perhaps, you don’t realize just how exciting Rock-Paper-Scissors (warning! Insanity behind link!) can be, and how deep the strategies can run!

Forza Italia!

Ben, what makes you think that? As far as I know there is no theorem

Hi Sean,

The theorem is due to John Nash. An easy-to-find reference is

J. Nash, Non-cooperative games, Annals of Mathematics 54(2), 286 (1951).

It’s on JSTOR but this commenting system seems to choke if I give a direct link. Amusingly, he solves a simplified 3 person poker game in the paper.

A few caveats: if there’s a rake, then the theorem does not apply. If your opponents are irrational, or rational but with limited computational power, then as you say, opponent modeling is desirable.

For an overview of the state-of-the-art in finding optimal solutions to Hold’em, have a look at the introduction to this paper by the Alberta group.

ben

http://news.cs.cmu.edu/Releases/demo/227.html

They will apparently be going up against the Alberta group in a competition in Boston next week:

http://www.aaai.org/Conferences/AAAI/2006/aaai06poker.php

. no decision tree you could unambiguously follow to guarantee the best possible outcome. (Indeed, if you had an opponent that used such a decision tree, you could in principle always beat them.) Unlike in chess, the computer can’t win by brute force; it needs to be clever enough to learn from the previous moves of its opponents to figure out how they are playing

.

In such cases there can still exist optimal mixed strategies, this is what Ben means. In practice one considers strategies that give probabilities for moves based on the information gained from the moves so far. However, this does not yield the most general mixed strategies.

Of course, in practice it’s far better to not to use the brute force approach. It isn’t practical and you should exploit the fact that your opponents are using strategies that are far from optimal.

You paint a much too grim picture of the cosmos…Bear in mind, we are making headway towards comprehending the “quantum card game.” My inspiration is derived from Wojciech Zurek’s work in quantum information theory. Unfortunately, Zurek seems to fall short of “factoring gravity into the quantum equation.”

From Pratchett and Gaiman’s Good Omens:

“God moves in extremely mysterious, not to say, circuitous ways. God does not play dice with the universe; He plays an ineffable game of His own devising, which might be compared, from the perspective of any of the other players, [ie., everybody.] to being involved in an obscure and complex version of poker in a pitch-dark room, with blank cards, for infinite stakes, with a Dealer who won’t tell you the rules, and who smiles all the time.”

Let me rephrase—in all of the examples you worked above, there were hands with competing cards…that is, the diamond flush, that the JdTd would be drawing to, is less likely to come if you know your opponent(s) has (have) a diamond. Just wondering—I’ve never been much for calculating numbers anyway, I generally get a ballpark estimate and go from there. (Perhaps it’s a second order effect.) But I will stick with my original estimates–put the three hands heads up, such that the A7 and 66 don’t take away any diamonds from the deck, and the JTs is 40% (ish), and maybe even a bit better, to win. Then the sixes, then the A7. And not to be picky, but you did say that “a standard hold em table has ten hands” in your original post…

Thank you for the academic discussion on poker—I was wondering when it would come up here!

A major software flaw in online Texas Hold Em was detailed here:

http://www.cigital.com/news/index.php?pg=art&artid=20

The MIT blackjack project was featured on this bbc Horizon program:

http://www.bbc.co.uk/sn/tvradio/programmes/horizon/million_prog_summary.shtml

When they tried it in Europe they got death threats.

A famous case involving online blackjack–a version called Caribbean 21–was when a player calling himself “Pirateofc21″ with an initial bankroll of $1000 blew it up into $1.3 million. Of course, they refused to pay and accused him of “cheating” but couldnt prove it or find any flaws in their own software. I think it went to court and they did pay. (It is worse for a casino’s long-term business to get a reputation for not paying up.) As far as I am concerned though exploiting a software flaw is fair game since that is the software they offer and it is their responsibility for how it functions. After all, they have absolutely no problems about bleeding money off you.

Interesting to learn that there is a poker research group at the University of Alberta. There is a chapter called ” Poker and Bluffing” in the famous book” Theory of Games and Economic Behavior” By Von Neumann and Morgenstern, but it’s a pretty hard book:)

Ben, what makes you think that? As far as I know there is no theorem

I believe Ben is referring to the classic theorem of von Neumann, that any two player zero-sum game, such as heads up poker, has a dominant strategy equilibrium. I’m not sure if the theorem has a generalization to an N player symmetric game. Further, I’m not sure if the game-theoretic idea of an equilibrium is what you want here. Don’t know enough about these things. So you should probably ask Ben.

my impression is that there was always a strategy that could defeat any known strategy

But there are strategies that minimize the maximum harm done to you, and if we consider non-deterministic strategies there exists a stable equilibrium of them. Non-transitivity doesnt change this. Consider rock-paper-scissors. If I’m allowed to only play one, I can always be defeated. However, the strategy “pick a random one with equal probability 1/3″, cannot be “defeated”. I can imagine that in a world tournament of super-expert RPS players, they would all just be playing randomly, since any deviation pure randomness would be punished. This means that rock-paper-scissors gets more boring as skill increases even faster than soccer, which judging by the progression of the World Cup is an impressive feat. I think you can see how this works in a poker situation, and already encodes the opponents behavior in some sense without having to dynamically evaluate it.

Oddly enough, card players appear to confront less uncertainty when playing with “quantum cards” than with “classical cards.” More specifically, the predictability sieve for a deck of playing cards is more refined in a “quantum-game-room” than in a “classical-game-room.” Nevertheless, gravity still remains “the wild card” for both groups of card games…

In the meantime, put gravity aside, shuffle the cards and enjoy the game!

JustAnotherInfidel, I’m not sure I get the question. If it were, say, Ah7c against JdTd, I don’t actually think the numbers would change much, to be honest — there would now be the possibility for a heart flush, for example. Of course, the hand would play out very different in a real game, since two people drawing to the same flush leads to very different betting.

There is no perfect strategy in Hold’Em… Unlike in chess, the computer can’t win by brute force; it needs to be clever enough to learn from the previous moves of its opponents to figure out how they are playing.

This is not true. There is an optimal strategy for Hold’Em: the difference from chess is that the optimal strategy is not deterministic. Obviously, it’s a hard computational problem to find this strategy (just as it is for chess), which is why current poker AI emphasizes opponent modeling.

There was a show on the History Channel, which detailed the MIT effort (led by a mathematician) to play Blackjack. Some new probability algorithm. One young guy was sitting at the pool, with hundreds-of-thousands of $$ in a duffel bag, contemplating his windfall. I think it broke up, after there was internal dissension..greed.

If you get rich, let me know & I’ll help you spend it. There’s a GNP stereo/computer store on Colorado Bl a couple blocks fro Caltech (started by a Caltech grad, who sold stuff to Caltech profs). I hear he got really wealthy by getting into the Stock Market. You need to hook up with that guy, & dabble in some mathematical-based Risk Management (aka “gambling”)

What if the other two hands weren’t taking away some of the outs from the diamond flush (best case scenario)?

As far as online poker goes I have been curious as to whether the software might have “tells”: that is, it might pause a little longer at certain times when it has to go into different or longer subroutines, or compute certain things depending on what hand it holds or what other players do and so on. Also, I have been curious as to how random the shuffling and dealing actually is since computers generate pseudo-random numbers from an iteration algorithm rather than pure random numbers. Any thoughts on any of this?