The solution is pretty impressive. There are more details here:
http://poker.srv.ualberta.ca/
You can play the bot (which seems to be down) and see how it plays and hand given any action. Interestingly, it doesn't 3-bet pre-flop - presumably constructing balanced ranges with 3-betting as an action is difficult.

However, you can quibble over how "solved" this actually makes the game. A standard test of this was also developed by the University of Alberta guys who built this bot. You search for the"nemesis" - the maximally exploitative counter-strategy to the solution,and see how much that wins against you. (The nemesis may not be very good against strategies other than your "solution" so may get no nearer a Nash equilibrium). The nemesis in this case is expected to win around 0.05 big bets per 100 hands (I think).

So the solution is pretty damn close to a Nash equilibrium.

I think the solution was developed assuming no rake. It may be interesting to re-run the algorithm assuming some level of rake to see if the solution changes much or not at all.

It's not clear to that the game-theoretical approach works multi-way. The key problem is collusion between two players. A classical example of this is in the book Maths of Poker (which is a wonderful book I wholeheartedly recommend.)

Suppose three players are playing 7 card stud. 1 player is all in. The other 2 players have enough for 1 bet each. The situation is on the final street. There is $10 in the pot, and bets are $1.

Player 1 (all in) has KKKK xxx
Player 2 has AAAA xxx
Player 3 has 3456 (all hearts) xxx

A Nash equilibrium is:
Player 2 checks
Player 3 bets when they have the nuts and otherwise checks.
Player 2 never calls and always folds.

Now suppose Player 3 throws a very small amount of bluffs into the mix (e.g. 1% of the time he bets it's a bluff). This may costs Player 3 a small amount of EV: Suppose Player 2 continues to fold. On a very small number of bluffs, Player 2 will hold the 2 and 7 of hearts and can safely call.

In the hands where Player 2 does not call and Player 3 bluffed, player 3 gets his bet returned and Player 1 wins the pot. So the impact of the bluff (assuming Player 2 folds) is to transfer EV from Player 2 to Player 3.

So suppose Player 2 starts to call some % of the time. In this case, on most calls, Player 2 will be wrong and Player 3 will not be bluffing. So Player 3 gains a lot, and again Player 2 loses.

By bluffing, Player 3 is implicitly colluding with Player 1. Whatever he does, Player 2 loses.

This is obviously an extreme example, but it demonstrates the issues with multi-way Nash Equilibria.

Who plays limit poker anyways (kidding)? I only enjoy no-limit because limit is far more dependent on the luck of the draw than the no-limit, man-to-man ultimate battle of wits.

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Thinking like a go player during a game of chess is like bringing a knife to a gun-fight. Thinking like a chess player during a game of go feels like getting knifed while you're holding a gun...

I'm not even aware that there are other several types of pokers. lol

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I'm a Chess intermediate and Go beginner player

Well, there are several main variants of poker (e.g. Texas Hold'em, Omaha, Omaha hi-lo, 7 card stud, 5 card draw, Razz, etc) and many many more lesser known variants. In addition there are several types of betting (fixed limit, pot limit, no limit, etc), which can be used with each variant (in theory at least, in practice some variant - betting -combinations are not feasible). Furthermore one can think of heads-up, short-handed (3-6 players) and full table (7+ players, upper limit depending on variant) as different "types". Putting everything together on can easily list hundreds of "types" of poker.

The original article talks about one specific variant (texas holdem), one specific betting (fixed limit) and one specific table size (heads-up). Hence the term "HU limit holdem"

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I seem to recall reading in one of David Sklansky's books that if you play according to solid math and game theory with a mathematical randomize to determine when you will always have a positive expectation. Not that this would be optimal play, because it ignores the mistakes that other players make and assumes that they will all be playing optimally against you. For example, you will earn much more when playing a table full of tight passive players by raising just about every hand.

Interesting.

Although personally, I have to say that I find such articles suspect, ever since I read the first "Chess Is Solved!!" pieces after the Deep Blue vs Kasparov match.

This particular article, while I can see what they did, I still think that some of the claims are fueled by sensationalism, at least from the mathematical standpoint. For example when they say: "in the long run, if you looked at all the hands that could happen and you averaged all of those, then the computer can’t be losing, at a losing rate — it has to be either breaking even or winning." In real life, this might be so, sure. But theoretically, you cannot make such statements.The best they can say is that they can prove the probram allying the absolutely optimal strategy, that's all.

By disregarding all that, I think that the only really interesting question is: Can I/somebody learn this method and become a "perfect" player in heads-up limit holdem? If the answer is "No", if the method is based on equations and/or data which are beyond human abilities to learn, comprehand, and memorize - then such program is practically useless, no more than a curiosity, not even approproate as a training partner.

If the answer is "Yes" - then we certainly have something important here.

But from what the article says, it seems this is not the case.

PS>
On a tangent - I remember when i was little, I read an article somewhere describing how to create a self-learning "computer" to play a very simple game, something similar to tic-tac-toe. The "computer consisted of a number of match boxes. Each box corresponded to a specific board position, and there were as many boxes as there are valid positions in the game. Inside each box, was a number of matches, each numbered and corresponding to each valid move in the specific position.

You play games against the "computer" by making your move, finding the box corresponding to the resulting position, and blindly drawing one of the matches from the box - this is the "computer"s move.

If the "computer" wins, you play another game. If the "computer" loses, you take the box corresponding to the last position before the loss, and remove the match corresponding to the move which lead to the loss. If the box gets empty, you move to the previous position and remove the match corresponding to the move the "computer" made - until you have a non-empty box.

After a few games, all the boxes/matches left are the ones correspodning to the non-losing positions/moves. And after that, the "computer" cannot lose anymore.

As a kid, this made an impression. It as at the time when computers were only something you read about in SciFi books.

For some reason, the approach the article hinted at reminded me of that.

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- Bantari
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WARNING: This post might contain Opinions!!

i love that machine learning by hand !
could easily be adapted ie to nim games and variations

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In theory, there is no difference between theory and practice. In practice, there is.

The casinos running the tables have solved poker long ago.

Actually, poker only exists in casinos as an attraction to supplement other games. The casino makes very little money off of it because it's a p.v.p. skill game which makes it almost impossible to manipulate. In other words, the casino can't make much money off of it because that would require risking a lot of money as well and they won't do that over a skill game (that's what house-odd games are for where the house is guaranteed to always win in the long run). All this to say, they probably haven't bothered to "solve" poker as its irrelevant.

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Thinking like a go player during a game of chess is like bringing a knife to a gun-fight. Thinking like a chess player during a game of go feels like getting knifed while you're holding a gun...

There is nothing to "solve" for the casino, since it is not one of the participants. It merely takes the rake (either percentage of each pot or hourly fee) and provides facilities and services (table, chairs, cards, chips, dealer, brush etc) for the players.

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That's my point, the casino is not a "player" in the game, they just take a small cut and there are (in the US anyways) quite heavy regulations on how much they can take (usually about 3% of the pot).

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Thinking like a go player during a game of chess is like bringing a knife to a gun-fight. Thinking like a chess player during a game of go feels like getting knifed while you're holding a gun...