I have seen countless misunderstandings of GTO (game theory optimal), which is totally understandable; it is a difficult concept. Thus I decided to take a shot at explaining it as clearly as I can while also being comprehensive. If I missed anything let me know!

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The key to understanding GTO is this: a GTO player can tell you their exact strategy and you still wouldn't be able to beat them in the long run because their strategy is perfect.

For a concrete example, let's say you open BTN to 2.5bb, I can tell you this is exactly how I'm going to react preflop. Let's say I call, the flop comes As9c8s, I check and you bet halfpot, I can tell you I'm going to do exactly this on the flop- for example if I have A9 I will mostly raise but sometimes call. Even though you know my strategy I am indifferent, I am going to play it regardless of what you do and you cannot beat me (OK this old sim might be inaccurate, just ignore that part. I'm quite dubious about the K9 fold but that's not the point of this post).

"OK, but what is GTO?" In poker we basically always use GTO to mean Nash Equilibrium. The equilibrium here basically is that all players are playing perfect strategy, and anyone who deviates from playing perfectly loses. The "perfect strategy" exists in pretty much every game and poker is no exception.

"Can anyone play GTO?" Nope! No one can play GTO (except at very shallow stacks) because it is way, way too complicated for our puny human brains.

"But solvers can play GTO?" Not perfectly. GTO is too complicated for our puny 2021 computers to solve (according to wikipedia, Checkers is the most complicated game to date that has been solved, and it took 18 years to do it). The solver just approximates GTO- for example, most commercial solvers will use static bet sizings determined by the user as a computational limitation and thus may differ from each other. That being said, the solver approximation of Pio, Monker, etc. is very good- for practical purposes, poker players will often just say that these solver outputs are GTO (and I will do so for the rest of this post).

"Just how good is GTO?" Way better than humans. We don't have rankings in poker, but Libratus beat some of the best HU players in the world in 2017 over 100k hands at a ~15bb/100 winrate. Again, Libratus is not GTO, just a really strong approximation. Using Chess as a comparison, Stockfish14 wins 4x as many game pairs in Chess vs. Stockfish13, and it was only released a mere 4 months after (July 2021 vs Feb 2021). Either one of them can already crush world champion Magnus Carlsen- the ELO gap between Stockfish to Carlsen is significantly larger than Carlsen vs. the weakest of the ~1300 chess grandmasters, which is already very large. From my research it seems that there is some debate on whether poker or chess is more complicated, but regardless of which one it is, they are both in the same ballpark of being super, super, super complicated. So in summary, the solver is already infinitely better than humans at poker but is likely to be improved even more in the future.

"I can exploit/beat GTO by doing XYZ!" No, you cannot. This statement is so absurd I almost didn't include it but I hear it frequently enough that it is worth addressing. It's human nature to think you are the **** but I'm sorry you are not that smart, in fact I would say you are extra not-smart if you believe this. Again, look back to the key concept. I can play against you and show you exactly what my solver says to do against any action you take, you can take as much time as you want to study it and make your decision to try to beat me and you will still lose (although, I imagine anyone who actually attempts this exercise will just end up quickly changing their tune and choosing to replicate the solver themselves).

"I still don't get it, if they tell me their strategy won't it be easy to call their bluffs or bluff when they are weak?" No. GTO is balanced by betting a perfect mixture of bluffs and value and also defending the perfect amount against bets so that it is indifferent to any strategy you choose to use, it can safely just do its own thing. As a minor addition to this point, GTO often uses mixed frequencies so you can't just know it won't have a certain hand just because it checked the flop or something (but even this isn't entirely necessary).

"How does GTO win money if it is indifferent to any action I take? Won't it just break even?" In my experience this is the #1 most common hangup people have with understanding GTO. I'll use Chess as a comparison again. The Chess engine Stockfish makes essentially the perfect move, it is indifferent and doesn't care what you do in response, it will still beat your ass because you're not going to make decisions as good as it can. So going back to poker, the solver saying it is "indifferent" is basically it saying "I don't care what you do, you can't beat me", but you can definitely still do things that the solver beat you even harder, such as open jamming 72o. Another thing that you can do to make the solver beat you even harder- and thus win lots of money against you- is by simply playing poker like a normal, or even elite, human player

"GTO only exists in heads up poker, not 6-max." Technically true if you use the definition of GTO as "an unbeatable strategy", it doesn't exist in multiplayer due to collusion. However, Nash Equilibrium does exist, and using that as a baseline is a good starting point. Again, if perfectly replicated it will win at an almost unfathomable bb/100 in any lineup where your opponents aren't all poker experts colluding against you. Of course, in heads up it is literally unbeatable in any way to defeat a GTO player.

