The Signal and the Noise

by Nate Silver

  The year 2003 was the start of the “poker boom,” a sort of bubble economy in which the number of new and inexperienced players was growing exponentially and even a modicum of poker skill could be parlayed into large profits. The phenomenon had two immediate and related causes. One was the 2003 World Series of Poker in Las Vegas, which was won by a twenty-seven-year-old amateur, a Nashville accountant with the auspicious name of Chris Moneymaker. Moneymaker was the literal embodiment of the poker everyman: a slightly pudgy office drone who, through a never-ending series of daring bluffs and lucky draws, had turned the $39 he’d paid to enter an online qualifying tournament into a $2.5 million purse.

  ESPN turned Moneymaker’s achievement into a six-part miniseries, played on nearly continuous repeat on weekday evenings until baseball season finally came along to fill the void. It was terrific advertising for the “sport” of poker, which until that time had a reputation for being seedy, archaic, and intimidating. Suddenly, every balding, five-foot-eight accountant who had long ago given up on his dream of being the next Michael Jordan or Derek Jeter could see in Moneymaker someone who looked just like him, who had a job just like his, and who in a matter of weeks had gone from rank amateur to the winner of the biggest poker tournament in the world.

  But the ESPN broadcasts presented a highly sanitized version of what reality actually looks like at the poker table. For one thing, out of the necessity of compressing more than forty hours of play involving more than eight hundred players into six hours of broadcasts, they showed only a small fraction of the hands as they were actually played. What’s more, because of the ingenious invention of the “hole cam”—pinhole-size cameras installed around the edge of the table beside each player—the cards of not just Moneymaker but those of each of his opponents were revealed to the home audience as the hand was being played out, giving the audience the feeling of being clairvoyant. Poker is a pretty easy game if you know what cards your opponent holds.

  Moneymaker was cast as the protagonist who could do no wrong. Hands that a sober analysis might have concluded he’d played poorly were invariably praised by the announcers—rash bluffs became gutsy ones, premature folds became perceptive ones. Moneymaker was not some slightly-above-average schmoe getting the cards of his life*1 but a poker savant who was cunning enough to have developed into a world-class player almost overnight.

  The viewer was led to believe that poker is easy to learn, easy to profit from, and incredibly action-packed—none of which is true. But that didn’t stop many of them from concluding that only a ticket to Las Vegas separated them from life as the next Chris Moneymaker. The number of participants in the World Series of Poker’s $10,000 main event exploded, from 839 the year that Moneymaker won it to 8,773 just three years later.

  I was one of those people.2 I lived the poker dream for a while, and then it died. I learned that poker sits at the muddy confluence of the signal and the noise. My years in the game taught me a great deal about the role that chance plays in our lives and the delusions it can produce when we seek to understand the world and predict its course.

  FIGURE 10-1: WORLD SERIES OF POKER MAIN EVENT PARTICIPANTS, 1970–2006

  The Start of a Poker Dream

  The other catalyst of the poker boom was the Internet. Internet poker had existed in some form since 1998, but it began to go mainstream in 2003 as companies like Party Poker and PokerStars became more aggressive about marketing their way through the legal quagmire of Internet gambling. Players from all around the world flocked to the online games, overcoming their reservations about the security and legality of the virtual cardrooms. They offered a whole host of conveniences: 24/7 access to every form of poker at stakes ranging from pennies a hand to hundreds of dollars; a faster-paced experience (a computer chip can shuffle much faster than a human dealer can—and there’s no need to tip it); the comfort of playing from home rather than at a smoky, decrepit casino.

