Physicists Playing Poker

By Sean Carroll | October 6, 2010 4:22 pm

Those of you who haven’t already seen it should check out the November issue of Discover, which features an article by a well-known science writer about physicists playing poker. This is not completely egregious, as big moneywinners like Michael Binger and Marcel Vonk are card-carrying (as it were) Ph.D. physicists. Vonk on the relative merits of hypothetically winning the Nobel Prize or the World Series of Poker: “I would choose to win the Nobel Prize. But, it’s close.”

Of course there’s always much more to a good story than can be squeezed into a print magazine. So if you want the background scoop, see Cocktail Party Physics. Where, unfortunately, I’m (accurately) quoted as saying something in an old blog post that really isn’t true:

“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.”

Two out of three sentences there are correct (which wouldn’t be such a bad average at a poker table, but is pretty lame in writing). The first sentence is right; what makes Hold ‘Em such a popular poker variant is that you know enough to do more than guess, but not enough to easily reduce the problem to a simple algorithm. But the second sentence is wrong, as written, at least under the perfectly reasonable reading that “win” includes “or tie.” One of John Nash’s major contributions to game theory was to prove, under reasonable assumptions, the existence of dominant strategies. Here, it’s not the opponents that are being dominated — it’s the other strategies a player might contemplate using. And “dominate” doesn’t mean “beat under any circumstances”; it just means “there is no alternative strategy that does better against every possible opponent strategy.” Since the rules of poker (integrated over all seats at the table etc.) are the same for every player, every player has the same dominant strategy — which means that there exists a strategy such that, if everyone used it, their expected returns would all be equal, and none of them could unilaterally change their strategy to improve on that expectation. Texas Hold ‘Em is sufficiently complex that the dominant strategy certainly isn’t known in closed form, but it does exist.

What I was clumsily aiming for in that sentence was the correct sentiment expressed in the last sentence. While a dominant strategy is in some sense “least bad” against the complete set of possible opponent’s strategies, it’s certainly not guaranteed to be the best against every specific opponent. If you know that your opponent deviates from dominant strategy in some particular way (not folding enough to re-raises pre-flop, for example), you will make the most money by choosing to deviate from dominant strategy yourself, in such a way as to take advantage of your opponent’s weakness. That’s the idea behind exploitative strategies, as advocated by Chris Ferguson in Jennifer’s blog post. Good poker is all about being exploitative. Any surprise that it’s a popular game among politicians?

  • max

    Sean, it’s great that you plug your wife’s articles and books, but don’t you think you should add a full disclosure note to let us know that she’s your wife? I know standards are different in print than they are in the blogosphere, but this still seems warranted.

  • Sean

    Not especially, no. I like to imagine our blog readers as a group of not-pathologically-uptight people.

  • Carl Brannen

    I’ve had the “dominant strategy” conversation with poker players. They were convinced it didn’t exist in that a computer could never beat a human at poker because the human would pick up on the computer’s strategy. I think they don’t understand random number generators; it’s humans that are the more predictable.

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  • mongoose

    The problem with poker at the top level now (and this is especially apparent at the WSOP main event) is that since its popularity exploded in recent years, bad players massively outnumber good ones, so even though any individual bad (i.e. loose) player is unlikely to win, one of them usually does due to pure luck.

  • Brian

    As a poker-playing physicist, the best book I’ve found for the mathematically inclined is Bill Chen and Jerrod Ankenmann’s The Mathematics of Poker.

  • Brian137

    …which means that there exists a strategy such that, if everyone used it, their expected returns would all be equal….

    I do not play poker, but I thought that some players were better at spotting various clues as to what the other players held – were better at “reading” their opponents emotions, voice and facial inflections, unconscious mannerisms, etc. If this is true, then two players, both playing optimal strategies, might still not have exactly the same prospects. Again, I do not plat poker, so maybe I am off the mark.

  • Dave Bacon

    A good summary of some game theory terms in poker and what they mean for the game is this post from 2+2:

    Also it’s not quite clear to me how to square all of this with information leakage (what Brian137 said): how does the game theoretic optimum deal with “tells”?

  • Brian137

    To further clarify what I meant in the last comment about “reading” an opponent, I meant not so much in terms of what the opponent would do as much as getting an idea what his hidden cards were and how he felt about his situation. Again, since I do not play the game, I could be immersed in unrealistic imaginings.

