Bluffing still matters, but the best players now depend on math theory
The World Series of Poker, 2010. Associated Press
More than 6,300 players, each paying an entry fee of $10,000, gathered in Las Vegas early this month for the championship event of the 44th annual World Series of Poker. The tournament ran for 10 days, and just nine players now remain. They will reunite in November for a two-day live telecast to determine who wins the first prize: $8.3 million.
Poker didn’t get this big overnight. In 2003, a then-record 839 players entered the championship for a shot at $2.5 million. The winner was an amateur with the improbable name of Chris Moneymaker. After ESPN devoted seven prime-time hours to his triumph, online poker took off and tournament participation ballooned, as did prize pools. The U.S. government’s ban on the major online poker sites in 2011 reined in enthusiasm, but the game has continued to grow in Europe, Asia and Latin America.
This growth over the past decade has been accompanied by a profound change in how the game is played. Concepts from the branch of mathematics known as game theory have inspired new ideas in poker strategy and new advice for ordinary players. Poker is still a game of reading people, but grasping the significance of their tics and twitches isn’t nearly as important as being able to profile their playing styles and understand what their bets mean.
In no-limit hold’em poker, the game used for the World Series championship, each player is dealt two private cards and attempts to make the best five-card hand that he can by combining his own cards with five cards that are shown faceup and shared by all players. Those cards are revealed in stages: The first three are the “flop,” the fourth is the “turn,” and the fifth is the “river.” Players can bet any amount they like at each stage.
Suppose you hold a pair of sevens, and before the flop is dealt you go all-in (bet all of your chips). One player calls your bet, and everyone else folds their hands. You both turn your cards face up, and you are happy to see your opponent show a pair of sixes. You are in great shape, since you have the better hand. But when the flop arrives, it contains a six, giving your opponent three sixes, and your own hand doesn’t improve, so you lose. Was your all-in play correct?
In terms of results, it wasn’t, because you lost all your chips. But according to the math of hold’em, a pair of sevens is favored to beat a pair of sixes 81% of the time. So if you can go all-in with sevens and get your bet called by players holding sixes over and over again, luck should even out, and eventually you will be a big winner.
Poker theorist David Sklansky once wrote that you should consider yourself a winner as long as you had the higher probability of winning the hand when all the money went into the pot. This attitude is consistent with the underlying mathematical reality of poker, and it can smooth out your emotional reactions to losses and wins. What matters is the quality of your decisions, not the results that come from them.
A few years ago, a young pro named Phil Galfond published a crucial refinement to Mr. Sklansky’s point. He showed that the right way to analyze a poker decision is to consider your opponent’s “range”—that is, the full set of different hands that he could plausibly have, given all the actions that he has thus far taken.
So if, for example, you believed that your opponent would only call your bet if he held sixes or a better pair, then at the moment he calls—before he turns up his cards—you should be unhappy. You want to see the sixes and be an 81% favorite, but you are much more likely to see a hand like eights, nines or higher, and against any of these your likelihood of winning is only about 19%. In fact, against this range of pairs from sixes up to aces, your “equity”—your winning chances averaged over all of those possible hands—would be just 27%.
Of course, in poker, you rarely know your opponent’s range precisely, nor does he know yours. In our example, if your opponent thinks you would never go all-in without at least a pair of tens, he probably won’t call you with anything worse than that. So his calling range depends on what he thinks your range could be.
In practice, this means that you should not make a particular play (such as an all-in bet) only when you have a superstrong hand, because this makes it easy for an observant opponent to deduce your range and fold with all but his own superstrong hands. If you sometimes make a strong play with weak hands—the ancient practice of bluffing—your opponent has a harder time narrowing your range down. This concept, known as “balancing” one’s range, supplements an expert’s intuition about when to bluff with logical explanations of why and how often it is the right play.
Calculating equities for ranges is too complicated to do while you are playing. Today’s top tournament players advise up-and-comers not to memorize formulas but to improve their feeling for ranges by playing with poker calculation apps that rapidly estimate odds by simulating thousands of hands.
Why this sudden leap forward in the strategy of a game that has existed for over a century? Computer analysis has contributed, just as it has wrought changes in backgammon and chess theory. But the real cause of the advances that have accompanied the poker boom has been the boom itself.
With 10 times more people seriously playing the game, the collective creativity and thinking power of the poker world has grown by at least an order of magnitude. The growth of poker theory is a perfect example of how innovation accelerates in interacting communities. Today’s poker players are in a world-wide arms race to discover new ideas and refine their playing styles, led by the younger generation of more mathematically minded pros. And collective progress comes from the application of collective intelligence: Putting more minds to work on a problem makes the discovery of new and better solutions much more likely.
1. Each player is dealt two private cards. The goal: to make the best five-card hand using the five faceup cards shared by all players.
2. Player A gets two sevens; Player B gets two sixes. Neither player knows what the other has yet, but a pair of sevens is favored to beatapair of sixes 81% of the time.
3. After the shared cards are dealt and the players reveal their hands, Player B wins with three sixes, beating the odds.
—Mr. Chabris is a psychology professor at Union College, the co-author of “The Invisible Gorilla: How Our Intuitions Deceive Us” and a chess master. He played in his first World Series of Poker this year.