at once random and unrepeatable, thereby perfectly illustrating the laws of chance. A roulette wheel consists of a precisely machined rotor that is balanced and spun on a steel spindle. Spaced evenly around the rim of the rotor are thirty-eight pockets numbered from 00 to 36. These alternate high-low, odd-even, and red-black, except for two green pockets, numbered 0 and 00, which face each other across the rotor. These green cups—and the fact that the payoff in roulette is less than thirty-eight to one—guarantee the house its advantage in what would otherwise be a fair game.

  The roulette wheel presides at the far end of a long rectangular table covered in green baize and marked with the squares and columns of a betting layout. Arranged sequentially in three columns that increment from left to right and top to bottom are the thirty-six numbers that lie scattered in more random fashion on the wheel. At the head of these columns preside squares for the 0 and 00, while at the foot and along the near side of the layout lie other squares representing bets on red or black, odd or even, the first twelve numbers, entire columns of numbers, and so on. Because bets can be split between numbers, columns, and rows, a roulette layout offers the discriminating investor thirteen different types of wagers.

  In the middle of a game, a roulette layout looks something like a Mondrian painting reworked by Jackson Pollock: dribbles of yellow, red, blue, and gold chips cover the white boxes etched on the baize and crisscross the lines between the boxes. Players stack chips into Pisan towers, change their minds, redeploy them from one box to another, and then, at the last minute, scatter a final inspiration of chips onto the green felt. Their work done, they stand back to watch its public reception.

  The odds, at least in American roulette, heavily favor the house. A winning bet straight up on one of the thirty-eight numbers (including the 0 and 00) is paid off at the rate of thirty-five to one. If roulette were a fair game, with no house advantage, the bet would pay thirty-seven to one, not counting the chip originally wagered. Every bet on the layout is similarly discounted, so that the house clears a cool 5.26 percent profit, except for the five-number bet that straddles the 0, 00, 1, 2, and 3, where the casino pockets a neat 7.89 percent profit. The odds in roulette compare unfavorably to blackjack, craps, and baccarat. In the last of these, for instance, the house advantage is 1.25 percent.

  It matters a great deal to the mathematics of the game that roulette wheels in Europe have thirty-seven pockets, rather than thirty-eight, the house having eliminated the 00. It was originally removed by Louis Blanc when he took over the gambling concession at Bad Homburg in 1840, and his promotional device has since remained a permanent feature on the European wheel. This, and other differences in how the game is played, narrow the house advantage to 1.35 percent, which explains why roulette is still the pre-eminent diversion in European casinos and others outside the United States.

  As John Scarne, magician and gambler-in-residence to the Hiltons, said of roulette, “This is the game which the handsome hero and well-dressed heroine have played for years in countless motion pictures, books and short stories. It has been publicized as the game of millionaire playboys, of kings and princes. It is celebrated in many stories of fortunes won and lost, of mathematical wizards who have spent years developing Roulette systems, and even in a song: ‘The Man Who Broke the Bank at Monte Carlo.’ Roulette is the world’s oldest banking game still in operation, and through the years it has given rise to many true stories, as well as much that is legend and myth.”

  Given the rules of the game, there are three conceivable systems for beating roulette. The first is mathematical: a pattern of numbers, or procedure for placing successive bets, that would give the player an advantage over the house. Another kind of system relies on biased wheels and their tendency to favor one number over another. A third approach, through measurement of its physical forces, tries to predict the actual outcome of the game.

  Roulette lends itself to the scrutiny of systemiers. Huddled at the wheel, scribbling figures into notebooks, these players resemble cabalists mooning over the number six. Gambling shops in Monte Carlo sell lists of roulette numbers recorded the previous day, and subscribers to the Revue Scientifique can receive their numbers monthly. Hundreds of books and articles describe “winning” schemes for mathematical prediction, but it remains indisputably true, after more than two hundred years of continuous play, that no such winning system exists. Edward Thorp is even more categorical in stating that “there is no ‘mathematical’ winning system for roulette and it is impossible ever to discover one.”

  Most of these systems employ a procedure known as “doubling up,” which is based on the idea that a loss on a one-to-one bet can be offset by doubling the wager in each successive round. If you bet one dollar and lose, and then bet two dollars and win, you will have ventured three dollars in bets and “earned” four dollars in return, thereby making a profit of one dollar. As simple as it sounds, this system has two flaws, the first being that it requires an infinite bank. It may be improbable that one would lose nineteen times in a row, but to double up on an original wager of one dollar would require, in this instance, a bet of $524,288 in order to make the expected one-dollar profit.

  The casinos, possessed of large but by no means infinite banks, protect themselves against doubling up with a simple counter-measure. They impose a house limit on bets, usually a thousand dollars, which effectively torpedoes the system at whatever wager would exceed the house limit. “What is perhaps truly amazing,” says Thorp, “is that this is also true for all mathematical systems, no matter how complex, including all those that can ever be discovered,” of which there are an infinite number!

  Systems for doubling up, doubling down, tripling up, and so on, make up a class of strategies known as martingales. The word comes from the French expression porter les chausses à la martingale, which means “to wear one
’s pants like the natives of Martigue,” a village in Provence where trousers are fastened at the rear. The expression implies that this style of dress and method of betting are equally ridiculous.

  Another popular mathematical system is named after Jean le Rond d’Alembert, co-editor with Diderot of the Encyclopédie. The d’Alembert system, also known as the gambler’s fallacy, operates according to the “maturity of chances” or “law of equilibrium.” These “laws” maintain that a long string of numbers in one color increases the likelihood of the other color appearing in order to “average things out.” This idea unfortunately contradicts the theory of probability, which asserts that every chance event is independent of the preceding and following events. Roulette balls have no memory, and their chance of landing on either red or black always remains an invariable fifty-fifty.

  Roulette systems based on the discovery of biased wheels have proved more fruitful. A British engineer named William Jaggers once hired six clerks to record a month’s worth of winning numbers at Monte Carlo. After calculating variances in their frequency, he and his staff cleared a tidy one million five hundred thousand francs by betting on the most favored numbers. They were stymied only when the casino redesigned its wheels with movable rather than stationary partitions between the cups. Switching these in the early hours of the morning, the croupiers were able to redistribute the variables on which Jaggers’s system had relied.

  Albert Hibbs and Roy Walford, friends from Cal Tech and fellow graduate students at the University of Chicago, managed a feat similar to Jaggers’s in 1947. Using a Poisson distribution to distinguish biased from unbiased wheels, they found that more than a quarter of the wheels in Nevada were sufficiently unbalanced to overcome the house advantage. In a well-publicized session that spawned a rash of imitators, Hibbs and Walford cleared seven thousand dollars at the Palace and Harold’s Club in Reno. Another graduate student at UC Berkeley by the name of Allan Wilson must hold the world’s record for gathering statistics on biased wheels. He made not a penny for his efforts, but he and a friend, working in twenty-four-hour shifts manned over a five-week period, recorded no less than eighty thousand continuous plays.