  • Sean

    Part of playing a dominant strategy is “do not reveal information to your opponents through tells.” If you violate that rule, your opponents will have a way to exploit you.

  • JollyJoker

    @Carl Brannen: They weren’t very good poker players. What they don’t understand is not randomness, but the concept of Nash equilibrium strategies. Any decent player would know about these (and that they’re by definition unexploitable) since there are many forms of the game where the equilibrium can be calculated, there are online tools that do these calculations and they’re being used for training by pretty much all winning sit-n-go players.

    @Mongoose: Bad players are not a problem for anyone that plays sufficiently many tournaments or cash game hands. Sure, the winner of the WSOP main event is unlikely to be a previously famous player, but that’s just good for the game. It’s still easier to win a tournament with 100 bad players than one with 100 good players.

  • Brian137

    Your comment #10 suggests the possibility of a theoretically dominant strategy that no one is capable of carrying out in practice or that some people are more adept at applying. Different players’ prospects might still be different. I would guess also that some players might be better at unerringly employing various cerebral aspects in a timely manner.

  • Jeff H

    Brian137, they are called bots. They’ve already been spotted playing small stakes poker at online sites with a small winrate. My guess is that they will be the death of online poker within the next decade. Live poker is a different matter of course.

  • bittergradstudent

    The slim margin that good players have over bad players is what frustrates me about poker. I don’t want to have to play hundreds or thousands of hands in order to sort the wheat from the chaff. Hold ’em also frustrates me in that the optimal strategy, for 90% of hands, is to throw your hand back.

  • Austin

    gradstudent, unless you’re playing 15 handed holdem, 90% is way too tight in practice and almost certainly not optimal. Today’s good players play 20-30% of their hands in 6-max and 15-20% in 9-max.

  • spyder

    I am fortunate to have a cadre of good friends who are professional poker players. The one thing i have learned is not to play. I don’t have the temperament, nor do i have the mathematical chops to play at a level i would want to play. It is a great lesson, and i have come to enjoy watching the games all the more because of it.

    I did notice, in your much better half’s post, that the mention of money on the table was only stated in passing. I would offer that the probabilities involving money are very important to understanding the game; afterall, it is what the game is about. Thus, the better minds, include in their calculations, the odds of: winning, winning other pots, the ability to get other money into the pots, the amounts of money on the table, the amounts of money that can come into the game (non-tournament play is where the real money is), and the capacity to cover with side bets of all manner of interests.

    Max, all you need to know is “the Spousal Unit”

  • Sili

    Sean, it’s great that you plug your wife’s articles and books, but don’t you think you should add a full disclosure note to let us know that she’s your wife?

    The Carroll-Oullette connection is hardly a secret nor news.

  • puzzled

    And “dominate” doesn’t mean “beat under any circumstances”; it just means “there is no alternative strategy that does better against every possible opponent strategy.

    I am sorry to be a pedant, but that’s wrong or at least a wrong use of pretty basic terminology. A dominant strategy is exactly a strategy that does best, no matter what the opponents do. Most games don’t have dominant strategies. What Nash showed that in a n-person game, there always exist at least one (and maybe more) n-tuples of strategies that are best responses to each other.

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  • Mark Weitzman

    Hi Sean – as a professional poker player dropout from Caltech’s Phd physics program I am looking to get back into physics and get my Phd. even though I am 53. I have spent the last 4 months reviewing QFT books by Peskin & Schroedr, Srednicki, and Zee, and this time unlike 25 years ago I understand them and I am doing all the problems in the books. What do you think are my chances and does anyone know of any Professors/Post-docs/Smart Grad students willing to tutor me for high wages.

  • jim dawson

    I want to go back to Max’s comment. I’m not remotely uptight, but Max was right. Whenever you talk about someone’s work, it is always proper to note any personal connections. And say ‘hi’ to Jennifer, an acquaintance.

  • réalta fuar

    I have to agree with Max and jim dawson on this and I’m definitely not uptight either. Hard to think of a venue where NOT noting personal connections is the way to go (and I suspect the great majority of readers here had no idea of the connection).

  • Sean

    With all due respect:

    1. This is a blog. Among the various features of a blog is an institutional memory; facts that have been mentioned many times previously may be taken as understood between the writer and audience, even if newcomers might need to catch up a bit.

    2. All one has to do is click the actual links to find very explicit statements of my relationship with Jennifer.

    3. Loud declarations that you are not uptight tend to convey the opposite impression.

    Let’s keep the remainder of the discussion to poker and physicists.