  I had a job not unlike Moneymaker’s, as I described earlier, working as an economic consultant for the accounting firm KPMG. One of my buddies at work suggested that we start a regular game, gambling just enough to make ourselves nervous. I had a smidgen of poker experience, having made a few four-in-the-morning sojourns to the Soaring Eagle Indian Casino in Mount Pleasant, Michigan. But I was rusty and needed practice and began to look online. A clunky, buggy site named Pacific Poker was making players an offer that was hard to refuse: $25 in real-money poker chips, with almost no strings attached.3

  I lost the initial $25 fairly quickly, but the players in the Pacific Poker games did not seem much more sophisticated than the mix of ex-convicts and septuagenarians who populated the games at the Soaring Eagle. So I deposited $100 of my own. Almost all professional poker players begin their careers on winning streaks—the ones that lose at first are usually sensible enough to quit—and I was no exception. My bankroll began to grow, by $50 or $100 a night at first and them sometimes by $500 or $1,000. After about three months, my winnings hit $5,000; I began staying up all night to play, taking a cab to work at the crack of dawn and faking my way through the workday. After six months and $15,000 in winnings, I quit my job, leaving the exciting world of international tax consulting behind to split my time between playing cards and working for Baseball Prospectus. It was liberating; I felt as though I’d hacked the system somehow.

  I have no idea whether I was really a good player at the very outset. But the bar set by the competition was low, and my statistical background gave me an advantage. Poker is sometimes perceived to be a highly psychological game, a battle of wills in which opponents seek to make perfect reads on one another by staring into one another’s souls, looking for “tells” that reliably betray the contents of the other hands. There is a little bit of this in poker, especially at the higher limits, but not nearly as much as you’d think. (The psychological factors in poker come mostly in the form of self-discipline.) Instead, poker is an incredibly mathematical game that depends on making probabilistic judgments amid uncertainty, the same skills that are important in any type of prediction.

  How Poker Players Predict Hands

  Good poker players are not distinguished by their ability to predict which cards will come next, as though they had ESP. Only the most superstitious or paranoid players believe that the order of the shuffle is anything other than random. Only the very worst ones will have failed to commit the most basic odds calculations to memory: that a flush has about a 1-in-3 chance of coming in with two cards to come, or that a pair of aces will beat a pair of kings about 80 percent of the time. The core analytic skill, rather, is what players call “hand reading”: in figuring which cards your opponent might hold, and how they might affect her decisions throughout the rest of the hand.

  This is an extremely challenging problem, especially in Texas hold ’em, the most commonly played variant of the game. In hold ’em, the cards are kept facedown and nothing is certain about an opponent’s hand until all the bets are in and the pot is pushed someone’s way. Each player begins the hand with one of 1,326 possible hand combinations. Everything from a majestic pair of aces to the lowly seven-deuce are among them, and nothing but the player’s love of money prevents her from playing the one hand as though it is the other one.

  However, players can use their hand-reading skills to develop what amounts to a forecast of their opponent’s likely range of hands. Poker players often talk about “putting opponents on a hand,” and sometimes they proceed as though they know exactly what two cards the opponent holds. But the best players always entertain numerous hypotheses, which they weigh and balance against the opponent’s actions. A good poker forecast is probabilistic. It should become more precise as the hand is played out, but it is usually not possible to predict exactly which of the 1,326 possible hands your opponent holds until the cards are revealed, particularly if your opponent is skilled and is deliberately behaving unpredictably.4

  Indeed, information is so hard to come by in Texas hold ’em that players begin to ma
ke estimates about their opponents’ range of hands even before any of the cards are dealt. In online games, this is often done through data mining: you’ll have statistics on how loose or tight, how passive or aggressive, each opponent’s play has been in previous games. In brick-and-mortar casinos, it is done through players’ past histories with one another—or, failing that, through what amounts to ethnic profiling. Players from Sweden, Lebanon, and China, for instance, have a reputation for being more aggressive than those from France, England, or India. Younger players are presumed to be looser and more aggressive than older ones. Men are assumed to be more likely to bluff than women. These stereotypes, like any others, are not always true: at the hold ’em games I used to play in at the Bellagio in Las Vegas, the best players were very often women, and they were good in part because they were much more aggressive than their opponents assumed. But poker players don’t have the time for political correctness. Even if the stereotype that women play more conservatively than men is false 45 percent of the time, the fact that it might be true 55 percent of the time gives them something to work with.