"But I can make even more exploiting the drunk fish than by playing GTO!" Technically true. If you played deepstack against my family, you will definitely make more bb/100 by playing 100% VPIP than by playing GTO preflop ranges. However if you're playing for real money at any realistic poker game, a GTO player will make more than you because its baseline winrate is so incredibly high. That being said, if you mixed the two and had a human assist the bot by making exploits such as folding AK to a supernit's 3-bet in a spot that the solver would continue, you could indeed improve it's strategy to be better than pure GTO. Worth mentioning, yes the solver does poorly in this exact spot by calling with AK vs AA/KK but will still crush the supernit in the long run for all the times it doesn't get 3bet because it is playing a range, not a single hand. In reality this can be dangerous tightrope walk because you SHOULD strive to exploit other players (since you cannot replicate GTO), but it is a much more common pitfall for a poker player to incorrectly deviate from GTO rather than incorrectly sticking to it.

"In spots where GTO uses a mixed frequency, it doesn't matter what I do?" Partially correct. In a vacuum, it doesn't matter- but again, the solver has to play a range, not each hand in a vacuum. Mixing is necessary for the GTO strategy to be unbeatable. I generally hate the comparison to Rock Paper Scissors because it tends to lead to lots of misconceptions (such as the misconception that it doesn't matter what you do), but in this case I will use it. So similarly, in a vacuum it doesn't matter what you throw in RPS, however mixing is obviously necessary you can't just play rock every time like Bart Simpson and say you're playing GTO. Remember, you should be able to tell your opponent your exact strategy and they can't beat you (so in this case I tell you I'm gonna throw rock/paper/scissors at 33/33/33% frequencies, good luck beating me!)

In practice, if the solver is doing something like checking 90% of its combos of AK on a certain flop texture and betting 10%, it can be beneficial to simplify your own baseline strategy to just checking 100% of your AK.

"How can I improve my poker play by using the solver?" In my experience this is a more of a UFAQ than a FAQ- an unfrequently asked question that needs to be asked more often. Solvers are misused constantly- I've seen so many people try to copy the results without really understanding why. I'm going to use an example from PLO here that I think clearly demonstrates why this is really dangerous. So someone might solve a KcQs4s board and see that it is cbet 90% of the time after 3betting from SB vs BTN. Then they might easily think "OK, so I'm just gonna cbet 100% on King Queen high boards, got it!" and implement that into their game. But then they get into a spot where they raise CO and get called by BTN and cbet that exact same board. OK here the problem- that spot is cbet 0% by the solver. In the first scenario you have a lot of KK/QQ and your opponent's range is very wide, but in the second scenario your opponent actually has just as many KK and QQ as you, and is in position with a SPR greater than 10%. Then they lose a lot of money misplaying the second board. Poker strategy is outside the scope of this post, but as a generalization if both ranges are roughly equal in equity and polarity and stacks are deep (such as in a single raised pot at 100bb), the OOP player will check the flop extremely frequently, which is why the second board is cbet 0%. So in summary here, the key to studying with solvers is to understand why they are doing what they are doing, not trying to blindly replicate the outputs.

Additionally, setting the inputs for the solver has a lot of room for error as well, see my example sim above on As9c8s where it was folding K9, that was because I didn't set the inputs properly so it gave me a bad output.

"How is GTO calculated?" The details are complex but I'm going to steal this post from tombos21 from 2+2 for this:
Start with two players, A & B, and completely random strategies.
Player A assumes player B's (random) strategy is fixed and adjusts to exploit it.
Now fix player A's new strategy and let player B exploit it.
Now fix player B's new strategy and let player A exploit it.
Repeat until equilibrium.

"Is there more than 1 GTO strategy?" Maybe someone else knows the answer but I don't actually know if we can prove it one way or the other in theory. In practice, the answer is yes- you can set a solver to play a flop betsizing of 75% or 33% in many spots and the EV of either one will often be very close to the same, so you are free to study and implement whichever one fits your playstyle more (although it would be good to know both since your opponents may lead you down the other game tree).

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I tried to make everything easy to understand while also being accurate enough, of course there are plenty of nuances I could not cover with this "short" of this post but hopefully this helps some people clear up any misconceptions they have of GTO. I'm almost certain I missed some stuff too so might add more later.

