Let's say I buy in to a large game, look down and see some middling cards, shove all-in, and then the outcome doesn't matter.

Then I come home, plug the hand in to a solver, and the solver recommends that the cards I played be raised all-in 0.01% of the time.

Can I then legitimately say that I played the hand in a perfectly GTO manner?

I feel like I'm trolling asking this question, but these GTO probabilities eventually have to collapse in to an actual decision, because you can't partially fold a hand or be in a quantum mechanics superposition of having raised both half-pot and full-pot. So I know it sounds stupid but how can all these different options be "correct?"

And the solver is running these simulations millions of times against an opponent that is also tabulating your results and playing back at you millions of times with that info. But this isn't how poker is played in the real world, where there's frequently a lot of people that you play a few hands against and then never see again.

So being a little more realistic, does this mean that if the solver recommends something that you should play the highest % option always as the first one against an unknown opponent?

It just seems so weird that if a solver recommends four different actions at four different percentages of the time, that all four could be chosen in one specific instance and still be "optimal." And because solvers suggest a variety of options at least some portion of the time, this would imply that people frequently make "optimal" choices unless they picked something a solver recommends exactly 0% of the time.

I dunno...it just goes so much against common sense but the fact remains that one specific hand can't be folded 33% in one instance. And also that if you start playing against new opponents...do you start over?

Alright I'm literally losing sleep over this so I'll wrap up. Perhaps part of the issue is that a specific hand can't have an optimal game theory; only as the number of trials approaches infinity can there be an optimal theory. But does this mean that the best possible play for one hand is completely different than GTO? I dunno.

It's easier to understand if you look at the logic of the entire range rather than analyzing hands in a vacuum. The truth is that many different strategies are playable, so long as your entire strategy is constructed correctly around that line.

Let's imagine you're shoving the river with either the nuts or air. I have a bluff-catcher. I need to call that bluff-catcher a certain % of the time to keep your bluffs indifferent.

If I ALWAYS call, you could value own me and stop bluffing. If I NEVER call, you could run me over with bluffs. So a mixed strategy becomes optimal. I need to *sometimes* call.

Notice how the hand itself isn't important. Calling/folding are the same EV. What actually matters is the overall range construction.

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A more common example is trapping. The solver will often recommend trapping with nutted hands at some frequency. Villain's playing a balanced strategy such that the EV of trapping = the EV of betting. If I were to simply NEVER trap, that leaves my checking range exposed which is exploitable. If I were to ALWAYS trap, that leaves my betting range too weak which is exploitable.

Either action is okay in a vacuum, both actions have the same EV, but the overall strategy needs to be constructed correctly to avoid becoming exploitable. That's why it mixes.

This is not the same thing as a superposition (which would mean you could raise, fold, do neither, or both at the same time). It's simply a mixed strategy.

Doing this (or even picking the least used frequency) would be fine against a GTO player that won't change their strategy. But it will lose badly against a player that adjusts to exploit your mistakes.

Here's how you wrap your head around it. Poker is not played hand-vs-hand. Realistically and strategically, it's played range-vs-range. A hand in a vacuum is completely meaningless without the context of the ranges and game state.

Thank you very much for the response, was really thinking about this one.

Fortunately I just hit a chapter in Maria Konnikova's book tonight that partially got to the answer. She's talking about her first trip to Aria, and there's one dude in a $25,000 buy-in tournament that's playing on his twitter feed and another that is an absolute model of zen-like focus.

The dude playing on his phone was just confident in plugging away at GTO, but it wound up costing him.

And also, while thinking about it, I realized I was bringing a card-counting mentality in to the whole thing. If the count is +4 and the player makes a big bet, gets an 11, doubles, then winds up losing the hand, they can confidently go to bed that night knowing that they absolutely made a +EV play. There's no psychology, there's no tells, cheating is nigh non-existent in major casinos, action z is mathematically correct when the true count is x and the dealer's card is y, end of story.

And any game like that will never rightfully be poker.

What does this mean?

Well said Tombos

Until a human can give off exactly zero tells they'll never be able to play a perfect game.

This is news?

Well Maria K says that even the pros pay too much attention to their phones and not enough to the game, so it's a leak that goes all the way to the top

I think people have alluded to this but think about the game as what would you do if you were to play the same formation and line a few million times ( cause at signifiacntly large sample size the outcomes converges to the the expected values- central limit theorem)
if you wanted to maximize your EV given perfect clairvoyance you have to play all hand in the certain % of the time. Surely, in any one instance the frequencies will collapse into a singular action but that's irrelevant cause over the million sample size if you were to keep track you would have been expected to have played that specific hand as per the GTO freq.
Obviously, no one has perfect clairvoyance. So the point of solver solutions is to provide a baseline from which you need to deviate based on what you observe. So you would start by playing pure strategies. Sooner or later your opponent and other at the table will figure out your pure strategies and they will adjust in a certain direction to exploit that 'pure' strategy. If you are aware of this then you would now stop doing that pure action and start mixing in other actions with that hand. and once your players observe this they will adjust and hence the cycle of adjustment keeps going on until it reaches an equilibrium where no one can really exploit anyone.
No one should be trying to mimic GTO. that's absurd and more importantly you might getting exploited unknowingly. The whole point of poker is to play in an exploitive way without getting exploited yourself. that's where the skill and EV is

If I may, let me state that all of this discussion of waves collapsing, superposition, and convergence is not wrong. It is just wrong-headed.

