08/27/2012 12:41 pm ET

Poker Science: Math, Game Theory Can Help A Gambler's Strategy, Study Says

By Jennifer Ouellette

Last week, a federal judge in Brooklyn overturned the indictment of a Staten Island man who ran poker games in the back room of a warehouse, on the grounds that poker is a game of skill, not chance — and hence, such games cannot be prosecuted under federal laws prohibiting illegal gambling businesses.

It’s just the latest sally in the ongoing debate over poker that’s been raging for more than 150 years. And it comes on the heels of a ruling last year by the Justice Department that 1962′s Wire Act applied only to sports betting, not poker. This is kind of ironic, since the Justice Department also shut down online poker in the spring of 2011, charging the men behind the three most popular online sites with fraud and money laundering.

Clearly, the issue is far from resolved, but John Pappas, executive director of the Poker Player’s Alliance, is encouraged by the latest ruling by Judge Jack B. Weinstein. “Today’s federal court ruling is a major victory for the game of poker and the millions of Americans who enjoy playing it,” he said in a statement. (The alliance is dedicated to decriminalizing poker.)

But wait! Via @Chemjobber on Twitter, I learned of a spanking new study by German researchers concluding that winning at poker is basically all about luck. They recruited 300 poker players, half self-defined “experts” and half “average,” sat them down at tables of six, evenly divided between expert and average players, and then had them all play 60 hands of Texas Hold ‘Em. Oh, and they fixed the deals, the better to measure the effects of luck.

Their conclusion, per Neuroskeptic: “Luck, rather than skill, was key in determining final balance, with experts taking no more, on average, than novices. Experts did play differently, on various measures, and seemed better able to cope with bad luck, losing less; but they also won less when given good cards.”

So are the German researchers correct that poker should thus be classified as gambling? Not necessarily. A 2008 study concluded that poker is a skill — students who received some basic pointers performed better while playing 1000 hands of poker than those who received no training at all. Still other studies support the German conclusion. Who are we to believe?

Neuroskeptic rightly points out a major flaw in the 2012 study, namely, the classification of “expert” players was based on self-reports. I would argue further that playing a mere 60 or 1000 hands of poker is an insufficient sample size, given the statistical complexities of the game. There are 52 cards, with more than 2.5 million possible five-card combinations. Texas Hold ‘Em uses seven cards so there are around 133 million combinations. Plus, you know, fixing the deals really messes with those probabilities.

Compare this to the sample size of the expert witness cited by Judge Weinstein in his massive 120-page ruling. Randal D. Heeb is an economist and statistician (and avid poker player) who analyzed 415 million hands online of no-limit Texas Hold ‘Em and found that the skill of a player “had a statistically significant effect on the amount of money won or lost.”

The many mathematicians and physicists who are aficionados of poker would agree with Heeb. I wrote a feature for Discover in November 2010 on poker-playing physicists, which included the Time Lord (a.k.a. Caltech physicist Sean Carroll, a.k.a. my Better Half), as well as string theorist Jeff Harvey, particle physicists Michael Binger and Marcel Vonk (both of whom have done extremely well on the professional circuit), and a former grad student of Harvey’s named Eduard Antonyan. It spawned an NPR piece for good measure. And I gathered all the material cut from the article into a massive blog post, which dealt explicitly with this question of whether poker is a game of chance or skill.

If poker is a game of chance, and hence gambling, why do physicists love it so much? Physicists hate to gamble. “I don’t like gambling at all,” Antonyan told me. “I don’t enjoy it and there’s nothing in it for me to compensate for the clear negative EV decision of gambling.”

Harvey’s not a fan, either: “Personally I don’t like to gamble on games where the house has the odds, but I’m not critical of people who do.” And while the Time Lord gamely learned craps with me while I was writing The Calculus Diaries (it was research, people!), he hasn’t been tempted to play craps since.

Binger doesn’t mind gambling, per se, but he learned the pitfalls of blackjack as an undergraduate, when he wrote a computer program to beat the game through card-counting (or, as the casinos like to call it, “cheating”) for his senior project. Then he tried to put his strategy into practice. He lost a pile of cash playing blackjack on an ill-fated trip to Reno, and was barred from six casinos in one day for card-counting in a desperate attempt to recoup his losses. “I realized I wasn’t going to get rich playing blackjack,” he recalled.

But poker was different: as he studied the game and pondered the underlying mathematics, Binger realized that poker could be a “beatable game.”

