Extremely brief answer, it would be "easy" to do with an infinitely powerful computer.

Longer: It's actually pretty easy (as in computers today can do it quickly) to compute a nemesis to a given strategy. A very surprising theorem is that if you start X playing strategy x, then calculate a nemesis y' for Y to play then calculate the nemesis x' of that strategy, and average that in with x to find X's new strategy. Then calculate the nemesis of that and average it with Y's current strategy, etc. This process will result in two GTO strategies. (Mathematically it's not quite "convergence" as they sometimes get worse, but they get arbitrarily close infinitely often.) This is the process actually used in all of the GTO strategy calculations for complicated games I'm aware of. See "Fictitious Play."

One thing that trips me up when talking GTO in 6 max games is when I get to these sort of examples.

Say you open the CO with 30% of hands.

Villain on button is a super nit who only 3 bets AA/KK, and he 3 bets you.

Now GTO would not take villains tendencies into account but would have us defending X% of our range vs the 3 bet, how does it not get slaughtered here trying to defend too much vs a very strong range?

Also how would a GTO strategy go about selecting which hands to open from different positions at 6 max?

by making money in all those situations where villain should have 3! K5s or AJo but didn't.

It loses exactly the same amount to AA/KK from a guy who never 3bets anything else as it does to a guy with a reasonable 3bet range. It doesn't exploit the guy being too tight by losing less to AA/KK, but it does profit off of guy failing to make other 3bets he should make.

+1 this is what people are constantly not getting. Same thing applies to the river, if we know villain has nuts all the time in a certain river spot, why would we still call some % according to GTO??? It's because he's missing value by not having enough bluffs in his range, exactly the same way you miss value by only 3betting KK+ pre

Could you give me a basic idea what sort of software/coding ability/processing power is required for doing this. I take it it's not something I can figure out by myself armed only with rudimentary excel skills.

Actually, you could probably do it with Excel, although perhaps the size of a strategy for a real game is a little too big. Certainly you could do it for any small game.

Really brief and hand-wavy description:

First, given a strategy for X, you can compute X's distribution at every point of the game tree. (Simple non-realistic example - if he raises preflop with only KK and AA, 50% of the time that he has each, and calls with everything else then you know his distribution after a raise is 1/2 AA and 1/2 KK, and his distribution after a call is the same as a random distribution minus half of the AA and KK.) So anyway, given the strategy, you find the distribution at every node of the game tree.

Then, given the distribution of your opponent on a decision at the leaf of the tree, it's really easy to decide what to do with each hand, right? (Call if you win often enough to win money, fold otherwise...)

So once you know what to do at each leaf, you can calculate your exact EV at each leaf - which lets you easily decide what to do at the nodes of the game tree right above the leaves - you choose the path that gives you higher EV. Carry this process up the tree, and boom, you have a nemesis.

To the GTO experts ITT:

How likely/probable is it that GTO strategy is something that can be grasped on a conceptual level by humans instead of some kind of weird mathematical distribution for various situations?
For example, how likely is it that the strategy for A8s OTB is something ‘intuitive’ such as to minraise 90%+ and do something else the other times instead of for example limp 12%, minraise 15%, make it 2.4398x 2%, make it 3.55x 7% of the time, make it 6.66x 8% of the time, make it 12x 1.5% of the time etc. Or how likely is it that the strategies for very similar hands (for example 75s and 76s on a 7KA flop) are completely different? Alternative, is it possible the strategies for HU NLHE 100BB deep and HU NLHE 99BB deep are completely different?

Basically, I’m curious whether the GTO solution (or a close enough approximation) is something that could be practically applicable by humans (‘I need to bet ¾ pot with this hand here the majority of the time and check about 20% of the time’) or if is it only a concept that could give insight in various situations but would not be actually playable in human vs human matches.

I can only guess, and I personally don't think it's useful to quantify the probability that a guess is true.

I would say that an exact GTO strategy will be fairly complicated. I would expect that raise sizes on various streets depend on the hands and the current SPR, and thus ultimately on the starting stacks. However it will have some logic to it, and so humans might be able to understand and implement reasonable approximations.

If I had to guess, I would expect a GTO poker program to do tons of stuff you don't see players doing now. Weird, and oddly specific, bet sizes, probably a lot of donk betting/not respecting the concept of initiative, possibly changing its play very dramatically based on tiny variations in stack sizes or board texture. The fact that players tend to fall into certain patterns now doesn't have much to do with heavily considered strategic reasoning as much as habit, guestimation, and often just laziness. Why do a lot of players open to 2x on the button, and 3x UTG? We certainly have a good reason to suspect that opening bigger from earlier positions and smaller from later positions is optimal, but why not 3.7x UTG and 2.19x on the button, for example? Probably because nobody is good enough to calculate a meaningful difference in EV between opening sizes separated by less than a big blind, and clicking the 2x button is way more convenient than calculating out and typing in a very specific number.

