The American Physical Society—the largest professional organization for physicists in the United States—once held its annual meeting in Las Vegas. From the city’s perspective, the meeting was a fiasco. The assembled physicists shunned the usual casino delights: showgirls, blackjack, roulette, craps, and copious amounts of alcohol. Plus they were lousy tippers. Vegas made so little money, legend has it, that the society was asked never to come back. The physicists could do the math, you see: They knew the odds were stacked against them in the casinos. That’s why physicists aren’t the gamblin’ kind.
Or so goes the conventional wisdom. Conventional wisdom hasn’t met my husband, Caltech cosmologist and poker fiend Sean Carroll, who happily spends hours on end in poker rooms. It started in 2004 after he readPositively Fifth Street, James McManus’s account of covering the World Series of Poker (WSOP) for Harper’s Magazine. McManus got so caught up in his reporting that he entered the tournament on a lark and ended up winning $247,760. Intrigued, Sean began lurking around poker rooms to watch the play, becoming a “railbird,” in poker parlance. He bought a few instructional books and played a bit online before venturing out to the Hollywood Park Casino just outside Los Angeles.
Hollywood Park has a seedy, vaguely disreputable vibe. The occasional fistfight breaks out late at night, and the fast rate of play can be intimidating for a new player (a “fish” or “dead money”). But there are also plenty of unskilled (often inebriated) players whose strategy seems to be “Call everything—you might get lucky!” That first night, Sean walked away $250 richer, and hooked on poker.
Later that year, Sean played in an informal poker tournament in Chicago (organized as a fund-raiser for presidential candidate John Kerry) and found that there were three other physicists among the participants. One of them, string theorist Jeff Harvey of the University of Chicago, won the tournament. He had learned the rules of the poker game being played from his teenage daughter just the week before, and he has been an avid player ever since.
One poker-playing physicist is a statistical anomaly; two is a coincidence; three, and it might just be a pattern. Michael Binger placed third at the 2006 WSOP main event, two months after earning his physics Ph.D. from Stanford University, and walked away with a cool $4.1 million. He has since played all over the world, racking up six tournament wins and an additional $2 million. At a tournament in San Remo, Italy, last spring, the final table included two more physics gurus: Michael Piper and Liv Boeree, former classmates at the University of Manchester. Piper placed fourth and Boeree won, pocketing 1.25 million euros—about $1.6 million—for her trouble.
Perhaps poker appeals to physicists because it is an intricate, complex puzzle, steeped in statistical probabilities and the tenets of game theory. The best players evince a rare combination of skills in math, strategy, and psychology. “Both physics and poker attract people who like to solve multifaceted problems,” says Marcel Vonk, a Dutch-born physicist at the University of Lisbon, who claimed his first WSOP winner’s bracelet this past summer in Las Vegas, beating out 3,800 players to win $570,960. “The skills required are similar: mathematical abilities, the ability to spot patterns and predict things from them, the patience to sit down for a long time until you finally achieve your goal, and the ability to say, ‘Oh well,’ and start over when such an attempt fails miserably.”
Most poker-playing physicists don’t consider poker to be true gambling. In craps or blackjack, Sean explains, you’re playing the casino (the “house”), and thanks to a slight statistical edge, the house always wins in the long run. What self-respecting physicist would accept those odds? But in poker you are playing against other people; casinos typically take a cut of the pot. Luck may be a factor, but poker is more a game of skill than of chance. “When played by a professional, poker is not gambling,” insists Eduard Antonyan, a former physics graduate student of Harvey’s. “While you can get unlucky for extended periods of time, eventually, if you’re good, you’re going to profit.”
Modern poker tournaments did not flourish until the World Series of Poker in Las Vegas took off in the 1970s, launching thousands of how-to books on foolproof poker strategies for amateurs harboring fantasies of beating the pros. With the introduction of online poker and the advent of cameras capable of showing a player’s hidden cards to a TV audience, poker has become a bona fide spectator sport.
The most popular form is Texas Hold ’Em. As in physics, the basic rules are simple. Each player is dealt two “hole cards” that only he or she can see. There is a round of betting, followed by the “flop”: three common cards dealt face-up in the middle of the table. There is another round of betting, and another common card is dealt (the “turn”). The players bet again, and a fifth and final common card is dealt (the “river”). The winner is the player who can construct the best five-card hand out of seven: their two hole cards and the five common cards.
The challenge lies in analyzing the probabilities. One well-established tenet of probability theory is the law of the sample space, the set of all possible outcomes of a random process. The probability of winning a roll of the dice, for example, is equal to the proportion of winning outcomes relative to all possible ones. A die can land on any one of its six sides, and those six potential outcomes make up the sample space. Place a bet on one such number and your chance of winning is one in six; place bets on three such numbers and your odds improve to three in six.
When rolling two dice, the reality is more complicated because not all outcomes are equally likely; different outcomes have different probabilities. It is another established tenet of probability theory that the odds of a particular outcome depend on the number of ways in which it can occur. To roll a 2, you would need to roll snake eyes (1+1). In contrast, there are three different combinations of dice that total 7: 1+6, 2+5, and 3+4. And because each die is distinct, you must also account for the combinations 4+3, 5+2, and 6+1. In fact, 7 is the most likely number to be rolled.