Fundamentally, poker is a game of skill and strategy, not a game of pure chance (although luck plays a role). Vonk has always loved games, but his love for poker rests on the combination of “math skills” and “people skills,” as he put it. “Good poker requires that you make sound game-theoretic decisions but there is still plenty of freedom to try and outsmart your opponents,” he said. “Other casino games miss that second element. All you can do in blackjack or roulette is make the best possible mathematical decisions, and even then, you will still lose in the long run. I have never been attracted to those games. It’s the fact that you play against other people that makes poker so interesting, and that makes it possible to actually be a winner at the game.”

"A Waterloo" by Cassius Marcellus Coolidge, circa 1906. Public domain.

Math skills help, but that’s not all it takes to be a poker badass. Binger said the probability and equity calculations and statistical analysis he applies give him an edge in the game. Vonk finds that his post-game analysis of how he played specific hands benefits from his mathematical skills.

But both Vonk and Binger admit that there are also plenty of other players who really don’t know much about the underlying math; they have a good feel, or instinct, for how to play the game.

“There are many people who hate math but are great poker players, but there are hardly any players who lack the people reading abilities and still manage to be good poker players,” said Vonk. “Mathematical knowledge can to a large extent be replaced by intuition and experience. After a player has played a million hands of poker, even if he does not know the math at all, he will have a decent feeling about when it is profitable to draw to a flush and when it is not.”

That said, knowing the math means you can acquire this kind of knowledge much more quickly, and those skills can give an edge in very rare situations that don’t often occur in a poker game. “To be a great player, you need both!” Vonk insisted. Chris “Jesus” Ferguson is one of the best players in the world, and definitely relies on math and game theory when he plays (his father is a UCLA mathematician, and the two men have written several papers together):

Antonyan estimated that the game of poker is “90% simple math/general strategy, and 10% understanding the dynamics of the table and/or the attitudes of one or more players towards you as they develop.” The math part rests on basic probability theory, and the probabilities of poker are a bit more complicated because there are many more possible combinations of hands — plus you’re working with incomplete information.

Vonk broke down the process to a few basic questions: What cards do I have? What range of cards do I think my opponent has? Given these, what is the probability I will win the hand after all cards have been dealt? And most important: given that probability, will I make money in the long run when I pay the bet? The best one can do, most of the time, is “make a very broad guess,” he says. Per the Time Lord (blogging way back in 2004):

“Texas Hold ‘Em is so popular because it manages to accurately hit the mark between ‘enough information to devise a consistently winning strategy’ and ‘not enough information to do much more than guess.’ The charm in such games is that there is no perfect strategy, in the sense that there is no algorithm guaranteed to win in the long run against any other algorithm. The best poker players are able to use different algorithms against different opponents as the situation warrants.”

To get a sense for how the probabilities can play out, consider the following three possible pairs of hole cards:

Jack-10 suited

Ace-7 unsuited

Pair of sixes

A game of Texas Hold 'Em in progress. Photo by Todd Klassy.

Sean posed this question on Cosmic Variance back in 2006: Which hand is most likely to win if you choose to stay in the pot all the way to the showdown, against other pairs of randomly chosen hole cards? The answer took a whole ‘nother blog post to delineate.

Mathematically, it depends on the number of opponents. The probability that you will win goes down as the number of opponents goes up, because there are more ways for you to be beaten. Some hands play well against very few opponents, while others play well against many opponents. It all depends on the circumstances.

Against one opponent, the sixes will win 62.8% of the time, versus 57.3% for Ace-7 and 56.2% for Jack-10 suited. Against four opponents, those odds are reversed: Jack-10 suited will win 27.3% of the time, versus 20.7% for Ace-7 and 17.9% for the pair of sixes.

Why does this happen? “Against only one randomly-chosen pair of hole cards, there is a substantial chance that the sixes won’t need to improve; likewise the ace can often come out on top just by itself, so the Ace-7 is second-best,” Sean explained. “But against four randomly-chosen pairs of hole cards, chances are excellent that someone will improve, and Jack-10 suited has the best chance.”

The probabilistic outcomes change again if we pit these three hands against each other, two at a time. In that case, sixes are slightly more likely to beat Ace-7, and Ace-7 is likely to beat Jack-10 suited, but Jack-10 suited is likely to beat a pair of sixes.

