Well in the 1-3 game, if you called against a stand pat with the 2 with any frequency at all, then the counter strategy is to only ever pat with the 3, and then the counterstrat has an instant edge. So the "solution" to the 1-3 game is then to never call against a pat with a 2 right?

But then If I have the 1, you either have the 2 or 3, and if I bluff every time I have the 1, you need to call me with the 2 some of the time to gain an edge don't you? But only calling with the 3 should break even.

So am i right that the solution to the 1-3 game is just to never call with the 2?

Thanks! I'd been trying to dig up his contact info. I just PM'ed Mike Johanson of U of A.

It doesn't adjust to your play. One reason to use it is that IGT might throw in some freeroll money for the publicity.

I know it doesn't adjust, I just don't know much else about it. Who developed the AI? Has it ever been entered in a competition? Have any academics done an exploitability analysis on it? I know it's been seen making some obviously poor plays though I do have the impression that it's pretty strong overall.

This thread is surprisingly excellent.

if ever a thread deserved a its this one.

Just because we don't know the solution to a problem doesn't mean that it doesn't exist. It can be proven that there exists a GTO Nash equilibrium for HU NLHE games - we just have not found it yet.

It can also be proven that there exists no exact solution to 3+handed games, but that there are good approximations that can lead to solutions.

Edit: darn, answered to early, people have already said what I said, sorry. Interesting thread, nonetheless!

GTO does not mean "most profitable". Sadly the O in GTO confuses many people.

inb4 obama posting itt

Didn't Hoss_TBF play a bot a few years back for x amount of hands. I can't find the details but i think it was this program

http://en.wikipedia.org/wiki/Polaris_%28poker_bot%29

I'm not an expert on game theory, and I have pretty much no draw poker experience, but I think there are some flaws in your logic here. First, you seem to be assuming that a GTO strategy is going to be very passive (either it calls down or it folds) when in reality it will probably be quite aggressive and putting you in lots of tricky spots. Second, you're neglecting what happens with the rest of your range when you don't pat all of it. Suppose you only pat with good made hands and never bluffs: it is entirely possible that, in hands where you pat, you win money in the long run vs. GTO play. However, because you are never patting with bluffs, you have more air in your non-patting range, it becomes weaker, and your strategy, as a whole, loses money.

Conversely, suppose you pat every hand. You will probably successfully bluff a GTO player out of more pots than you should. This doesn't mean that you are beating it, however, because when it *does* call you down, your range will be so weak that you are getting killed, and, on the whole, you are losing money.

In other words, it's easy to tune a strategy to beat, or even crush, GTO play in certain specific scenarios, but by doing so, you hurt yourself in other scenarios in a way that doesn't favor you. It's a bit like how a maniac will shove all in with air ten times in a row and keep picking up the pot and laughing to himself about how exploitable everyone is... until the eleventh time when someone wakes up with a hand and he bluffs his stack off.

Another Mike Johanson quote (from the same HULHE thread cited earlier):

It's really important to appreciate the distinction between "We know a GTO strategy exists" and "We know what the GTO strategy is". It is known that there is a GTO strategy for heads-up poker (as sauce and ike explained, that is Nash's proof). Additionally, because poker is a game of hidden information, we know that the solution is a Nash equilibrium in mixed strategies, in other words, what the strategy gives us at each step is a weighted distribution of actions which we sample from randomly (I don't remember the name of the proof of this and can't be arsed to look it up). We do not currently know the GTO strategy for any real heads-up poker game. If we did, the game would be declared "solved", as has happened for RPS, noughts and crosses, and less-trivially drafts, for example.

Now when people say "would a GTO bot adjust"? A lot of further misunderstanding creeps into the debate. A GTO strategy for a game is by definition a total strategy for the game - that is, it specifies exactly the optimal thing to do at every possible decision point in the game based on all available information. So it would of course do something differently in a draw game if you drew two versus if you drew one - it would do the optimal thing in each case.

