The Numbers Behind NUMB3RS (27 page)

LORDEN'S STORY

At this point, we can't resist recounting the experience one of us (Lorden) had with the Thorp system.

In the summer of 1963 I was on vacation from graduate school, back home in southern California working for an aerospace company. I was fascinated by Thorp's book, particularly the part where Thorp explained how the “gambler's ruin problem” sheds light on the very practical issues of winning at blackjack. I was familiar with the problem from my mathematics studies as a Caltech undergraduate, but I had not heard of the Kelly gambling system or the other money management rules that Thorp explained.

What these rules reflect is that there is an important but little understood corollary to the well-known principle “You can't beat the odds in the long run.” Many years later, in a public lecture at Caltech, I demonstrated this fact by involving the audience in an elaborate experiment.

I programmed a computer to print out 1,100 individual “gambling histories,” one for each audience member, mathematically simulating the results of making steady bets on a single number on a roulette wheel, five days a week, eight hours a day, for an entire year. In spite of the 5.6 percent casino advantage at roulette, about a hundred members of the audience raised their hands when I asked, “How many of you are ahead after three months?” At the end of the lecture, the woman who got a framed certificate attesting to her performance as “best roulette player” had won a tight competition. There were three others in the audience who, like her, actually made a profit playing full-time roulette for a year! (As a seasoned presenter, before asking the computer to run the simulations and print out the results, I had calculated the probability that no one in the audience would come out a winner, and it was acceptably small.)

If random chance fluctuations can sometimes forestall for such a long time the inevitable losses in playing roulette, then perhaps it is not surprising that the flip side is also true. If I played blackjack with a winning percent advantage using Thorp's system, I still had to face the prospect of losing my meager stake before reaching the promised land of long-term winnings.

Of course, Thorp's book explained all of this and emphasized the usefulness of the Kelly gambling system, a strategy invented by a physicist at Bell Laboratories in the 1950s, which instructs that you should never bet more than a certain percentage of your current capital—typically about the same percentage as your average percentage advantage over the casino. In theory, this strategy would completely eliminate the possibility of “gambler's ruin.” Unfortunately, casino games have minimum bets, so that if your capital ever gets down to, say, five dollars, betting a small percentage of it is not allowed. Playing one last hand at that point will of course give you a good chance of losing your whole stake—a genuine case of gambler's ruin.

TEAMS TAKE ON THE CASINOS

The initial response of the casinos to the success of Thorp's book turned out to be just the first round in an ongoing war between math-types and the casinos. Students of mathematics and its profitable applications quickly realized that multiple-deck blackjack, in spite of obvious disadvantages compared to single decks, has some very attractive and exploitable features. For one, it's easier to disguise card counting with multiple decks, because whenever the composition of the remaining cards becomes favorable for the player, it tends to stay favorable—perhaps for many hands. Fluctuations in the player versus dealer advantage are dampened by the presence of many cards remaining in the deck.

Also, blackjack players started playing in teams, something else that needed the much longer play cycles of multiple decks. One of the pioneers of team play was Ken Uston, who gave up his job as vice president of the Pacific Stock Exchange to devote himself full-time to winning money at blackjack. His book
Ken Uston on Blackjack
popularized methods of team play against the casinos that greatly enhanced the potential of card counters to extract profits.

In its simplest form, team play involves members pooling their money and sharing the net proceeds of their individual wins and losses. Since it can take many hands for a small percentage edge to turn into actual winnings, a team of, say, five players, playing as one, can improve their chances significantly, since they can afford to play five times as many hands than if they played individually.

Moreover, teams can avoid detection much more effectively by adopting the classic economic principle of specialization of labor. What Uston proposed was “big player teams,” an idea credited to his mentor, a professional gambler named Al Francesco. Here is the idea. A casino can detect card counters because they need to change the size of their bets—suddenly changing from making small bets when the odds are on the casino's side and then placing big bets when the remaining cards are in their favor. But by playing as a team, one player can avoid detection by not betting anything unless the deck is sufficiently favorable, and then making only big bets.

The idea is for some team members to act as “spotters.” Their job is to play quietly at several tables, placing small bets, all the time counting cards out of the shoe at their table. When one of them sees a favorable deck start to emerge, they signal the “big bettor” to come over to that table and take advantage of it. The big bettor thus moves around from table to table, making only large bets (and generally raking in large wins), leaving a particular table when the counters signal that the deck has turned unfavorable. The small bets made by the spotters have little effect on the team's overall winnings or losses, which come predominately from the big bettor. The main risk in this strategy is that someone watching the big bettor moving around in this fashion can recognize what's going on, but in the hustle and bustle of a large busy casino with literally dozens of blackjack tables, a skilled and experienced team can often play their well-choreographed game all night long without being detected at all.

The potential for generating steady profits from this sort of team play began to attract considerable interest among mathematics students at many universities. For most of the 1990s, teams from MIT, in particular, became highly effective in raiding gambling casinos in Nevada and other parts of the country. Their winnings were not consistent (chance fluctuations always play an inescapable role), their stealth techniques and disguises were not always effective, and their personal experiences ranged from inspiring to abysmal. But overall they gave the casinos quite a run for their money. Many of these exploits were chronicled in a popular book,
Bringing Down the House
by Ben Mezrich, in magazine and newspaper articles, in a television documentary (in which your other author, Devlin, became the only mathematician to play a James Bond role on the screen), and in the recent movie
21
(which is the alternative name for the game of blackjack).