The sixes are the best starting hand all by themselves. For one of the latter two to win, favorable community cards must appear on the flop, turn, or river. The only way for the Ace-7 to beat paired sixes is for either an ace or a seven to turn up — or, less likely, for just the right combination of four cards to land on the board to make a straight or flush.

Pit those same sixes against Jack-10 suited, and the situation is reversed. In that scenario, there are more ways for Jack-10 suited to improve. The cards are “connectors,” so there are more possible cards that would give low straights (7-8-9) and high straights (Q-K-A), plus the hole cards are suited, making it much easier to make a flush.

So Jack-10 suited will usually beat a pair of sixes. But it won’t usually beat Ace-7 if the ace is of the same suit. For instance, if four more suited cards come up, the Jack-10 suited will have a flush, but the Ace-7 will have a higher flush, and will win the hand.

See? Poker is a very complicated game, even more so once you add in player behavior during the various rounds of betting. If determining the edge and the odds were all it took to succeed at poker, probability theory would suffice, and one could fairly deem it gambling. If it were a purely logical game like chess, it would merely require impressive feats of calculation to determine the winning series of moves.

But poker is a game of limited information, where players must deduce what cards their opponents are likely to have based on their knowledge of the odds and clues from other players’ behavior. There may not be a single answer. As Harvey put it in the Discover article: “Chess is like classical mechanics. Poker is like quantum mechanics. In chess, there is only one right move. In poker, there is a probability distribution of right moves.”

Like Poker for Chocolate: playing for Hershey's minis at Science Online 2012 in Raleigh, NC.

Harvey admitted that one of his classic errors is “calling when I think I am beat for other reasons (betting patterns, tells),” but he calls anyway because “the math says I should. At times like that, I need to pay less attention to the math.”

Human beings aren’t always predictably rational, particularly when it comes to poker: if you assume your opponent is skilled and rational, and he isn’t, your strategy could backfire and fall victim to “beginner’s luck.”

I found this enlightening analysis over at, outlining the different between an optimal strategy and an exploitive strategy (Ferguson’s favorite) in No-Limit Texas Hold ‘Em. (Note the very specific circumstances described throughout: change even one element and it might call for a different strategy.)

“Let’s say you’re playing no-limit hold ‘em against a calling station who never folds pre-flop no matter what the bet is, but will sometimes fold after the flop if he misses completely. He just insists on seeing the flop. Now say you’re dealt two aces and you each have a few thousand blinds in front of you. The optimal strategy is probably to make a small raise, both building a pot and disguising your hand. But with this player in the game, a much better play is to move all in, knowing he’ll call you.”

“To take maximum advantage of this terrible opponent, you need to employ an exploitive strategy. The optimal strategy would still win you money but against bad players, other strategies might win you more money. … An optimal strategy is designed to protect you against opponents who play well. But when we can find ways to do better than optimal strategy against certain players, we do it.”

The article also mentions mathematical/computational giant John von Neumann, who with Oskar Morgenstern (an economist) wrote the definitive treatise on game theory and poker in 1944: Theory of Games and Economic Behavior. It offered an intriguing insight into the art of the bluff: you should always bluff with your worst hand, not a mediocre “bubble” hand.

If betting is slow, it might be worth calling, or “limping” into the game, with a mediocre hand, because your chances of winning are pretty good against other mediocre hands (assuming someone isn’t “slow-playing” pocket aces). A bad hand won’t win unless everyone else folds, so an aggressive raise is the best strategy.

Indeed, there are rare cases where game theory dictates you should fold pocket aces before the flop when playing a tournament. In non-tournament play, the goal is not just to win the hand but to make the most money. In a tournament, you want to outlast your opponents to win it all. That might entail intentionally opting not to maximize your monetary gains on one specific hand to remain competitive in the tournament. You sacrifice short-term gain to achieve the long-term goal.

It doesn’t pay to be too consistent, either. I once played poker with a group that included Harvey. I consistently bet when I had a strong hand, checked when I had a “bubble” hand, and folded when I had a bad hand. So when I scored with pocket aces and nothing but rags (low cards of varying suits) after the flop, I pushed all-in, going heads-up with Harvey.

He had pocket Queens, a strong hand — unless one’s opponent holds pocket aces. It’s hard to fold pocket Queens but that’s just what Harvey did. He correctly analyzed his chances, based on my all-too-predictable style of play. I won the hand, but didn’t win much money because Harvey folded before he’d committed many chips to the pot.