Good post!

The answer to "Is there more than 1 GTO strategy?" is yes. For example, on 22333 with P1 having {AA,JJ,TT}, P2 having {KK}. There's ton of combinations of how to reach optimal bluff ratio with {JJ,TT} together

I do have a disagreement though with the emphasis on how playing GTO (Nash Equilibrium) in a real game for real money will overperform your whacky exploits.

GTO brings something really cool to poker which is objectivity, something that exploits lack, but ultimately, the best players are good at roughly navigating the GTO framework but excellent at deviating and punishing everyone elses mistakes.

At the end of the day, it's just a matter of where you put the emphasis, and I think the end goal should always be to take the highest EV decision in a vaccuum, while you use all of your GTO knowledge to disguise your strategy enough to not get caught.

I do agree with this as well perhaps could have worded that exploit section better

One thing that I have learned about poker is that there are very few absolutes, even when people try to bring solver work into the mix.

Obviously not folding AA pre-flop in a cash game is an absolute.

However particularly post-flop, there can be a rationale behind betting a particular hand, but at the same time there can be just as much logic and justification to check it instead and let your opponent do the value betting for you, and it is hard to know necessarily which answer is correct unless you know your opponent's strategy.

The same thing with playing draws passively, as while you lose fold equity, you get to draw for cheaper and hope to get there in smaller pots.

I’ve read various summaries of GTO over the years (I believe) as well as more extensive explanations (I know). This reads well as a summary, for sure, especially for someone who doesn’t directly use a solver, but applies solver-based strategy. The K high flop/cbet example for PLO is one of the more counterintuitive ones, which I struggle with in game, me still wanting to cbet as CO v BN. In fact, the mere idea of a 0% cbet frequency in this spot makes me realise how many gaps there probably are in my strategy, overall. Much of the skepticism of GTO likely derives from the fact that players realise/believe a solver will show how much they don’t know, which, in turn, leads them to devise shabby heuristics to justify this lack of knowledge.

I appreciate all of the time and effort you have put into this. You are obviously very knowledgeable and no doubt an excellent player.

"GTO is perfect, but no one can play it and no one does play it." Agreed. And this doesn't mean we can't learn a lot from these sims. I think GTO information is a very valuable learning tool and should help guide our strategy but I agree that no one actually plays GTO. Nor do I think we should play a strict GTO game.

You say "the perfect strategy exists in every game and poker is no exception." Perhaps, but it is true that at this point in time we do not know what the perfect strategy is in a true, multiway poker game with infinite bet sizes and different stack depths, i.e., a real game, correct? Suffice it to say that a complete solution to NLH has not been found.

Chess is not a good comparison to poker for game theory solving purposes for 2 reasons. Chess is a game of perfect information, poker is a game of imperfect information. There is a limit to the number of moves available in a game of chess that makes it easier to solve than multiway NLH.

You say, "you can try to beat a GTO player (by deviating from his GTO strategy) and you will still lose." First of all, different GTO players use different GTO strategies. You can have 1 player running stats for a 2.5x open raise and another guy running stats for a 3.5x open raise and they will have different strategies but they will both tell you they are playing GTO. Second, as we already covered, no one actually plays GTO. Third, if you don't play differently from your opp's GTO strategy, you are just supposed to break even, no? Fourth, when you say "lose" you mean lose EV, right? If I deviate from your GTO strategy and refuse to play as your precise strategy pair, then I lose EV, but I may win actual money. You may win EV all night but lose actual money.

I have got to go. I'll add some comments on the rest of this later. It is fascinating stuff. I am really just playing devil's advocate. No disrespect intended. Quite the contrary, I have a lot of respect for all of you knowledgeable GTO folks.

Really good post overall. I would only add two things:

First, to what you said about not knowing if we can prove whether there is more than one Nash equilibrium in theory: I am almost certain that it is far beyond humans' current theoretical capability to know whether more than one Nash equilibrium exists. But one important comment about that as well. In HU, even if there is more than one Nash equilibrium, they must have the same EV for both players. This is a mathematical fact. So if you are playing HU, and you could somehow know that your own strategy is GTO, there is no reason to worry that someone else might be playing "different GTO." In multiway, however, this is far from the case, at least in theory.

Secondly:

The bolded is sort of true but is not the whole story.

Sort of a semi-example/thought experiment here is let's say you sit down in a 6-handed game and somehow you and all your opponents know how to play GTO. So you all play GTO and you all get some fixed EV per orbit. Let's just simplify and assume it's 0.