A few months in to my study of modern poker theory, I did a trial of PIO. I spent several hours learning the software, plugging in some situations, and looking at the results. The tool is fantastic. I did not buy it. I came to believe that, for me at least, using a solver would not nearly be the most efficient use of my time. Presuming that my goal was to improve my game and maximize my ROI.

I learned that my goal was not to "play GTO", it was to use GTO to improve my game. I let the pros use the solvers, and watch them explain the important concepts. My advice, FWIW.

1. The solver is playing the solver. The solver is check-calling middle pair on the turn in a particular spot because the solver is finding bluffs that most humans would never consider. In real life his 2-barrel range is crushing yours. You are lighting money on fire. "But the solver said. . ." is not a good enough excuse. Some things haven't changed since I learned the game from Doyle and TJ. Play the player, not the solver.

2. You're playing against human beings (at least as far as you know) so play, and learn, like a human being. Learn how to construct your own ranges, don't try and memorize a chart. Learn the types of hands to potentially 3-bet with by relative position, and why. Learn how to figure range advantage and nut advantage. Learn how to play the blinds. Learn how to play short stacked. And on and on. Just don't waste your time trying to decide if QJs in the CO vs. UTG at a given SPR is a raise or a call. Because either way it's probably close.

3. Review your hands to refine your ranges and find your leaks. Study where the EV is.

I note that some of the top players actually do try to randomize mixed strategies. I'm not there yet, and neither am I playing against many opponents where it might matter. Someday I might get the hang of this game. In the meantime, I'm still happy to take the money.

So looking back at my OP a month later it's kinda dumb, although at the time I was having like a philosophical epiphany about the whole thing. After all "playing perfect GTO" might be great and all but it's better to win ;-)

There are some slightly more practical issues:

--I want to bring up again the issue...If a solver recommends doing something 80% of the time, is it likely that against an unknown player we should take that action, or are people just so different from solvers that we should ignore them entirely?

--What are the most common fish mistakes? I'm guessing that as DangerousDan mentioned, the solver finds bluffs that people would never dream of, and therefore certain spots may be underbluffed.

--Will running pre-flop GTO also get a player wrecked without knowledge of how to play it post-flop?

I do want to start working randomness in to my play, I definitely think it'll make me harder to play against. Although when I play drunk people early in the morning their actions aren't particularly subtle...but against the good sober players I need to start mixing it in.

If a solver mixes 80/20 with some hand, then both options have the same EV (Against GTO). So it's important to realize that solvers mix to remain unexploitable.

It's more important to look at your overall range construction rather than tunnel-visioning the exact frequencies of one hand. Mixing one combo 90/10 or 50/50 is absolutely meaningless without context.

You need to develop a solid baseline strategy to play against unknowns, and mixing is useful for that purpose. From a more pragmatic perspective, it's possible that one action significantly outperforms the other, but if you always cbet top pair you'll never find that out.

Common fish mistakes include but are not limited to:

• Overvaluing any Ace
• Playing their hand without considering their range
• Castrating their checking lines and being way too value-heavy in their aggressive lines
• Using bet sizes that don't make sense for the board/situation
• Extremely top-heavy preflop ranges
• Overplaying medium-strength hands, and/or not understanding appropriate value thresholds
• Using linear ranges instead of polarized ranges. In other words, their bluffs are medium-strength hands at some point.

If there are streets remaining, for example a spot on the flop where you're supposed to call certain hands at a certain frequency, your overall strategy will lose EV if you don't mix frequencies correctly (even if your opponent doesn't deviate to exploit you), due to future runouts.

Solvers mixing to remain unexploitable is more of a misconception than a rule imo and you'll see many spots where the EV of a certain action is essentially or actually (can't say "actually" because all solver solutions are just really good approximations) the same EV but the action isn't chosen. Again, in these cases deviating from the mixed frequencies will cause your overall strategy to lose EV, even though that individual combo won't be directly losing EV. Mixing mistakes can still cost you EV without someone deviating to exploit those mixing mistakes.

ev of each action is well defined given the opponents strategy over every node remaining in the tree. your own strategy has absolutely no impact on the ev of each action (given that the opponents strategy is locked)

against a perfect gto opponent, you could play a strategy with zero mixing and breakeven.

No, there being streets left to act doesnt make mixing frequencies bad lose versus Nash Equilibrium. This is the actual common misconception.

Money can't time travel to know if you played the right frequencies in the last 100k hands.
The only connection between two individual hands is the brain of the people playing them and how they can use information from one to play differently in another.

The overall frequencies for your entire range are the important thing, IMO. The actual hands are less important than the percentages of your range that are strong, marginal, weak, draws, etc. It's about balance. When we get check raised on the flop, if we fold every time then our opponent can bluff with abandon, and if we call every time they can take us to value town.

