Beating the system: Academia goes to the casino
August 29, 2008
Up until August of 1957, I do not remember ever having played a card game for money. This had nothing to do with morals. I had been an undergraduate mathematics major at Harvard, going as far as getting a Master’s degree before switching to theoretical physics. During this whole time I had been doing mathematics day and night, so playing a card game like bridge or even poker, where some mathematical skills were involved, was the last thing I wanted to do. Better to go to a movie.
In the spring of 1957, when I was completing a two-year post-doctoral program at Harvard, I was approached by a recruiter from Los Alamos to go there as a summer intern. Many members of the Harvard physics department had been to Los Alamos during the war, and one of them must have recommended me.
In light of the then-raging Cold War, the weapons laboratories such as Los Alamos and Livermore were expanding and actively looking for qualified people. I had no particular interest in working on nuclear weapons, but I did have a great curiosity about Los Alamos. So I accepted the Los Alamos offer, which was conditional on my being able to get a so-called “Q-clearance” ” a rigorous security clearance needed by anyone working in the technical divisions of the laboratory.
Soon after I got to the New Mexico laboratory it became clear to me that I was going to have no assignment and would have nothing whatever to do with the weapons that were being designed there. It was also made clear that, while I had a Q-clearance, information was only shared on a “need to know” basis. Since I did not need to know anything, I was not told anything.
I had a colleague, also a post-doctoral from Harvard, in the same situation. We shared an office, and I proposed that we work on a problem in elementary particle physics that I had thought of but lacked the mathematical skills to do myself. During most of that summer we happily worked on our problem while, no doubt, bombs were being designed all around us. We wrote up our paper, and the head of the Theoretical Division, Carson Mark, encouraged us to publish it and identify the work as having been done at Los Alamos. He wanted the image of the laboratory to be something other than a bomb factory.
During the summer I made friends with a more senior physicist named Francis Low. He had just been made a professor at MIT and was spending the summer in Los Alamos with his family. Francis also was not working on weapons, so I was surprised when, in the middle of August, he announced that he was going to Mercury, Nev., to see some tests.
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My first thought was, “I want to go, too.” Francis suggested that I ask Carson, which I did. He was agreeable if I paid my way, which I was more than willing to do.
Mercury, where the tests took place, is about 65 miles northwest of Las Vegas. As far as I could later determine, everyone connected with these tests, from the scientists to the soldiers on maneuvers, went to Las Vegas to play blackjack. And blackjack is perhaps the most purely mathematical of card games.
As I learned, in 1956, four men ” Roger Baldwin, Wilbert Cantey, Herbert Maisel, and James McDermott ” had published a paper in the Journal of the American Statistical Association, called “The Optimum Strategy in Blackjack,” which was followed in the fall of 1957 by their 92-page monograph, “Playing Blackjack to Win: A New Strategy for the Game of 21.”
Until 1956, there were probably as many strategies ” essentially all losing ” as there were players. But Baldwin et al. had found the Rosetta stone. They had done their original calculations on primitive electro-mechanical hand calculators. But Los Alamos had what was probably at that time the most powerful electronic computer in the world ” the so-called MANIAC ” used primarily to design weapons. It was elementary to program this computer to run blackjack hands by the tens of thousands to verify that the scheme actually worked. I have read some accounts of blackjack history with nary a mention of Los Alamos. I suspect that this activity at the laboratory was not widely known.
The net result of this was that a little pocket folder was produced from which one could readily read off the basic strategy. I use the term “basic strategy” deliberately. It soon became the basis of more advanced strategies.
My friend Francis had taken the trouble to read the original article and noticed that if you followed the basic strategy you broke about even with the casino. One may recall A. J. Liebling’s gambler’s prayer: “Dear Lord, help me to break even, because I sure need the money.”
In fact, the original authors said that the casino would come out marginally ahead, while later calculations using computers showed that the player would come out slightly ahead.