Now let's say that one of the GTO players gets up and a new player sits down who does not play GTO. You and 4 of your opponents continue playing GTO and this player deviates. Let's even say this player deviates in a way that causes them to lose, which in theory is not guaranteed--that player could break even. So just as a thought experiment, say that player loses 3 big blinds per orbit.

I think it is obvious that those 3 big blinds will not necessarily be distributed equally to the winrates of the GTO players at the table, right? This would be because of positional assymetry. So your new winrate will be a function of both how the player plays, and what seat they're at.

What I think is not obvious is that it is theoretically possible that one or more of the GTO players' winrates could also go below 0 once this new player sits down. And no collusion--not even a second non-GTO player--is required for this to be true.

More concretely, instead of 0/0/0/0/0/0 with all GTO, it could now go 1/.5/.5/.5/.5/-3, but it could also go 3/1/.5/-.5/-1/-3. Both of these--and many others--are theoretically possible. And again, this is before you have a second player deviating, and before you consider possible collusion.

Yeah good clarification if they deviate from GTO to a different GTO in heads up, you guys both break even. Indifference is the key here, both players are indifferent.

I shared this on reddit as well (with most of the comments there being so, so wrong, so good job to 2+2), but u/usernamchexout rephrased what I said in an eloquent way IMO so I'll share that here:

We don't know 100% yet that chess is a draw, only 99.9%. I just wanted to use the analogy anyway. Even if it isn't a draw, then a 2-game match would still be a tie if both players alternated being each color. The bot would go 1-1 vs itself, whereas a human vs the bot would go 0-2 or 0.5-1.5.

Since poker is at least as hard to solve as chess, your chance of holding your own against a hypothetical gto bot is the same as your chance of doing so vs Stockfish, or worse.

The one difference is as you mentioned poker does have randomness. So you can win a hand literally no matter what you do, whereas you can never win a game vs. Stockfish. However, as poker players we can't control the randomness, so we like to consider "winning" the hand whoever plays it better, not whoever gets pushed the chips in the end, and in that sense, GTO always wins vs. a human.

Yep- again the key is indifference. I intentionally made this the only word I bolded besides the subheaders/intro in the entire post to stand out, but of course the post is super long and I probably should have put this at the top. But yeah both GTO players are indifferent to their opponent's strategy.

Regarding the multiway stuff, I used collusion just as an example to make it more clear why an "unbeatable strategy" cannot exist in multiplayer, but everything you said is also true as well. Looking back at it I definitely should not have worded it that way ("due to collusion"). Great clarification

You say, "However if you're playing for real money at any realistic poker game, a GTO player will make more than you because its baseline winrate is so incredibly high." I am skeptical about claims of incredibly high winrates.

How do you measure a baseline winrate for a GTO player? What do you think a GTO player's winrate looks like, approximately, as compared to very good winning players who don't play GTO?

I can tell you that we see posts on 2+2 in which players are complaining about playing "perfect" GTO and not showing a positive winrate. These are the "what am I doing wrong?" type of posts. The player will post screenshots of their stats and say "see I am VPIP'ing 21/PFR'ing 18 and 3 betting 6% (or whatever the stats may be) just like I am supposed to be doing, how come I am not winning?"

We also regularly see posts in which players claim winrates over small samples such as 10k hands and very good veteran players will say, "the sample size is meaningless. If you can make 4-5bb/100 over 100,000 hands, you are doing well."

So, what is an incredibly high winrate that should be expected by players attempting to implement GTO strategies? Are we actually seeing this, in reality? Is this verifiable, somehow? Or is it just anexcdotal stories about Doug Polk or whoever? Thanks. I am enjoying the conversation.

That isn’t an answer to the question. You don’t know that this spot occurs in the game tree with those ranges.

I think the answer is likely yes for most NLHE formats and stack sizes due to that type of situation, but it’s not certain.

Personally I would guess if you replicated Monker/Pio/GTO+'s strategy using reasonable sizing options in the 2/5 NLHE Zoom pool on Ignition it would win at 30bb/100. No one really knows for sure though, and in no way is my opinion an authority here. What we do know for 100% certainty is it will be higher than any human player in that pool since it is a reasonably tough game (but as stated in the OP, could have a winrate lower than a human player in a very very soft game).

