by Nate Silver
You can see how complicated poker decisions become. Some of these possibilities imply that we should continue to bet our hand as aggressively as possible. Others imply we should take a more cautious approach, while still others would mean we should be preparing to fold.
Just as we are contemplating this thorny decision, the dealer puts out the ideal card and makes our life easy. It is one of the two remaining eights in the deck, the 8, giving us three of a kind. The only way we are beaten is if the Lawyer started out with a pair of nines or a pair of kings and made a higher set on the flop, and has played them passively in order to trap us. (Poker players term this “slowplaying.”) Still, we should not be thinking so defensively. In sorting through his possible hands, we should have the better one perhaps 98 percent of the time. So this time we make a relatively large bet: $100 into the $135 pot.
The Lawyer calls us once more. Because he is likely to have folded out his weaker pairs and weaker draws, we can now narrow his hand range even further. In fact, of the 1,326 hands that he might have started out with, not more than about seventy-five are all that likely at this stage. Often, he will have a pair of kings, a hand that we were previously worried about but now beat. To the extent we are concerned, it’s mostly about another club coming, which could still give him a flush.
Instead, the final card is the harmless-looking five of spades, which does not complete the flush:
K 9 3 8 5
We bet $250 into the $335 pot, hoping that the Lawyer will call us with a worse hand. Suddenly, however, he springs to life. “I’m all-in,” he says to the dealer in a voice just barely louder than a whisper. He neatly pushes his remaining chips—about $1,200—into the pot.
What the hell just happened? We now need to put our Bayesian thinking skills to the test. If our forecast of his hand range is off, we could easily make a $1,200 mistake.
We look at the board and realize there is one exact hand, one from among the 1,326 random combinations, that seems most consistent with his play. The specific hand is a seven and a six of clubs (7 6). It is a suited connector, so we think he would have called with it before the flop. On the flop, this hand made a flush draw with four clubs, and we didn’t bet enough to push him off it. On the turn, the hand missed its flush but nevertheless became stronger: the 8 that made our hand three-of a-kind gave the Lawyer the possibility of making a straight with any ten or five. If that was indeed his hand, the 5 on the river made his straight, which beats our three-of-a-kind and would explain why he is now betting so boldly.
So should we fold? Even if you have never played poker, it is worth pausing for a moment to consider what you’d do.
The answer is that you should very probably not fold. In fact, against many players, you should be pleased that more money is going into the pot.
The solution comes from Bayes’s theorem. It’s true that the all-in bet is an extremely powerful move—it conveys much more information than his calls before. But before the Lawyer went all-in, we would have assigned a very low probability—perhaps 1 percent—to his holding exactly the seven and six of clubs, just one possible hand out of the myriad combinations. Unless we are extremely confident that the 7 6 is about the only hand that he’d make this play with, folding could be a huge mistake. Our hand only needs to be good about 35 percent of the time to make the call correct mathematically.
In fact, there are some alternate possibilities for his hand. The Lawyer could have a set of threes or possibly a set of fives, which still lose to our set of eights. He could plausibly have made two-pair with a hand like K 5. Some players would play a pair of aces in this way. In his Bayesian model of our hand range, the Lawyer might reasonably figure that hands like this are better than ours even though they are not—good enough to go all-in—and he might be willing to get a lot of money in with them.
There are also a couple of hands apart from the straight which would beat us. If the Lawyer was slowplaying a set of nines the whole way, or a set of kings, he’ll now get our money. This is counterbalanced by the possibility of a complete bluff. If the Lawyer had a flush draw that missed, the only way he can win the pot is by bluffing at it.
As Arthur Conan Doyle once said, “Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth.” This is sound logic, but we have a lot of trouble distinguishing the impossible from the highly improbable and sometimes get in trouble when we try to make too fine a distinction. All of the opponent’s hands are in some way highly improbable at this stage; this has been an unusual hand. It is a matter of weighing improbabilities against other improbabilities, and the calculation weighs against the hypothesis that he has 7 6 exactly. If we ran the possibilities through a computer, it might think there’s something like a two-thirds probability that we still have the best hand (figure 10-5).
In practice, poker players might differ a lot in how they assess the probabilities for his hand. Skilled poker players are probably better than 99.9 percent of the population at making reasonably good probabilistic judgments under uncertainty. In fact, I don’t know of a single game or intellectual exercise that better refines these skills. However, when I posted this hand on Two Plus Two, an online forum for professional poker players, assessments ranged from that we were nearly certain to have the best hand to that we were nearly certain to be beat.6 My view is that both these assessments are overconfident. We should not proceed as though we don’t know anything about the opponent’s hand, but in general our predictive errors come in thinking that there is more certainty in the world than there really is. In this case, seeking to put the opponent on an exact hand would imply a fold, while a fuller assessment of the probabilities—coupled with the favorable adds from the pot—means that we should call instead.