Therein lies the secret of the house advantage. In craps, for instance, it is no accident that the “losing” roll is 7 once the game gets under way. Even a slight edge of 1.4 percent (a measure of the house’s advantage in a particular bet) is enough to tip the scale irrevocably in the casino’s favor. Play craps long enough and eventually you will lose everything.
The probabilities in poker are much harder to calculate. The deck has 52 cards, so 2,598,960 five-card hands are possible. That number comes from a simple statistical formula of factorials; it is (52 × 51 × 50 × 49 × 48)/(5 × 4 × 3 × 2 × 1). Texas Hold ’Em further complicates matters because you use the best of seven cards (133,784,560 potential combinations) to develop your five-card hand. And your information is woefully incomplete: You can only guess which cards an opponent holds based on your analysis of body language, nervous tics, betting patterns, and so forth.
So while math skills are critical for analyzing the odds of winning a given hand, it is unwise to put too much stock in the numbers. As Harvey—ever the string theorist—puts it: “Chess is like classical mechanics. Poker is like quantum mechanics. In chess there is only one right move. In poker there is no single right move. There is a probability distribution of right moves.”
Enter game theory, an approach to devising a “mathematics of life” pioneered by the brilliant mathematician and computer pioneer John von Neumann in the 1920s. He was fascinated by the art of the bluff, famously observing, “Real life consists of bluffing, of little tactics of deception, of asking yourself what is the other man going to think I mean to do.” He used hands of poker for his analysis, modeling strategic interactions between two players, each of whose actions depended on determining what his opponent was likely to do. With economist Oskar Morgenstern, Neumann published the definitive treatise on this topic in 1944, “Theory of Games and Economic Behavior.”
Ideally, say physicist/poker players like my husband, you want to employ a mixed strategy in poker based on the probabilities. If your hole cards are pocket aces (a pair) before the flop, you may raise 80 percent of the time and check 20 percent of the time. If your paired 6s are going head-to-head against ace-7 unsuited (in different suits)—a situation in which your odds of winning are about 50 percent—game theory dictates you should bet half the time and fold the other half. Varying your behavior in accordance with the odds has the added benefit of throwing your opponents off-guard.
Chris Ferguson—who holds a Ph.D. in computer science from UCLA and has written academic papers on poker and game theory with his father, UCLA mathematician Thomas Ferguson—won the 2000 WSOP, relying heavily on game theory. Some viewed this as a triumph of math over intuition and experience. But Neumann’s model had one serious limitation, especially as poker caught on with the masses: Traditional game theory is best equipped to handle highly simplified situations, where players are perfectly rational; in the messy real world, human judgment remains crucial.
Deciding how much to bet to maximize one’s winnings is also math intensive, but a simple equation exists to determine that: the Kelly Criterion, named after John Kelly, a physicist who worked at Bell Labs in the 1950s. You simply divide your “edge” by the odds to find what percentage of your bankroll you should bet each time. In poker, the edge describes the amount you expect to win, on average, if you make the same wager repeatedly under the same probabilities; the odds determine how much profit you make if you win. Even if the odds are in your favor, you don’t want to bet your entire bankroll; one stroke of bad luck and you’ll lose everything. Play it safe and bet too little, however, and your return won’t be sufficient to make up for the inevitable losses.
The trick is figuring out the values for the edge and the odds. For poker, it is very difficult to do this with sufficient precision, even if your name is John von Neumann. Fortunately, several effective computational models have been developed; most serious, mathematically inclined players find their optimal balance through a mix of theoretical research and practical, hands-on experience.
Still, the player has to prepare for the variables. “You need to start out with a sufficiently large bankroll to weather the inevitable statistical fluctuations,” Harvey says. He estimates 15 to 20 times the amount of the buy-in is a good rule of thumb, “because you can easily lose two or three buy-ins just due to variance, even if you play perfectly.” Antonyan once lost 30 buy-ins in a row.
Every player has a trove of such “bad beat” stories. My husband Sean once ran into a string of bad beats at the MGM Grand in Las Vegas. Twice he had a straight—putting him ahead of another player who had three of a kind at the turn—only to be beaten by a full house on the river. When Sean finally drew a full house, his opponent snagged a coveted four of a kind. I found him that evening in the Bellagio’s Baccarat Bar, nursing his wounded pride with a dry martini.
Statistical anomalies are inevitable; that is the hallmark of true randomness. Michael Binger ran afoul of the odds at the 2010 WSOP during an early shoot-out event with 10 players. He made it to the final two and had a substantial 3:1 chip lead over his opponent, an amateur who had a “rather recklessly aggressive style of play,” as Binger recalls. Lady Luck seemed to be smiling on Binger at first. He was dealt the ace and king of clubs—an excellent starting hand—so he raised. The other player went “all in,” and Binger called, only to find his opponent was holding a pair of 3s. When the flop cards were turned, Binger got another king, but the third card was another 3, giving his rival three of a kind (or “trips”) to win the hand.