So what's happening today in blackjack casinos? Almost certainly there are some unchronicled math-wise card counters still playing, but the casinos' countermeasures now include some high-tech machinery: automatic card shufflers. In the early 1990s a truck driver named John Breeding had the idea to replace the shoe with a machine that would not only hold multiple decks but allow played cards to be shuffled back into the deck automatically and frequently. This lead to the development of Shuffle Master machines, now visible in many casinos, which, besides relieving the dealer of the time-wasting burden of shuffling, also relieve card counters of their potential for profits. The latest versions of these machines, called CSMs (for “continuous shuffling machines”), effectively approximate “dealing from an infinite deck,” a feature that makes card counting useless. In the subculture of people who play blackjack professionally, these machines are dubbed “uncomfortable shoes.”

Single-deck games still exist, but in a disturbing recent trend the casinos have been transforming them into “sucker propositions” by changing the 3:2 bonus for a natural to 6:5. This shifts the advantage a whopping 1.4 percent in favor of the casino, turning the game into little more than a salutary (and possibly expensive) lesson for the sort of person who doesn't read the fine print. (And if you don't think a 1.4 percent advantage to the casino is “whopping,” you should stay well away from the gaming tables!)

The “Double Down” episode of
NUMB3RS
hinged on the idea that a rogue genius mathematician was hired as a consultant to the company that manufactures the shuffling machines, and he intentionally used a poorly chosen algorithm to control the random mixing of the cards inside the machine. He then hired mathematics students and armed them with the instructions needed to decode the pattern of the cards dealt by the machine, enabling them to anticipate the sequence of cards as they came out. The writers helped themselves to a little dramatic license there, but the point is a good one. As Charlie observes: “No mathematical algorithm can generate truly random numbers.” Poorly (or maliciously) designed algorithms intended to generate random numbers can indeed be exploited, whether they appear in cell phones, Internet security, or at the tables in a casino.

A FOOTNOTE: MATHEMATICIANS AND THE GAMES THEY CHOOSE TO PLAY

Thorp himself never made a huge amount of money from his casino method—apart from the royalties from his bestselling book. But he did go on to become wealthy from applying his mathematical expertise to a different game. Shortly after his stunning success in transforming blackjack, he turned his attention to the stock market, wrote a book called
Beat the Market
, and started a hedge fund to use his mathematical ideas to generate profits in stock market trading. Over a nineteen-year period, his fund showed what Wall Street calls an “annualized net return” of 15.1 percent. That's slightly better than doubling your capital every five years.

Nowadays, Wall Street and financial firms and institutions are heavily populated with “quants”—people trained as mathematicians, physicists, and the like—who have made the study of the mathematics of finance and investment into a hugely profitable enterprise.

You get the idea.

LORDEN AGAIN: CALTECH STUDENTS TAKE ON THE CASINOS

Some years ago, about a decade after Thorp's book came out, I had an experience that brought home to me just how seriously the casinos took the threat mathematics posed to their business. By then, I was back at Caltech, my alma mater, as a professor, my brief student foray into casino life long behind me. My specialty was (and remains) statistics and probability, and I would occasionally hear stories about friends of friends making killings at the casinos. I knew of the improvements in blackjack card counting that Thorp and others had made, such as the “hi-lo” count, where the player keeps a single running count, adding 1 for every “ten” or ace coming out of the deck and subtracting 1 for every low card (2 through 6). The greater the count in the positive direction, the fewer “tens” or aces remained in the deck, favoring the player who could hit on 17 with a reduced chance of going bust. These new strategies were not only more powerful but also easier to use than Thorp's original tens strategy.

One day a senior came to my office at the beginning of his last spring term to ask me to give him a reading course in probability theory. He wanted to probe more deeply into some topics (specifically, random walks and fluctuation theory, for those who know what these terms mean) that were only touched upon in the standard courses that I taught. I should have guessed what he was up to! After a few once-a-week meetings, at which the student and I went over some fairly advanced techniques for calculating probabilities and simulating certain types of random fluctuations, I began to catch a whiff of a more than purely mathematical purpose: “Do you have any special
practical
interest in these topics?” I asked him.

With that slight prod, the student opened up and told some tales that gave me, I must admit, considerable vicarious pleasure. He and a classmate, both seniors required to take only a very light load of course-work, had been spending most of their days and nights in Las Vegas playing blackjack. They sought out single-deck games, which were still available at high-minimum tables, and they played with stacks of “quarters”—$25 chips. (My student came from a wealthy family.)

As young men playing for very high stakes, they were subject to intense scrutiny and had to go to enormous lengths to avoid being detected and barred from play. They feigned drunkenness, showed extreme interest in the cocktail waitresses (not feigned), and played with seeming lack of interest in the cards while secretly keeping their counts. They planned their assaults on the casinos with considerable care and cunning.

Every week they would pick four casinos to hit and would play blackjack for four days, sleeping on a schedule that made each day twenty hours long instead of twenty-four—a cycle that enabled them to face each eight-hour shift of casino personnel only twice in that week. The next week they would move on to another set of four casinos, taking pains never to return to the same casino until at least a month had gone by.

Being barred from play was not the only risk they faced. As Thorp's book described, some casinos were not above bringing in “cheat dealers,” specialists in techniques such as “dealing seconds”—giving the player a hit with the
second
card in the deck if the top card would give him a good total. (The dealer has to peek at the top card and then execute a difficult maneuver to deal the second card instead.) Playing at a popular and very swank casino in the wee hours one morning, my student and his friend noticed that the appearance of a new dealer at the table occurred sooner than normal—a dangerous sign, according to Thorp. Suitably wary, they decided to play a few more hands and see what would happen.