Schrödinger’s Poker Hand
If this hand came up in a televised tournament on ESPN that showed us each player’s hole cards, the analysis from the commentators might be different. They might assert that the fold was obvious if they knew that the opponent held 7 6. In a parallel universe where the hand played out exactly the same way but the opponent had 3 3 instead, they’d tell us how thrilled we should be to get more money into the pot.
In a televised game in 2009, two world-class players, Tom Dwan and Phil Ivey, played a hand in which the pot size eventually reached more than a million dollars.7 In the hand, Ivey caught a miracle card on the turn to make him a 5-high straight. Unfortunately, the same card also gave Dwan a 7-high straight,* the only possible better hand. “If anybody can get away from this, it’s Phil Ivey,” one of the announcers said, implying that it would be a sign of superior poker talent if he folded. In fact, throwing away the hand would have been a terrible play. Given what Ivey knew at the time, and how aggressively he and Dwan play, he should have expected to have the best hand at least 90 percent of the time. If Ivey hadn’t lost all his chips on the hand, he would have been playing badly.
While television coverage has been a great boon to poker, it leaves many casual players with misleading impressions about the right way to play it, focusing too much on the results and not enough on the correct decision-making process.
“It’s not very common that you can narrow someone’s holdings down to one hand,” Dwan told me. “Definitely much less common than most pros and TV shows would have you believe.”
Making Ourselves Unpredictable
Dwan was once better known by his online screen name “durrrr,” which he selected because he figured it would put the other players on tilt if they lost to him. Dwan deposited $50 at the online site Full Tilt Poker at age seventeen, later dropping out of Boston College to play poker full-time.8 He rose through the ranks to become the apex predator in the online-poker food chain.9 Millions of dollars flowed through him each month; sometimes he lost it but more often he won.10
By the time I spoke with him in 2012, Dwan was widely considered among the best no-limit hold ’em players in the world.11 He has a reputation for being creative, aggressive, and most o
f all, fearless. In 2009, he challenged any player in the world, except for his close friend Phil Galfond, to play him head-to-head at very favorable odds. Three strong players eventually took him on and Dwan won two of these matches.
Yet for all his apparent bravado—Dwan is fairly low-key in person—12 his approach to thinking about poker and the world in general is highly probabilistic. He profits because his opponents are too sure of themselves. “It’s important in most areas of life to come up with a probability instead of a yes or no,” he told me. “It’s a huge flaw that people make in a lot of areas that they analyze, whether they’re trying to form a fiscal union, pay for groceries, or hoping that they don’t get fired.”
Dwan seeks to exploit these tendencies by deliberately obfuscating his play. If the most important technical skill in poker is learning how to forecast your opponent’s hand range, the next-most-important one is making your own play unpredictable. “The better people are, the less certain you’re going to be about what they have or what they’re doing or what their range is,” Dwan says. “And they’ll be more apt to manipulate that to take advantage of your judgments.”
While I’ll never be the player that Dwan is, I took advantage of this in my own way during my days as a poker professional. In the soft online games of the mid-2000s, I could make money by playing conservative, tight poker, but I soon discovered that a more aggressive style could make me even more. The idea was to find the blind spots that my opponents might have in estimating my hand range.
When you raise before the flop, for instance, the opponent will typically put you on big cards like those containing aces, kings, and queens. You will have those hands sometimes, of course. But I would also raise with hands like the ones we were worried about the Lawyer having, hands with small cards like 7 6. What I found is that when big cards came on the board, like an ace or king, the opponent would often give me credit for catching those cards and fold. If smaller cards came instead, meanwhile, I’d often have made a pair or some kind of good draw. Sometimes, I’d even make an unlikely-seeming hand like a straight with these cards, which could send my opponents into tilt. One interesting thing about poker is that the very best players and the very worst ones both play quite randomly, although for different reasons.* Thus, you can sometimes fool opponents into thinking you are a weak player even if you are likely to take their money.
Eventually, some of my opponents caught on to my more aggressive style, but this wasn’t all bad. It meant that they were more likely to call down when I did have a “predictable” hand like a pair of kings, making these hands more profitable for me.
In fact, bluffing and aggressive play is not just a luxury in poker but a necessity—otherwise your play is just too predictable. Poker games have become extremely aggressive since I stopped playing regularly five years ago, and game theory13 as well as computer simulations14 strongly suggest this is the optimal approach. Blitzing your opponent with a deluge of possibilities is the best way to complicate his probability calculations.
Sometimes you may also be able to identify situations where your opponents’ intuitive estimates of the probabilities are too crude. Whenever a poker player thinks that his opponent might never play a certain hand in a certain way—never bluff in a certain situation, for instance—that’s when you have the opportunity to exploit him by confusing his sense of the improbable and the impossible.
“There were a bunch of things I did that I knew were extremely suboptimal but made me a really large amount of money for a long portion of time,” Dwan told me. “It’s only in the last few years people finally started realizing and getting better.”