Ok, finally read this long thread to the end. And a few very good posts in there, seriously!

a) As many have noted, durrrr is completely wrong. You can prove that there are (is a) solution(s) to a theorem without solving it. I do hope he takes on a bet, though.

b) Game Theory Optimal in general terms for poker sakes can be summarized as the balanced strategy that maximizes value. In that sense, you strive for unexploitability from your opponent. We had this discussion some time ago when people were wondering if c-betting 100% HU in LHE was GTO (probably not, but it is at least balanced and unexploitable). http://forumserver.twoplustwo.com/35...ology-1218514/

c) afaik (and I am NOT up to date with this) there still isn't a complete GTO solution for HU LHE even, though there are pretty large solution branches of the game tree tackled. Naturally it follows that NLHE HU has not been solved.

ok so GTO is a strategy employed in a way which isn't exploitable no matter what the opposition does? for some reason i just can't get my head round this even being possible, i get that 2 player games like chess is solveable (no incomplete information/luck involved?) but how is a hu game like poker which includes incomplete information (how does it calculate card removal? -mucked cards/villain's holecards) wouldn't this be important in figuring out what the most optimal line would be in any given spot that isn't exploitable?

also how do you confirm that a bot is actually playing 100% perfectly without a human actually knowing what a GTO game looks like?

(assuming this lhe bot is really good which i guess is the case ) won't the biggest edge be the bot/computer playing their A game (this "apparent" GTO strategy) no matter if it's losing or winning? is 100k hands even enough to determine if this GTO bot is even profitable vs a human or vice versa?

I was definitely wrong. Thinking about things in terms of state was confusing the matter. The state itself doesn't change but the action of discarding introduces new strategies that could potentially exploit a strategy that does not consider it.

I think a lot of other people were also confused about this and it's not easy to explain in a fashion that gets by the intuitive but incorrect issue that it must be making 'assumptions' about your range if it does respond to somebody drawing. But it's not really making assumptions about your range so much as compensating for the fact that the the max potential strength of your range has been increased.

I think the stand pat issue in draw is actually a great way to elaborate upon that. If I'm not mistaken them we get the very counter intuitive result that the GTO play for the bot against a person who stands pat would be very near identical to somebody who draws 5. But further more there seems to be an even more counter intuitive issue that the bot may need to actually treat the person's range who draws 5 as very slightly stronger than the person who stands pat since if the person only draws 5 when they have unpaired/low cards below 'x' then card removal would result in a presumably very slightly stronger random 5 card draw versus the initial deal. I realize that sounds crazy so I'd appreciate some feedback there as I'm not exactly ignorant on this topic but obviously also not quite educated enough!

Durrrr is obv wrong itt: poker is solvable; a GTO solution exists (though we haven't yet found it); you can't beat it by "i can play similarly back but exploit occasional spots and you won't be sure if I'm doing that"

I would be very surprised if Galfond or Jungleman came in here and confirmed this: Not that I doubt durrrrs integrity, I just think there must have been some misunderstanding, perhaps caused by the lack of understanding this demonstrates: btw, on the final point in that quote, incomplete information doesn't prevent a game from being solved, it just means that the GTO solution will include randomization.

Regarding the challenge, 2 possibilities:

(1) Bot strategy is not disclosed in advance: Durrrr has (ignoring variance miracles) no chance (don't waste your money!)

(2) Bot strategy is disclosed to Durrrr (a reasonable time) before the match: Durrrr can beat it for a small amount (ex rake) if he bothers to do the work (and has the processing power) in advance to identify the optimum counter-stategy (this assumes he is allowed to use computer help when he plays btw).

I'm not sure exactly how one goes about proving that a strategy is GTO, but I imagine it involves absolutely immense amounts of computerized number crunching to analyze every possible situation. HU limit is close to solved because it has relatively few situations, but the number explodes exponentially as you add in more players or variable bet sizes. I think it's likely that we won't see someone solving GTO NLHE within our lifetimes (though we will probably see bots more than good enough to beat the best humans).

This is like betting you can make one and one add to two.

I think that's wrong. Someone who has a legitimate pat hand (straight or better) will never draw, which skews the range for a pat hand after the initial deal upwards enough to offset the card removal effect in any mixed strategy that includes either standing pat or drawing.

Interesting discussion btw.

You don't have to consider specific opponents when playing GTO. You only have play different depending on what your cards are and te bet to pot size ratio.

Edit: and your opponents previous action

... and a few other factors

durrrr i like you but please don't waste your money against the limit bot. you stand no chance.

wow this thread really delivered.
and is uber interesting

I understand there are still very large simplifications/groupings to reduce the game tree by orders of magnitude.