  • Paul Stankus

    When you can’t usefully address a question, a well-known strategy is to change the subject up to a higher-level question!

    I don’t know of any unbeatable algorithm for Hold ‘Em poker, though I concur with the Nash argument that there must be one. In fact, Hold’Em is one the simplest poker games, in a tree-branching sense, and so I wouldn’t be surprised if it’s one of the easier ones to solve. The higher-level question is, what procedure would you use to analyze any arbitrary poker-like game to find its unbeatable strategy? Does this lead to a meta-strategy for solving games in general?

  • spyder

    The vast majority of professional poker players (i would say all, but can’t prove it) earn their money from playing non-tournament poker, especially online. Online poker allows the pros to choose to compete as themselves and as aliases, sitting in on tables of other pros and novices. Two of my friends only play online now, because the money is excellent. Poker is, after all, about winning money (even the bizarre side bets break down to dollars when paid off).

    Poker, unlike other gambling games under house rules, can be played without losing money to the house. This makes all of the money available to be won by those playing. There are also several different “official” poker games; some quite complicated, some more easily accessed (one of my professional friends has invented and sold two new game variants and earned a great deal of money just from that enterprise). I do not think that a comprehensive unbeatable algorithm can exist for poker, even for the simplest variants, because the game involves human beings playing from irrational perspectives. But i do think professionals’ dominant strategies perform extraordinarily well when pros play against novices. That is one of the reasons big poker casinos have professionals on call when novice high rollers come to town.

  • bittergradstudent


    There is almost certainly a Nash equilibrium solution for the game–an optimal strategy to use if everyone else knew the rules of the game, and was deeply versed in the strategy of the game.

    This does not mean that someone who knows a bit about human psychology and real-life strategy won’t dramatically outperform the Nash equilibrium. This is demonstrably true for much simpler games than poker, where the Nash equilibrium is directly calculable. For instance, take the parlor game where you ask everyone to guess a number between 0 and 100, and the winner is the one who gets the closest to 2/3 of the average. Very few people correctly guess the Nash equilibrium value of 0. Therefore, when playing this game in real life, guessing zero is probably a poor play.

  • grumpy

    the Holdem game tree is gigantic and only coarse approximations to two-player games can be solved exactly with today’s computing power. There is a rather large CS group at U Alberta working on this and related problems–probably others in academia as well.

    Also, I agree that Sean is saying “dominant strategy” when he probably means Nash equilibrium strategy.

  • bellatrix

    Ah, another Caltech physicist playing poker, maybe you should come to our home game some time :)

    The true Nash equilibrium hasn’t been found for any poker games, although end-game SnG strategy and HU LHE are very close. That nobody plays close to “optimal” makes exploitative strategies much more profitable in poker at the moment. There have been suggestions that there is no true global Nash-equilibrium in poker for 3+ handed games, although I’m certainly no expert in that. The factor of betting sizes makes NLHE, PLO or others unsolvable for now.

    And +1 for Bill Chen’s book!

  • Scott Humphreys


    The idea that bad players are a problem to me is quite silly, i often hear
    poker players complaining when a ‘bad’ player wins a hand. They often
    ridicule someone for poor play and i just want to strangle them. If someone
    is truly ‘bad’ then they are giving their money away to the whole table on
    a long term basis.

    If you are only able to beat people who are ‘good’ (predictable) then maybe
    you’re not as good as you think you are. I am not a good player but i can
    spot a sucker pretty quick, and you have to adjust your game accordingly.
    Ive been at a table and i sat behind a poor player and i played any hand
    that he played regardless of my cards and won pot after pot while ‘better’
    players only played when they had a hand.

  • William Hill Poker

    Hi- Very good writeup. The player i had seen with the brain power exerted on a regular basis at his day job, the challenges of the WSOP didn’t seem to faze Marcel Vonk. The 36 year-old theoretical physicist from Holland. Any way very good site, I’m subscribed to your RSS feed now so I’ll check in more often!


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About Sean Carroll

Sean Carroll is a Senior Research Associate in the Department of Physics at the California Institute of Technology. His research interests include theoretical aspects of cosmology, field theory, and gravitation. His most recent book is The Particle at the End of the Universe, about the Large Hadron Collider and the search for the Higgs boson. Here are some of his favorite blog posts, home page, and email: carroll [at] .


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