Dwan’s main game, no-limit hold ’em, is especially fertile for such a strategy because you potentially control, through the size of your bets, the amount of money at stake on each decision. Some choices that Dwan makes involve no more than $100, while others might be for stakes of $10,000, $100,000 or even more. Make a few extra decisions right in million-dollar pots, and the collective sum of what you do for $100 at a time hardly matters at all.
I mostly played limit hold ’em instead, where the betting increment is fixed on each round. (Until very recently, this was the most popular game outside of tournaments; ten years ago, there were often no more than two or three no-limit games running anywhere in the United States.15) Limit poker offers fewer opportunities for creativity. Still, until practice caught up with theory, I had a couple of very successful years by exploiting an aggressive approach. In both 2004 and 2005, I made an income from poker in the six figures, with my cumulative profits from the game peaking at about $400,000 overall.
The Prediction Learning Curve
The difference between Dwan and me is that, while he is willing to take on almost literally any other player for any stakes at any time, I was merely in the upper middle class of poker players and needed to be in a game with some bad ones to be a favorite to make money. Fortunately, there were plenty of these bad players—what poker players call fish—during the poker boom years.
There is a learning curve that applies to poker and to most other tasks that involve some type of prediction. The key thing about a learning curve is that it really is a curve: the progress we make at performing the task is not linear. Instead, it usually looks something like this (figure 10-6)—what I call the Pareto Principle of Prediction.
FIGURE 10-6: THE PARETO PRINCIPLE OF PREDICTION
What you see is a graph that consists of effort on one axis and accuracy on the other. You could label the axes differently—for instance, experience on the one hand and skill on the other. But the same general idea holds. By effort or experience I mean the amount of money, time, or critical thinking that you are willing to devote toward a predictive problem. By accuracy or skill I mean how reliable the predictions will prove to be in the real world.
The name for the curve comes from the well-known business maxim called the Pareto principle or 80-20 rule (as in: 80 percent of your profits come from 20 percent of your customers16). As I apply it here, it posits that getting a few basic things right can go a long way. In poker, for instance, simply learning to fold your worst hands, bet your best ones, and make some effort to consider what your opponent holds will substantially mitigate your losses. If you are willing to do this, then perhaps 80 percent of the time you will be making the same decision as one of the best poker players like Dwan—even if you have spent only 20 percent as much time studying the game.
This relationship also holds in many other disciplines in which prediction is vital. The first 20 percent often begins with having the right data, the right technology, and the right incentives. You need to have some information—more of it rather than less, ideally—and you need to make sure that it is quality-controlled. You need to have some familiarity with the tools of your trade—having top-shelf technology is nice, but it’s more important that you know how to use what you have. You need to care about accuracy—about getting at the objective truth—rather than about making the most pleasing or convenient prediction, or the one that might get you on television.
Then you might progress to a few intermediate steps, developing some rules of thumb (heuristics) that are grounded in experience and common sense and some systematic process to make a forecast rather than doing so on an ad hoc basis.
These things aren’t exactly easy—many people get them wrong. But they aren’t hard either, and by doing them you may be able to make predictions 80 percent as reliable as those of the world’s foremost expert.
Sometimes, however, it is not so much how good your predictions are in an absolute sense that matters but how good they are relative to the competition. In poker, you can make 95 percent of your decisions correctly and still lose your shirt at a table full of players who are making the right move 99 percent of the time. Likewise, beating the stock market requires outpredicting teams of investors in fancy suits with MBAs from Ivy League schools who are paid seven-figure salaries and who have state-of-the-art computer systems at their disp
osal.
In cases like these, it can require a lot of extra effort to beat the competition. You will find that you soon encounter diminishing returns. The extra experience that you gain, the further wrinkles that you add to your strategy, and the additional variables that you put into your forecasting model—these will only make a marginal difference. Meanwhile, the helpful rules of thumb that you developed—now you will need to learn the exceptions to them.
However, when a field is highly competitive, it is only through this painstaking effort around the margin that you can make any money. There is a “water level” established by the competition and your profit will be like the tip of an iceberg: a small sliver of competitive advantage floating just above the surface, but concealing a vast bulwark of effort that went in to support it.
I’ve tried to avoid these sorts of areas. Instead, I’ve been fortunate enough to take advantage of fields where the water level was set pretty low, and getting the basics right counted for a lot. Baseball, in the pre-Moneyball era, used to be one of these. Billy Beane got an awful lot of mileage by recognizing a few simple things, like the fact that on-base percentage is a better measure of a player’s offensive performance than his batting average. Nowadays pretty much everyone realizes that. In politics, I’d expect that I’d have a small edge at best if there were a dozen clones of FiveThirtyEight. But often I’m effectively “competing” against political pundits, like those on The McLaughlin Group, who aren’t really even trying to make accurate predictions. Poker was also this way in the mid-2000s. The steady influx of new and inexperienced players who thought they had learned how to play the game by watching TV kept the water level low.
FIGURE 10-7: THE PARETO PRINCIPLE OF PREDICTION IN COMPETITIVE ENVIRONMENTS