There's an optimal calling percentage, which is an important thing we can learn from a solver. This frequency is effected by things like raise size and how the board interacts with each player's range.

But if our human opponent only raises with two pair plus, it's reasonable to fold hands that can't beat that range. We just need to remember that if our opponent picks up on what we're doing, they can then change their strategy to bluff raise and take advantage of our folds.

The question to ask yourself in a spot where the solver says you should call sometimes and fold sometimes is this: Is my opponents ratio of value hands to bluffs higher or lower than what the solver suggests it should be?

If I think they have more value hands than they should then I always fold those mixed frequency calls. If I think they have more bluffs than they should then I always call with them.

lets say its a SRP sb rfi 3x bb call, flop Qh7d2c, oop bets 1/3, IP has many indifferent calls/raises/folds, lets say IP pure folds all indifferent BDFD floats, like J5s, J6s, J4s, 95s, 84s, but pure calls all offsuit indifferent floats like 65o 54o, A6o, K8o, J9o, T8o, T8s nbfd, IP will naturally have less flushes on future runouts and more air, won't this lower the EV of IP on turns and rivers regardless of what oop does?

It's true that that range will have different EV from optimal even vs GTO on some particular run out, but when you add EVs across all runouts you'll get the same number.

In your example on BDFD runouts ip will possibly have lowere EV due the lack of flushes, but when BDFD doesn't get there he has less air in his range.


It's bit surprising to me that some successful high stakes regs don't understand the indifference.

IP will have less air on non flush runouts.

Basically you can look at this in a very simple way. EV of range = sum of EV of each hand.
If you make indifferent plays with every hand you play, even at the wrong frequencies, the sum of those EV ends up being indifferent with the EV of playing perfect frequencies.

Obviously people don't play Nash though, and most people try to exploit, so keeping your frequencies not looking ******ed has a lot of value, specially vs stronger opponents. But RNGing everything even in spots where villain couldn't possibly gather or infer enough information to make a good exploit is superfluous at best imo

This is a very common misconception, even among high-level players.

Jarretman, you should check out this thread which covers the topic in more detail: https://forumserver.twoplustwo.com/1...23/?highlight=

The players below are playing HU against a fixed GTO strategy:

Player A overfolds. They fold any hand that would normally be indifferent/mixed between folding and some other action.

Player B overcalls. They call any hand that would normally be mixed/indifferent between calling and some other action

Player C overraises. They raise any hand that would normally be mixed/indifferent between raising and some other action.

All players will have exactly the same EV against GTO.


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I'll explain it to you the way Aner0 explained it to me:

You aren't playing "range vs range" against the opponent unless their strategy is dynamic. If their strategy is fixed, then you're playing "your hand vs their strategy" in a vacuum.

Here's an actual demonstration of this concept, taken from this post:

Ok, I have a question here. I've been running some Sims recently, and one thing I have noticed is that for flops where I start nodelocking different strategies, then the total EV (sum of IP and Oop EV) changes based on what I'm assuming is total expected pot size.

For example, let's say for whatever flop, I force Oop to range check. If I then force IP to range check, the sum of EV for all potential runouts is (slightly) higher than if I force IP to range bet. This sum is always smaller than original pot size minus current rake. This only occurs with rake. If the same situation occurs in a rake free pot, the sum of EV is the same whatever the flop actions (and equals pot size otf).

Basically, and whilst it's clearly possible I'm doing something wrong in my Sims, how does this factor impact the point you're making? As far as I can tell, the difference in total EV is due to the expected final pot size, and the rake taken out of it. So, overfolding will yield a higher sum of EV than overcalling or overraising. Surely, the fact that there are different total EV means that all 3 strategies may perform differently.

As a side note, I was unable to lock the strategies in for turn and river for either player. I wasn't able to figure out how to lock them for all different runouts. So the "GTO" strategy would change in response to opponent's actions.

Fantastic question.


So as you have already noticed, Pot = OOP EV + IP EV + Rake EV. In your example, when both players are forced to rangecheck that results in lower Rake EV, so the sum of OOP+IP EV is larger.

Nash Equilibrium stipulates that no player can unilaterally change their strategy to increase their own payoff. It says nothing about decreasing the opponent's payoff. Your mixing mistakes can influence the rake, which will either add to or decrease villain's EV.

Let's assume villain is playing a fixed strategy:

If we call all our 0EV bluff-catchers, our expectation is still 0EV, however, we reduce villain's EV and increase the rake's EV.
If we fold all our 0EV bluff-catchers, our EV remains the same, but villain gains EV and we reduce the rake's EV.
If we raise every hand that's a mixed raise, our EV still remains the same, but villain's EV is reduced and the rake's EV is increased.

Seems wild at first, but you can test this out with simple raked toy games. Haizemberg93 likes to talk about rake as if it's a third player, because you see similar effects where one player deviating in multiway spots can unfairly shift the other equilibrium player's payoffs. You can see the same effect in HU raked pots. Mixing mistakes won't effect your EV against a fixed strategy, but it can shift EV between villain and the rake.

Makes sense if we're assuming a true Nash equilibrium. So that means finding the mixing errors is the way to exploit solvers?

Thanks for the in-depth reply btw, much appreciated.