In many threads about GTO play it's stated that this strategy is unexploitable and guarantees "even" EV at the worst. But is that really true? Isn't GTO in fact the sum of actions that will get you "exploited the least" (against a player who has a perfect style against you), but not UNexploitable. To avoid getting exploited you have to give up some equity not only to an imperfect player (++EV to assure +EV), but also accept a small disadvantage against a perfect player (-EV to avoid --EV), right?

I decided to post here in the HU limit forum because it seems to me that this is the game in which GTO theories has the most relevance. If moderator wish to move it to "poker theory forum" or elsewhere, I don't mind.

I'm by no means an expert in game theory. I've got some rudimentary understanding, enough to answer your question, I think, but take this for what it's worth.

GTO play is an approach to play that leaves an opponent unable to improve his/her win rate by adopting a strategy other than the same GTO strategy you are playing. He may not do any worse with a different strategy, but he can't do better.

In HUHU play, this assures that a GTO player will, at worst, break even (assuming no rake). If two players (A,B) play identical GTO strategies, then they will obviously break even. If player B can alter his strategy to win more, then player A wasn't playing GTO, by definition.

This basically relies on the definition of GTO to prove that it will (at worst) result in break-even results (on average) for HUHU. Proving the existence of a GTO strategy is a different kettle of fish, and much more difficult. I believe the existence of a GTO strategy (but not the details) has been established for HUHU poker through mathematical methods. I could be totally wrong about this. I do not have the faintest idea about the status of proof of the existence of any GTO strategy for 3 or more players.

With the above in mind, the elements of a GTO strategy can be guessed at. One such element is concealment of hand strength. The discussion of balanced ranges is an attempt to conceal information from opponents in order to avoid being exploited on the basis of that information.

I should also reinforce, as in this thread, that GTO is not likely to be positionally static. Rather, it will be a mix of strategies that depend on whether you are SB or BB (in the HUHU case). Neither position is guaranteed a non-negative EV, but the joint strategy is guaranteed at least a zero EV in a HUHU non-raked game (since unraked poker is a zero-sum game).

Edit: Actually, in thinking about the concealment issue a bit more, I probably should have said partial concealment. You give up information about your hand when you check-raise. It suggests a stronger than average hand. But a balanced x/r range will be ambiguous as to whether you have a somewhat strong hand (in the model one-street-2-bet game it's placed below check/call but above check/fold) and a very strong hand (above the bet-call range in the one-street-2-bet game).

OK, but is it proven, mathematicaly (I understand it isn't proven in detail) that there must be a way of playing HU limit that can't be exploited and depending on opponent at least will break even? Or is it more a theoretical discussion? I would have suspected that a strategy which main goal is to lose as little as possible against a perfect player (one who always comes up with the best counter-strategy) must be exploitable, since it's fixed (which in itself sounds non-optimal) and the opponent should be able to adapt. I mean, theoreticaly he knows how you are playing...

I have no objection to the thread but I wonder if the quoted statement is true.

Anyway, I think Bryce's old post is still the best explanation of GT terms found on the forums and perhaps answers the OP better than I can.

I thought it must be more relevant to limit HU because it's here where a true GTO strategy will be discovered first (if ever). NLHE multiway should be both more complicated to solve and probably also uninteresting from a player perspective becasuse even good players make so many theoretical mistakes so a GTO style would be far behind the explotive style (while in limit GTO has more relevance because players usually aren't doing so many obvious mistakes and are generally harder to exploit).

I have read Bryce's post but it didn't give me an answer to my question.

"Game Theory Optimal (GTO): A strategy that yields the highest possible EV (or: “is optimal”) if your opponent always chooses the best possible counter-strategy."

"So long as your opponent always chooses the optimal counter-strategy to whatever strategy you choose no strategy on your part can have a higher EV than this"

Still wonder: will EV be positive (or at least not negative), not only "the highest possible"?

Yes, at least that's the claim in The Intelligent Poker Player (and here). The reference in IPP is to the work of von Neumann and Morgenstern. The claim is basically that any two-person zero-sum game can be played unexploitably by choosing a minimax criterion with respect to betting choices (maximizing the minimum benefit). I don't have the original sources, and the proof might be beyond my math skills in any case(von Neumann was a genius), but I'm convinced that a proof exists. Actually, the claim in IPP is that there is at least one solution for any finite number of players in a finite game.

Not necessarily positive, but at least non-negative.

Thanks, changed it (even though I believe it's almost the same in practice)...

I just don't think it's so clearly evident that HUHU players aren't making a significant number of exploitable mistakes. In fact, they arguably have the opportunity to make more mistakes (though smaller ones than those made in big bet games) since they typically face many more decision points per hand. To say it differently: just because a game has a solution doesn't mean that humans will ever be able to successfully emulate it. In general, I think the degree to which GTO lines have been understood or adapted by HU LHE specialists continues to be exaggerated.

On the other hand, as is pointed out in the Bryce article, being able to identify GTO lines can help you to better find exploitative plays, so knowing something about how to solve certain mini-games should be helpful in almost any circumstance (although, as I've argued elsewhere, this isn't strictly necessary when playing really bad opponents).

From the same article:

Thanks, somehow missed that.

Thank goodness for that! I'm playing nano-SNGs and I don't think I'd win at all if I had to understand GTO.

(Actually, I lie. I've picked up more than one useful concept from GTO discussions. It's just that I'm sure others have picked up infinitely more, so thank the creator for creating enough bad opponents!)

This thread can a win a prize for being the thread with maximum number of smart words that make absolutely no sense. OP and beeker, please read Phil's book, it will help.

??

Actually, I have read Phil's book - most of it - and some passages more than once. Perhaps I'm a slow learner. No - I am a slow learner. But I'm not a BAD learner. I don't see much that I said that could be interpreted as inconsistent with Phil's work. (The early stuff I wrote about not knowing the status of a proof for 3+ players I corrected later itt.)

Any pointers?

I am not sure we're talking about the same Phil's book