On the other hand, if you were prepared to do more work and follow a more advanced strategy you could beat the casino by a couple of percent. Francis made an estimate that ran something like this: Suppose I can play a new game every three minutes ” not an unreasonable assumption for casino blackjack ” and suppose I play for an hour. That is 20 games. Suppose that on the average I bet $10 a game. (In those days a $10 maximum was substantial.) Then I stand to make about $2 an hour if my advantage is one percent. This is a lot less than the dealer was being paid, but in principle one had the satisfaction of beating the house.
That satisfying, but not really profitable, basic strategy had taken several years to develop.
In 1953, the above-mentioned Baldwin, who had a Master’s degree in mathematics from Columbia, was a private in the army. He had been drafted and because of his technical background was assigned to the Aberdeen Proving Ground, not far from Baltimore, which did research on ballistics and the like.
He liked playing cards, and one of the games he played the day of the revelation was “dealer’s choice.” The dealer chose blackjack, and Baldwin quickly learned the rules. One thing he learned was that in casino blackjack the dealer is an automaton. That is, the dealer makes no choices; he simply follows the house rules. To understand this let me explain a bit of how blackjack is played in a typical American casino.
The players, from one to seven, sit around a table with the dealer at the head of the table. I will consider the case where a single 52-card deck is used. In a modern casino several decks are used, which increases the advantage to the house. The dealer shuffles the deck, and one of the players cuts the cards. A single card is “burned,” placed face up at the bottom of the deck. In the basic strategy, where you do not count the cards that have been played, it does not matter whether this card is made visible or not. In the more advanced strategies it does matter slightly, and one wants to know what this card is. In any event, the burned card is not played, so effectively the deck has 51 cards. One card matters, but not much. Better not to make a fuss and get unwanted attention.
The dealer then proceeds clockwise around the table, distributing two cards face down to each player. The dealer also receives two cards, one of them face up. Now the play begins.
Each player plays in turn before the dealer plays his hand. This is what gives the casino its advantage. The object of each player is to have cards that add up to a higher number than the dealer’s, provided that the total is 21 or less.
On the deal, with the ace counting by choice as 1 or 11, and all face cards counting as 10, no one can have a higher number than 21. Being dealt a 21 ” “blackjack” ” wins unless the dealer also has blackjack, which is a “push” with no money exchanged. (Incidentally the name “blackjack” comes from a defunct 19th-century practice in American casinos of paying a high premium if the player was dealt the ace of spades and a jack of spades or clubs.) A blackjack pays off 1.5 times your original bet.
If you don’t have a blackjack, when it comes your turn, you can ask for an additional card ” a “hit” ” and, at your discretion, additional hits after that. But if you go over 21, you go “bust” and lose without the dealer having to beat you. This is where the asymmetry between the players and the dealer resides.
Once everyone else has played their hand the dealer turns his second card face up and plays his hand. It is here where the dealer becomes an automaton. If his total is 16 or less the dealer must take hits until the total is 17 or more, at which point the dealer “stands.” There are some nuances involving aces, but this is the general idea. If the dealer goes bust then all the surviving players win.
This was explained to Baldwin, who immediately realized that there must be an optimal strategy. He must also have realized that mimicking the dealer-as-automaton will not work. It turns out that if you do this, the order in which the hands are played, with dealer playing last, gives the casino an advantage of something more than 5 percent.
Baldwin realized that working out all the permutations and combinations of the various hands was going to be a serious matter. So he enlisted his fellow soldiers: Sergeant Wilbert Cantey; Herbert Maisel, who also had a Master’s degree in mathematics and later taught at Georgetown University; and James McDermott who, like the rest, had a Master’s degree in mathematics. McDermott and Maisel were also draftees. Cantey had been in the reserve, which is why he was a sergeant.
The group got permission to use the base’s calculators after hours. When I recently talked to Maisel he told me that Baldwin was definitely their leader. The “four horsemen,” as they are known to blackjack aficionados, worked in their spare time for the next year and a half perfecting their strategy, which resulted in the 1956 article mentioned before.
Maisel told me that they were never aware of anything that went on with the MANIAC computer at Los Alamos, and he was surprised when I told him about it. Maisel never used the scheme in a casino, but Baldwin did. He told me that he spent a week or so in the summer of 1954 in Las Vegas seeing what the actual casino rules were. He visited all 16 that then existed.