  Once the game begins, these crude assumptions are supplanted by more reliable information: how the player has played previous hands at the table that day and how she is playing this one. The process is fundamentally and profoundly Bayesian, with each player updating their probabilistic assessments after each bet, check, and call. If you doubt the practical uses of Bayes’s theorem, you have probably never witnessed a poker game.

  * * *

  A QUICK PRIMER ON TEXAS HOLD ’EM

  It is easy to access the rules of Texas hold ’em online or in other books, but I’ll point out a few basics for the uninitiated, to introduce you to the terminology used in the next few pages. These rules are simple compared with those of other card games. But much as is the case in chess, the relatively simple rules create a game with exceptional tactical and strategic depth.

  The game begins when two personal cards (called down cards or hole cards) are dealt facedown to each player. A round of betting ensues at this stage. These personal cards then start to be combined with a series of community cards (also called the board) that are dealt faceup and shared between all players at the table. Each player seeks to formulate his best five-card poker hand between his down cards and the community cards. The community cards are revealed sequentially, with a round of betting separating each stage. The first three cards are revealed simultaneously and are called the flop (one of the many pieces of colorful vocabulary that poker players use). The fourth community card, called the turn, is revealed next. Finally the last card, the river, is exposed, and a final round of betting takes place. More often than not, all players but one will have folded by this point. If not, the players’ down cards are finally flipped faceup and the best hand at the showdown wins the pot.

  The ranking of poker hands goes thusly:

  A straight flush (K Q J T 9)

  Four-of-a-kind (7 7 7 7 2)

  A full house (Q Q Q 5 5)

  A flush (A J 9 4 2)

  A straight (8 7 6 5 4)

  Three-of-a-kind or a set (9 9 9 A 2)

  Two pair (A A 3 3 7)

  One pair (K K 9 8 6)

  An unpaired high card (A Q 8 5 3).

  Otherwise-tied hands are awarded to the player with the highest-ranking cards: for instance, an ace-high flush beats a 9-high flush. When pairs are of the same rank, the tie is broken by the third-highest card (or kicker). For example, the hand (8 8 K 7 5) beats the hand (8 8 Q 7 6) since the kicker is a king rather than a queen.

  * * *

  A Not So Simple Poker Hand

  Say that you’re playing in a $5/$10 no-limit hold ’em game at the Bellagio.* The first few players fold and you find yourself looking at some decent cards, a pair of eights (8 8). So you raise to $25 and are called by just one player, a sixtysomething man whom we will call the Lawyer.

  The Lawyer is a nice guy—a little chatty in between hands, but quiet enough once the cards are dealt. We have learned that he has done fairly well for himself as a partner at an intellectual property law firm on the East Coast. You might imagine him wearing a polo shirt and periodically texting his friend to check on tee times. He had one beer when the cocktail waitress first came around before switching to coffee. He is not likely to be intimidated by these medium-size stakes and isn’t acting like it.

  When the Lawyer first sat down at the table with us, we had been torn between two hypotheses about him: first, that he might be playing to show off a little, and therefore might make a few wild moves and bluffs, and second that he might take a more legalistic and by-the-book approach. Our subsequent observation has confirmed that the latter profile is more likely to be correct: he seems to be a mediocre player, avoiding catastrophic errors but without applying much finesse. The Lawyer is by no means the worst player at the table, but he’s probably not a long-term winner. Still, we haven’t played against him for all that long and aren’t completely sure about any of this.

  So what do we know about the Lawyer’s hand so far? The only thing we know absolutely without doubt is that he cannot hold any hands that contain the 8 or the 8, since those cards are in our hand. Unfortunately, that reduces the number of starting hand combinations only to 1,225 from 1,326, each of which was equally likely to be dealt when the hand began.