I have to say sorry to those guys, but anyone who is claiming to play perfect GTO is not very smart, period. It's no wonder they are losing at poker.

agree entirely with this post

there's absolutely no chance GTO would win at 30bb/100 at midstakes, maybe 10 to 15 and i think it's honestly pushing it given that gto is going to be paying off nits constantly and isoing fish with AQo+. I think it would more realistically win at lower than 10bb 100 and maybe even lower than the best regs of the stake.
Of course the bot would rape those players whenever they get into a hand against each other, but that's not where most of the winrate is made in midstakes

30bb / 100 is overestimated. There is actually research where professionals have played against bot. And bot had 5bb / 100 WR vs 5 pros. (https://www.vip-grinders.com/super-p...uribus-locked/ )
This is without rake, so I wouldn't be surprised if an extremely strong pro can overperform a solver in real 6-max poker.

Great post, Aesah! Can we get this stickied?

I want to add something about mixed strategies that was enlightening for me.

When a GTO strategy mixes a hand between two actions, those actions theoretically have exactly the same EV (because, why would you intentionally choose a worse strategy?). The reason it mixes is to remain unexploitable against all possible counter-strategies.

If you were playing a fixed GTO bot, and your hand was theoretically 5% fold, 30% call, 65% raise, you could pick either fold/call/raise 100% of the time and never lose! This is because GTO doesn't adjust to exploit your imbalances. An adaptable opponent could adjust to exploit you and gain from these mixing mistakes. So we can classify mistakes into two different categories:

1) Pure mistakes - Loses EV against a fixed, unchanging GTO strategy. In other words, playing an action that ought to never be played.
2) Mixing mistakes - Only loses EV against opponents that adapt to exploit you.
Read more here

Of course, there can always be exploitative justifications, so the word "mistake" is relative. But that's beside the point.


To understand this, you need to realize that if your opponent is playing a fixed strategy then your range doesn't matter at all. You are simply playing your hand in a vacuum vs their strategy.

Here's an example.

To quote Aner0:

I've studied Pluribus's play for many hours years ago . Apparently Pluribus was developed in 8 days and costs $150. Analogously, I think this is equivalent at best to Deep Blue in Chess which was ~2700 ELO. For comparison Stockfish 14 is ~3600 ELO. So I think of Pluribus as 2700 ELO while true GTO is 3600+ ELO.

I must caveat again as I have already earlier in this thread on this topic, I absolutely am not an authority on this and this is just my guess. However the one thing I do feel very strongly about is that you cannot extrapolate to say "GTO will not crush because Pluribus did not crush".

Yes, I oftentimes hate the RPS comparison since it leads to many misconceptions, but its a good example here where we actually can play rock 100% of the time vs. a fixed GTO RPS strategy

I don't see why we can't. If you have something that plays close to GTO you hit the point of diminishing returns and you gain very little EV by getting closer to true gto.
You can see this with solver using 3 or 17 sizes dosen't gain much EV.
You have any links that confirm that cost and time of building?

What Aesah is saying (and I agree with) is that we actually have no idea whether the bots that currently exist are playing close to GTO. So we do not know how close to diminishing returns the bots are.

While this is true, if all players involved are playing a GTO strategy, in theory GTO would break even (well, actually lose a bit due to rake). What is weird though is that also in theory, one player in a multiway game deviating from GTO could actually cause some of the GTO players to lose, even though they wouldn't lose against the deviating player. This is due to the fact that his sub-optimal play could hurt some positions while helping others.

Again, this is in theory, and seems unlikely in practice. But hey, this is the theory forum.

I just used wikipedia, but it links here if you want more details https://www.science.org/lookup/doi/1...cience.aay2400

Again, using the Chess analogy, when Deep Blue was at 2700 ELO many people thought that was gonna be the pinnacle of Chess AI, and it was unfathomable to many people that any bot would ever be higher than 3000 ELO. Then when AlphaZero beat Stockfish8 at ~3200 ELO in 2017, many more of the people who DID think Deep Blue would be beaten eventually thought that OK thats it for sure now, these chess engines are basically perfection, they are definitely unbeatable. Now Stockfish14 is ~3600 ELO: the gap between it vs. Stockfish8 is now equal to the gap between Stockfish8 and the best human player in the world. Finally, I expect that Stockfish14 will probably be demolished by whatever Chess AI is released in 2025.

So once again, I think poker right now is in the Deep Blue stage of AI (at best). And as I will say as I have in every post on this topic, I'm not an AI researcher/expert here. Just from my understanding of what I've seen.

yep! CallMeVernon's post #9 has some more detail expounding on this