The optimal strategy can also depend on what type of poker is being played: your strategies will be different for No-Limit Texas Hold-Em, for a No-Limit tournament, a Limit Texas Hold ‘Em “ring game,” and different again for online poker. “The only people playing online are serious,” said Harvey. “They also use software to keep track of their opponents’ statistics, which is consistent with the rules of the site, even if it seems a bit like cheating.”

Here’s the difference: in a live game, you have to remember/keep track of opponents’ style of play yourself, i..e, when they raise and in what position. The online software can analyze thousands of hands being played at the same time, and that larger sample space makes for a more accurate statistical analysis.

“It’s much more about modeling, statistical analysis and game theory at that level,” said Harvey. “I’d have to spend as much time learning and playing poker as I do on physics.” And to date, he’s been unwilling to do that, unlike Binger, who left physics after scoring big in the 2006 World Series of Poker. He won $4 million with his third place finish, and more than $2 million since. (You can follow his exploits on Twitter: @mwbinger.)

The mathematicians have had a good run when it comes to analyzing poker, but the Time Lord is (rather cheekily) on record predicting that physicists will prove to be the better poker players in the future. His reasoning? No-Limit Texas Hold ‘Em is such a complex system that “we cannot derive a dominant strategy in a closed form.”

“Game theorists and mathematicians study simplified systems about which they can actually prove theorems,” he explained. This is a decent strategy for two players going heads-up, but for a full table, pre-flop, “it becomes a question of which approximations to make and which models to choose for your opponents.” Physicists, let’s face it, are often pretty adept at choosing the best models.

He also had a corollary: “Phenomenologists and astrophysicists will be better poker players than string theorists.” Take that, Jeff Harvey!

There’s a saying that Texas Hold ‘Em consists of long stretches of boredom punctuated by three minutes of sheer terror. Poker never lacks for suspense: you can play a hand flawlessly from a probabilistic standpoint, but there is still the possibility you’ll lose; statistical anomalies do happen. In poker, they’re known as “bad beats.” Even with pocket aces and a flop of 9-9-2, your chance of winning a heads-up showdown against pocket queens is only 92%. Harvey once faced just that scenario – and a third queen appeared as the very last card. His opponent “sucked out on the river.”

Temperament matters too. Poker requires nerves of steel, and an emotional equilibrium that Harvey, for one, admitted he does not possess. “You need to be unflappable. Bad luck can’t bother you. It’s too easy to get ‘tilted,’ and start playing looser, more erratic, or too passive.” More often than not, he said, “My emotions get the better of me.”

How can you possibly take into account all those confounding factors in a small study involving 300 players and 60 hands of poker, where the deal is fixed? Quite frankly, you can’t. As Dedonno and Detterman concluded in their 2008 paper, “The reason that poker appears to be a game of luck is that the reliability of any short session is low…. [O]btaining accurate estimates of poker ability may not be easy. Luck (random factors) disguises the fact that poker is a game of skill. However…. skill is the determining factor in long-term outcome.”

Having a good poker face won’t hurt your chances either.

Adapted from an October 2010 blog post from the archived Cocktail Party Physics blog.


Dedonno, M.A. and Detterman, D.K. (2008) “Poker Is a Skill,” Gaming Law Review 12(1).

Ferguson, Thomas, and Ferguson, Chris. (2003) “On the Borel and von Neumann Poker Models,” Game Theory and Applications 9 (2003), 17-32.

Ferguson, Thomas, and Ferguson, Chris. (2007) “The Endgame in Poker,” Optimal Play – Mathematical Studies of Games and Gambling, 79-106. Stewart Ethier and William R. Eadington, eds. Reno, NV: Institute for the Study of Gambling and Commercial Gaming.

Ferguson, Thomas, Ferguson, Chris, and Garwargy, Cephas. (2007) “U(0,1) Two-Person Poker Models,” Game Theory and Applications 12, 17-37.

Fiedler, Ingo C. and Rock, Jan-Philipp. (2009) “Quantifying Skill in Games—Theory and Empirical Evidence for Poker,” Gaming Law Review and Economics 13(1): 50-57.

Meyer, G., von Meduna, M., Brosowski, T., and Hayer, T. (2012) “Is Poker a Game of Skill or Chance? A Quasi-Experimental Study,” Journal of Gambling Studies, Online First, August 15, 2012.

von Neumann, John and Morgenstern, Oskar. Theory of Games and Economic Behavior. Princeton, NJ: Princeton University Press, 1944.


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