A GTO bot will have fine tuned its strategy down to the most minute details, with bet and raise sizes optimized down to the penny, in ways that a human player will never be able to approach. Luckily, a truly GTO bot capable of anything more complicated than heads-up limit is vastly beyond our computing capacity for the foreseeable future (though I think it's certainly possible we will see non-GTO, but still very very good, poker bots crushing high-stakes NLHE in coming years).

is it strange that i get a boner whenever fitzcat is posting?

^^^ yes

many of the people explaining GTO have stated that it takes advantage of opponents not doing certain actions as wide/balanced as they should (only 4-betting with the top of their range, etc). i understand how this would help GTO but after doing basic calculations for roshambo i'm either confused, making bad calculations, or misinterpreting the result

Roshambo Example
Let's say that a win constitutes +1, tie=0, loss =-1

Fred Flinstone plays Rock 100% Paper 0% Scissors 0%
The GTO strategy is Rock 33% Paper 33% Scissors 33%
The exploitative strategy is Rock 0% Paper 100% Scissors 0%

GTO EV: (.33*1)(0)+(.33*1)(1)+(.33*1)(-1)=0
Exploitative EV: (1*1)(1)=1

The exploitative strategy is positive EV, while the GTO strategy (by definition) is 0 EV. The GTO strategy is also 0 EV against the exploitative strategy.

Based on this, why would we want to implement a GTO strategy?
Answer: GTO cannot be exploited; even the most optimal exploitable strategy can be exploited.

So I guess I'm missing the big piece of this: why is GTO so important from an EV perspective? If it is breakeven against all strategies, sure it might not be exploitable, but that implies we cannot ever be +EV, correct?

Any response within the context of the example provided or general theory would be much appreciated

bolded is false in poker.

The short answer is, i think, that Roshambo is too simple a game for the more interesting aspects of the GTO solution to emerge. If you play about with other games like the {A, K, Q} game or (0,1) games, then more interesting elements are more apparent, I think, and thus some strategies will be break even vs GTO without themselves being GTO, and others will actively lose value versus GTO.

If you consider a game like Roshambo where there's a 4th option that loses to everything, then the GTO strategy would be break even against all strategies that play without the fourth option, but would be +EV against any strategy that included it. Obviously no one in their right mind would ever play the fourth option, but in a game like poker there are likely to be such suboptimal strategies that are far less obvious in their flaws.

(seriously, check out the stuff on GTO in the book mathematics of poker. I've owned it for like 4 years, but this thread made me finally read it, and it's really good and illuminating)

Even shorter answer is:

You can not make a mistake playing vs a GTO-Roshambo strategy due to the structure of the game.

For poker this does not hold true (see the now famous 'fold pre 100%' strategy earlier in the thread for proof).

Pics of Asian waitress?

Possible, impossible. At least it is impossible for it tomatter

I'm surprised you say this. If this were true, then the EV of 'the strategy of only 3betting AA'= 'the EV of the strategy of only 3betting AA/KK' versus the GTO strategy. I'm pretty sure that isn't true though, 3betting both AA and KK should increase the EV of a strategy substantially against GTO in nlhe, at least according to my experience and rough simulations I do. The more generalized version of this point is that some if not most deviations from GTO cost money versus the GTO strat in poker.

I say most because there are certainly important counterexamples. On the river we are indifferent versus GTO when calling bluffcatchers, etc.

ROFL wat?

It's quite clear from this thread that only nosebleed players seem to have even the mildest grasp of this.

It's also quite clear to me from this thread, knowledge of GTO strategies and the like, and the extent to which it matters when making a great poker player, are marginal at best.

The point Ike was making is that once Hero's strategy is fixed (at GTO), then Villain's EV of any particular line with any particular hand is also fixed.

So if Villain 3-bets AA vs a GTO-playing Hero, that has the same EV no matter how he plays his other hands. If his other hands are also 3-bets at equilibrium, then 3-betting only AA loses him money with the other hands but doesn't gain him any more with AA.

I think you're misunderstanding me. I agree with everything in your post. I'm not saying it has the same EV against the strategy as a whole. I'm sure it does better against the strategy as a whole. I'm saying that when you get dealt AA and 3bet it, you beat the GTO bot for exactly the same amount whether you 3bet other hands or not.

just for a comparison. in chess we have some position with table-base-solutions that humans dont really understand. like this position:

not even magnus carlsen would win from that position. poker will have tons of solutions like that.

or these:


we humans are just clueless to what is going on the first few moves...

right, because GTO isn't exploitative

i'm not completely sure what you mean here, as i don't really see what the analog in poker would be for those intricate series of moves.

it is very likely true that humans can't arrive at GTO fullscale (HUNL or HUNLE) poker without computer aid.

on the other hand, i think it's unlikely that those solutions would seem "crazy" or totally incomprehensible to professional players. some of them would be surprising at first, but in most if not all cases i would bet they would make sense after some thought and analysis. this is the case in my experience with toy games as well as the play of bots like polaris.