  However, the Lawyer has given us some other information: he has chosen to call our bet. That means his hand is likely to be at least decent: most players, including the Lawyer, fold a clear majority of their hands against a raise before the flop. It also means he is unlikely to have an extremely strong hand like a pair of aces, since he called not just in lieu of folding but also in lieu of reraising, although this possibility cannot be discounted completely.*

  We can start to form a probabilistic, Bayesian assessment of what hands the Lawyer might hold. From past experience with players like this, we know that his range of hands is likely to include pairs like 9 9. It might include some hands with aces, especially if both cards are of the same suit (like the hand A 5), meaning that it can more easily make flushes. Likewise, it might include what are called “suited connectors”—hands like 6 5 that have consecutive cards of the same suit and which can make both flushes and straights. Finally, he might call with hands that contain two face cards like the king of clubs and the jack of diamonds (K J).

  If we had enough time, we could enumerate the Lawyer’s hand range completely, with all 1,326 possibilities assigned a probability from 0 percent to 100 percent given his action so far, as in figure 10-2a. This is how a computer, which can sort through the possibilities very quickly, might think about his play.

  FIGURE 10-2A: PROBABALISTIC REPRESENTATION OF OPPONENT’S HAND RANGE

  A matrix like this is far too complicated to keep track of under real-world conditions, however, so what players seek to do instead is break their opponent’s range into groups of hands that might function in about the same ways as one another (figure 10-2b). In this case, the group of hands that would most concern us is if the Lawyer began the hand with a pair higher than our eights.

  Fortunately, the probability of this is low: a player is blessed with a pair only rarely in hold ’em. Indeed, when the cards are first dealt in hold ’em, the chance of starting out with a pair of nines or better is only about 3 percent. However, we need to update our estimate of this probability because of his call: the Lawyer is throwing out many of his worthless hands, increasing the chance that he has a strong one. According to our estimate, the chance he holds a pair higher than our eights has already risen to about 6 percent given how he has played the hand thus far.

  The other 94 percent of the time, the Lawyer starts out with a worse hand than ours. The problem is that there are five cards still left to come, and while it is relatively hard for them to improve our pair (we would need to catch one of the two remaining eights in the deck), he could more easily make a higher pair, straight, or flush.

  The Lawyer takes a swig of coffee as the
dealer arranges the flop cards on the center of the table. They consist of two clubs—a king and a three—along with the nine of hearts.

  K 9 3

  These cards have not improved our hand. Our hope is that they have not improved the Lawyer’s either, in which case our pair of eights will still be best. So we make a relatively modest bet of $35 into the $65 pot. The Lawyer pauses for a moment and calls.

  This call is not great news for us, as we can see by further refining our estimate of his hand range. The key, following Bayes’s theorem, is to think in terms of conditional probabilities. For instance, if the Lawyer started with a hand like K J, now giving him a pair of kings, how likely is it that he’d call us again? (He’d almost certainly at least call having made top pair, but might he have raised instead?) What about if he began the hand with a smaller pair than ours, like 7 7—how likely is it that he’d call rather than fold? If we had the luxury of time, we could go through all 1,326 hand combinations individually and revise our estimates accordingly (figure 10-3).

  Our actual estimates at the table will not be quite as precise as this. Still, we can come to some broad probabilistic characterizations of his hand range given his call. About 30 percent of the time, the Lawyer’s hand connected strongly with the flop and he has a pair of kings or better—a good hand that is unlikely to fold without a lot of pressure. There is also about a 20 percent chance that he has a pair worse than kings but better than our eights. These hands beat ours as well, but the Lawyer might be more likely to fold them later if we keep betting aggressively.

  Meanwhile, there is about a 25 percent chance that the Lawyer has a draw to a straight or a flush, which is behind us for now but has many ways to improve. Finally, there is a 25 percent chance that he has a worse pair than ours or has almost nothing and is continuing in the hand solely in the hope of bluffing us later; these are the most favorable cases.