The scene now shifts to California. Robert Sorgenfrey, who was a mathematics professor at UCLA and an avid bridge player, had come across the Baldwin et al. article. He knew a young mathematics instructor named Edward Thorp, a former physics major who had switched to mathematics and taken his Ph.D. at UCLA. Thorp had decided to take a Christmas vacation with his wife in Las Vegas. The Thorps were not gamblers but had been to Las Vegas before, since the prices of hotels and meals were reasonable.
Sorgenfrey called Thorp’s attention to the new blackjack strategy, and Thorp decided to try it out. He purchased 10 silver dollars and after some considerable efforts at the table, lost most of it and quit. When he got back to UCLA, he decided that the strategy needed improvement. During the summer he moved to MIT, where he became an instructor, and it was there that he devised the improved strategy.
Thorp’s advance changed the way casino blackjack is played. He realized that if you altered your bet as a function of what cards had been played in previous hands, you could alter the odds in your favor. This, of course, required card counting.
First I will need to explain how this differs from the basic strategy of Baldwin et al. To explain their strategy, I will again consider a situation of one player and a dealer using one deck. I should note that the last time I played blackjack was in an Indian casino along the superhighway that now leads from Santa Fe to Los Alamos. It was fairly early in the morning, and I was the only player. The difference was that the dealer used several decks that are dealt out of what is known as a “shoe.” I did not last long.
Ignoring the matter of the burned card, when the deal is done you have two cards and the dealer has two, with one of the dealer’s cards face up. The knowledge of these three cards is all the information you have.
You must now make one of four choices. Two of the choices I have already mentioned: staying put or taking a hit. But there are two others: “splitting” a pair and “doubling down.” Suffice it to say that these extra options make the process even more complicated. But, in the end, the basic strategy tells you exactly what to do in every situation, as a function of what you are holding and the dealer is showing.
For example, if the dealer is showing a 7 and you have a hard 17 or more, you stand, and so on. If you do all this, and live by the system, you can beat the house by a fraction of a percent. A lot of work for rather little reward, at least financially. This is what Thorp thought.
However, he realized that in the course of an actual game information was revealed that the basic strategy ignored. If you took advantage of this information and changed your bet accordingly from hand to hand, then you could actually beat the house.
In the course of the game what is revealed are the cards that have been played, as well as the number of cards that remain to be played. Continuing to assume that one deck is being used, if you see that the four aces have been played, you know that in the next hand you cannot be dealt an ace, which lowers your chances, so you might want to lower your bet. (The lack of aces lowers your chances by something like 3 percent.) If all the aces are still available you might want to raise your bet, and so on. We are now entering the realm of card counting.
One might at first think that card counting would require an eidetic memory. That would help, but there are simplifications that enable mere mortals to play the system.
In a general way, low cards are friends of the dealer, who must take a hit on any total of 16 or less. The friendliest of all the cards are the 5s. There is no way that the dealer hitting on 16 or less can go over 21 by drawing a five. Thus the simplest card counting system is to count the 5s that have been played, along with the number of cards that remain. More sophisticated is to assign a number to each card that has been made visible. Cards from 1 to 6 are assigned plus 1, cards 7, 8, and 9 are counted 0, and the rest are assigned minus 1. After each round the counter adds up the score and, keeping track of the remaining cards, bets accordingly.
By June 1960, Thorp had the strategy worked out. He presented a paper at a meeting of the American Mathematical Society in Washington, D.C., which he called “Fortune’s Formula: A Winning Strategy in Blackjack.” This was a combination of the basic strategy and counting the 5s.
Before he left for Washington he received a phone call from a reporter at the Boston Globe who somehow had seen the abstract and wanted to do a story. The Globe even sent out a photographer. One must understand that blackjack is the most popular gambling game in casinos and that there are a lot of casinos, more all the time. As a result, the notion that there is a winning strategy in blackjack is like announcing that there is gold in California: exciting to many people, but not necessarily rewarding. Most people don’t understand that these strategies are statistical and that winnings of a few percent are possible if you are prepared to play for a long time, during which you have a good chance of losing all your money.
