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!