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 â" and hence, such games cannot be prosecuted under federal laws prohibiting illegal gambling businesses.
Itâs just the latest sally in the ongoing debate over poker thatâs been raging for more than 150 years. And it comes on the heels of a ruling last year by the Justice Department that 1962â²s 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 Playerâs Alliance, is encouraged by the latest ruling by Judge Jack B. Weinstein. âTodayâs federal court ruling is a major victory for the game of poker and the millions of Americans who enjoy playing it,â 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 âexpertsâ and half âaverage,â 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 âEm. Oh, and they fixed the deals, the better to measure the effects of luck.
Their conclusion, per Neuroskeptic: âLuck, 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.â
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 â" 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 âexpertâ 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 âEm 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 âEm and found that the skill of a player âhad a statistically significant effect on the amount of money won or lost.â
The many mathematicians and physicists who are aficionados of poker would agree with Heeb. I wrote a feature forÂ 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 Harveyâs 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. âI donât like gambling at all,â Antonyan told me. âI donât enjoy it and thereâs nothing in it for me to compensate for the clear negative EV decision of gambling.â
Harveyâs not a fan, either: âPersonally I donât like to gamble on games where the house has the odds, but Iâm not critical of people who do.â And while the Time Lord gamely learned craps with me while I was writing The Calculus Diaries (it was research, people!), he hasnât been tempted to play craps since.
Binger doesnât 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, âcheatingâ) 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. âI realized I wasnât going to get rich playing blackjack,â he recalled.
But poker was different: as he studied the game and pondered the underlying mathematics, Binger realized that poker could be a âbeatable game.â
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 âmath skillsâ and âpeople skills,â as he put it. âGood poker requires that you make sound game-theoretic decisions but thereÂ is still plenty of freedom to try and outsmart your opponents,â he said. âOther 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. Itâs 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.â
Math skills help, but thatâs 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 donât know much about the underlying math; they have a good feel, or instinct, for how to play the game.
âThere 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,â said Vonk. âMathematical 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.â
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 donât often occur in a poker game. âTo be a great player, you need both!â Vonk insisted. Chris âJesusâ 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 â90% 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.â 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 â" plus youâre 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 âmake a very broad guess,â he says. Per the Time Lord (blogging way back in 2004):
âTexas Hold âEm is so popular because it manages to accurately hit the mark between âenough information to devise a consistently winning strategyâ and ânot enough information to do much more than guess.â 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.â
To get a sense for how the probabilities can play out, consider the following three possible pairs of hole cards:
Pair of sixes
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 ânother 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? âAgainst only one randomly-chosen pair of hole cards, there is a substantial chance that the sixes wonât need to improve; likewise the ace can often come out on top just by itself, so the Ace-7 is second-best,â Sean explained. âBut against four randomly-chosen pairs of hole cards, chances are excellent that someone will improve, and Jack-10 suited has the best chance.â
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 â" 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 âconnectors,â 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 wonât 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 playersâ behavior. There may not be a single answer. As Harvey put it in the Discover article: âChess 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.â
Harvey admitted that one of his classic errors is âcalling when I think I am beat for other reasons (betting patterns, tells),â but he calls anyway because âthe math says I should. At times like that, I need to pay less attention to the math.â
Human beings arenât always predictably rational, particularly when it comes to poker: if you assume your opponent is skilled and rational, and he isnât, your strategy could backfire and fall victim to âbeginnerâs luck.â
I found this enlightening analysis over at Cardplayer.com, outlining the different between an optimal strategy and an exploitive strategy (Fergusonâs favorite) in No-Limit Texas Hold âEm. (Note the very specific circumstances described throughout: change even one element and it might call for a different strategy.)
âLetâs say youâre playing no-limit hold âem 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 youâre 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 heâll call you.â
âTo 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. â¦ 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.â
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 âbubbleâ hand.
If betting is slow, it might be worth calling, or âlimpingâ into the game, with a mediocre hand, because your chances of winning are pretty good against other mediocre hands (assuming someone isnât âslow-playingâ pocket aces). A bad hand wonât 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 doesnât 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 âbubbleâ 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 â" unless oneâs opponent holds pocket aces. Itâs hard to fold pocket Queens but thatâs just what Harvey did. He correctly analyzed his chances, based on my all-too-predictable style of play. I won the hand, but didnât win much money because Harvey folded before heâd 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 âEm âring game,â and different again for online poker. âThe only people playing online are serious,â said Harvey. âThey also use software to keep track of their opponentsâ statistics, which is consistent with the rules of the site, even if it seems a bit like cheating.â
Hereâs the difference: in a live game, you have to remember/keep track of opponentsâ 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.
âItâs much more about modeling, statistical analysis and game theory at that level,â said Harvey. âIâd have to spend as much time learning and playing poker as I do on physics.â And to date, heâs 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.Â (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 âEm is such a complex system that âwe cannot derive a dominant strategy in a closed form.â
âGame theorists and mathematicians study simplified systems about which they can actually prove theorems,â he explained. This is a decent strategy for two players going heads-up, but for a full table, pre-flop, âit becomes a question of which approximations to make and which models to choose for your opponents.â Physicists, letâs face it, are often pretty adept at choosing the best models.
He also had a corollary: âPhenomenologists and astrophysicists will be better poker players than string theorists.â Take that, Jeff Harvey!
Thereâs a saying that Texas Hold âEm 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 youâll lose; statistical anomalies do happen. In poker, theyâre known as âbad beats.â 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 â" and a third queen appeared as the very last card. His opponent âsucked out on the river.â
Temperament matters too. Poker requires nerves of steel, and an emotional equilibrium that Harvey, for one, admitted he does not possess. âYou need to be unflappable. Bad luck canât bother you. Itâs too easy to get âtilted,â and start playing looser, more erratic, or too passive.â More often than not, he said, âMy emotions get the better of me.â
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 canât. As Dedonno and Detterman concluded in their 2008 paper, âThe reason that poker appears to be a game of luck is that the reliability of any short session is lowâ¦. [O]btaining accurate estimates of poker ability may not be easy. Luck (random factors) disguises the fact that poker is a game of skill. Howeverâ¦. skill is the determining factor in long-term outcome.â
Having a good poker face wonât 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) âPoker Is a Skill,â Gaming Law Review 12(1).
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Ferguson, Thomas, and Ferguson, Chris. (2007) âThe Endgame in Poker,â Optimal Play â" 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) âU(0,1) Two-Person Poker Models,â Game Theory and Applications 12, 17-37.
Fiedler, Ingo C. and Rock, Jan-Philipp. (2009) âQuantifying Skill in Gamesâ"Theory and Empirical Evidence for Poker,â Gaming Law Review and Economics 13(1): 50-57.
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