Thorp reports that after his talk in Washington he was asked to give a press conference, which was followed by radio and television interviews and then hundreds of letters and phone calls containing all sorts of offers. He finally chose offers from two professional gamblers; in his book, “Beat the Dealer,” he called them “Mr. X” and “Mr. Y.”
The three of them headed to Las Vegas, with X and Y supplying a $10,000 grub stake. In his book “Fortune’s Formula,” William Poundstone reveals that X and Y were Manny Kimmel and Eddie Hand. Kimmel was a shady bookie who made a fortune in parking lots and funeral homes. Hand was a gambler, then well known in Las Vegas casinos. It is very unlikely that either one had any understanding of the mathematics of what Thorp was doing.
By this time Thorp had improved his system. Instead of counting 5s, he was counting 10s. The presence of 5s substantially increases the dealer’s chances, while the presence of 10s somewhat enhances the player’s. But there are sixteen 10s and only four 5s, so the cumulative effect of counting 10s is better.
Thorp gives an amusing account of his stay in Las Vegas with X and Y. It fell into a pattern. When he started to win, the casinos simply kicked him out or changed the rules so as to diminish his chances by, for example, introducing new decks in mid-play. This slows the game down, so the casino is not anxious to do this very often, but it is better than being cleaned out.
As a rule, the casinos did not give reasons for throwing him out. They felt that something was going on that they did want to be a victim of. Thorp’s ejection was done with the greatest of good manners. He and X and Y were sometimes offered free dinners and unlimited drinks, but told to go away.
During this expedition with X and Y, the maximum that was ever at stake in a single game was perhaps $2,000. At the $500-limit tables, he sometimes had two or three multiples of $500 down, either from pair splitting, doubling down, multiple hands, or a combination of these. Nonetheless, the three of them came away with a profit of some $11,000.
This was peanuts compared to what Thorp must have made on his book, which was a national bestseller. A second edition appeared in 1966.
Between the editions a computer scientist at IBM named Julian Braun ran the strategy on the latest IBM main frame computer. He used Thorp’s annotated program and suggestions to take the calculations out to greater accuracy, possible because of the increase in computer power in the intervening years.
I do not know if it produced better results than the MANIAC at Los Alamos, but Thorp used Braun’s results in his 1966 edition. Some years later a mathematician named Peter Griffin was able to use a computer to do an exact calculation. Thorp’s book has now sold more than 600,000 copies.
Thorp’s publishing success was followed, some decades later, by Ben Mezrich’s best-selling book “Bringing Down the House.” The book, which purports to be nonfiction, is loosely based on reality ” and it was followed by the Hollywood film “21,” which is loosely based on Mezrich’s book.
Before I get into that, let me describe the apparent facts. MIT has something called an Independent Activities Period, which encourages its students to do things outside the curriculum. I think of this as a somewhat more adult version of the programs designed to keep kids off the street by offering them after-school activities.
When I was a student at Harvard I took a few courses at MIT, and nothing would surprise me about MIT undergraduates, or so I thought. In January 1979, an Independent Activities course was offered called “How to Gamble if You Must.” This must have seemed like a wonderful idea to someone. If you are going to practice a vice, at least do it intelligently.
Several MIT students attended this course and learned about card counting in blackjack. Flush with this knowledge, some of them headed to Atlantic City and got their clocks cleaned. Most of the group decided that this was enough, but a couple persisted and even taught a second version of the course in January 1980.
About this time one of them, J. P. Massur, had an accidental meeting in a Chinese restaurant with a recent Harvard business school graduate named Bill Kaplan, who had in fact been running his own blackjack team in Las Vegas.
In Mezrich’s book ” one is tempted to say novel ” Kaplan has become Micky Rosa, an MIT graduate and former instructor. In the film, he has become a full professor of mathematics, played by Kevin Spacey. (The real Bill Kaplan looks about as much like Kevin Spacey as the film resembles real events.)
In another bit of Hollywood shape-shifting, the film’s protagonist is an MIT student called Ben Campbell, played by Jim Sturgess. In the book, that character was called Kevin Lewis. In fact, the real “Ben Campbell/Kevin Lewis” was an MIT student named Jeffery Ma.
Viewers of the film will recall the scene in which Campbell is being indoctrinated by the unsavory MIT professor played by Spacey. (MIT refused to allow the film to be shot on its campus, so Boston University was used.) The indoctrination consists of Spacey shouting amid a background of general chaos numbers of the count, which Campbell is supposed to keep straight. It went by so fast in the film that I could not make heads or tails of what was going on.
In any case, if we can slip back into reality again, Bill Kaplan agreed to come with the MIT group on a second foray to Atlantic City. He did not do this for intellectual reasons. Kaplan was making a business of sponsoring blackjack teams and had just parted ways with his previous team.
What he observed in Atlantic City was another disaster. Each member of the team seemed to have an individual counting strategy and spent a great deal of time arguing about which was best.
Kaplan said that he would back the team, provided that they would change their modus operandi. For example, there would be one counting system and that would be the one that Kaplan told them to use. The new team, with a capitalization of $89,000, started playing in August 1980. By two months later the team members were earning about $80 an hour of play, while Kaplan and his other investors were raking it in by the carload.
Oddly, in reading what I have been able to find, I do not see that Jeff Ma ” Kevin Lewis/Ben Campbell ” played an especially important role. He was certainly never a medical student, as the film portrays. He seems to have taken the money he earned and bought a town house in the South End of Boston and invested in a bar.
The MIT team grew to some 80 players before it broke up in 1993. By this time most of them had been banned from most casinos, and the casinos were using multiple decks hidden in a “shoe.” Some of the team went into real estate, thinking there was more money to be made there. Maybe they have now returned to blackjack.
Incidentally, early in the movie, a copy of Thorp’s book is shown on a table in passing.
As for Thorp, he became a professor of mathematics, ending his academic career in 1983 at the University of California at Irvine. By this time he had employed the skills he had acquired in the study of blackjack to play the stock market, becoming one of the most successful money managers who has ever practiced that craft. He was associated with various hedge funds, which managed hundreds of billions of dollars. He always beat the market, and in some years he even beat Warren Buffet.
Perhaps the best known of Thorp’s funds was the one he founded with a partner, James Regan, called the Princeton-Newport Partners. Regan was in Princeton and Thorp in Newport Beach, Calif. In its heyday the fund managed money for institutions such as Harvard. Its heyday came to an abrupt end in 1988, when the fund was investigated by Rudy Giuliani, who threatened to use the then-new RICO law to indict the company.
It turned out that the Princeton wing had been engaged in “stock parking.” This is the practice of holding stock for someone else, so that tax advantages can be gained. In this case, the “someone else” turned out to be people such as Ivan Boesky and Michael Milken, both of whom ended up in jail. The company was never indicted, only threatened with indictment unless Regan agreed to “rat out” Milken. He didn’t, and there was a trial of five people from the Princeton office. Thorp had had nothing to do with all this, but the fund was dissolved. Since then he has run his own funds.
Thorp has obviously made far more money in the stock business than he ever made ” or could have made ” with the theory of blackjack. That is doubly true now that the casinos have declared all-out war on “card counters,” including changing the way the game is played.
I have never understood the casino paranoia about “counters.” It contrasts vividly with what happens with the slot machines. If someone hits the jackpot the casino practically declares a national holiday. They want people to know that there are winners so that the flotsam and jetsam are sucked in. Somehow they feel differently about blackjack.
But the vast majority of blackjack players do not understand the system and do not realize how fast the casino game moves and how much distraction there is. They also do not understand the statistical nature of the system. As an illustration I will end this with an absolutely guaranteed way of making money in a casino.
Go to a roulette wheel and put down $2 on red. If it comes up red, you have won $2. If it comes up black, put $4 on red. If it comes up red, you have won $2. If it comes up black, put $8 down, and so on. If the wheel is “rational,” it eventually has to come up red, and you will win $2.
But keep in mind that $2 raised to the 20th power is $1,048,576. To paraphrase what Lord Keynes might have said: “Roulette wheels can remain irrational longer than you can remain solvent.”