Ok, so lets say it's SB vs BB.

SB raises to 3bb and BB calls. There is no rake in this game.

The pot on the flop is 6bb.

If SB bets 4bb's, then to make an automatic profit on the cbet, the BB will need to fold more than 40% of the time.

Now lets say that the BB folds 45% of the time, and calls the cbet 55% of the time and always wins the pot.

This should mean that the SB is printing money as the BB is not defending enough as per GTO.

However, in Pokertracker or Hold'Em Manager, the SB would be making a loss in this scenario. This is because it only counts half the money in the middle as money that is there to be won.

Isn't this where GTO concepts like cbet defence fall apart, because they only look at one street and say that you have to defend X% in order to stop your opponent printing cash. In reality, this isn't the full picture because if you fold more than GTO says, then the times you don't fold you are more likely to win the pot plus more later money than your opponent, since your opponent still has his 100% range, (as he's attempting to exploit BB), and the BB has a much narrower and stronger range since he folds to most cbets.

Apologies if I am not explaining myself well, but I see this as another reason why GTO is not that great and doesn't give the whole picture.

Nice bait.

Whitebait.

White Magic

The issues you're getting at are a problem with people's tricks for trying to approximate gto play, not with gto play itself. The whole "don't let villain auto profit with air" thing is often completely wrong as a way to think about and solve for gto strats.

So what would be a better way to think about not letting villian exploit us??

Could you elaborate on why this is the case? I don't think I've ever read a convincing argument as to why "don't let villain auto profit with air" is a good or a bad approach.

Yes, I'd like to know why you think that. I've often thought it's a good place to start... because it's actually relatively easy to implement.

Of course SB loses money when he raises preflop, get's called, and then misses the flop and is holding air. This is not GTO falling apart.

Try your scenario again with BB folding slightly too much preflop and slightly too much on the flop.

I made a video about a lot of these issues once, but really quick, the argument basically goes like this --

If we always call, villain can exploit us by not bluffing and if we always fold, villain can exploit us by never bluffing, so to keep him from exploiting us, we need to call just enough so that his EV of bluffing and not bluffing with air are equal.

So then we assume he has some air hands that always lose the pot if they don't bluff and also lose if they bluff and get called, and then we call enough to keep these indiff to bluffing.

First of all, the original indifference argument can break down for a lot of distributions (eg what if villain doesnt even hold any air or not much). Second those ideas about the EVs of bluffing and not bluffing are pretty much always wrong. ATC have like 20perc chance to pair up if they check back and get a couple free cards, bluffs and esp semibluffs can still win a lot if they bluff and get called, etc. And it turns out that these things can have a v big effect on the estimated gto strats. So the naive estimate found by calling to make complete air indiff is usually completely wrong.

Would it follow then that the less equity our opponent's bluffs have, the closer our calling frequency should approach a static symmetrical range optimal calling frequency? And also that the more equity our opponent's bluffs have, the less we should call compared to a static symmetrical range game?

I thought it is generally true that hero is able to prevent villain to bet atc. Yet there might be some occasions where distributions are so skewed that this is not possible.

And to prevent atc attack hero needs obviously take into account that villain has some equity if hero is calling. How to do that is hard (for me) to assess as hero also can benefit from free cards.

Should we raise more often if the bluffs have more equity?

I believe so.

Doesnt the fact that bluffs have 20% equity should imply that we can actually defend a little bit less then the exact threshold and still not getting exploited by any two?

I believe the fact that value hands usually have less than 100% equity offsets this effect. Probably not equally though. GT before the river is hard.

sent from phone

The problem here is that you're using monkey math, and only looking at one branch of the overall strategy.

In order for the BB to have 100% equity when he calls the SB's cbet (which is actually impossible in real poker, but not important to get the point), he must be folding X% of his hands PF. Every time the BB folds PF, the SB is gaining 1bb and this must be considered in the overall EV of the strategy.

It is from the BB folding PF that the SB is going to be printing money, because in order to have the equity distributions that you're talking about the BB is going to have to fold a ridiculous amount PF.


In regards to the EV of the Flop cbet:

If villains folds 40%:
(Probability * Win) – (Probability * Lose) 0.40 * 6 - 0.60 * 4 = 2.4 - 2.4 = 0bb

If villain folds 45%:
(Probability * Win) – (Probability * Lose) 0.45 * 6 - 0.55 * 4 = 2.7 - 2.2 = 0.5bb

This

So to solve for GTO play you'd have to work backwards from the river, then? And isn't the idea that betting a certain amount on the flop and needing to generate a certain of folds what makes a play profitable "in a vacuum" and not because of GTO considerations?

And in the given example, the idea that the BB would win 100% of the time he defends is unrealistic and too simplistic, since he'll have a certain amount of equity vs SB's range that would have to be taken into account in GTO calculations?

OP is making a simple mistake that actually doesn't have a lot to do with game theory. When you analyze your "EV" two different concepts commonly get confused, the EV for the hand as a whole, and the EV for that particular decision. The EV for the hand as a whole includes play on all streets and the posting of the blinds as part of its calculation. The EV for a specific decision does not, that type of calculation has no memory of who actually put money into the pot. If you yourself put 3 bb into a 6 bb pot at one point, or if a random stranger generously contributed all 6bb of that pot it, the "EV of bluffing" won't change. Also, even when you're exploiting your opponent by getting him to fold too much, his EV will still be positive.

To determine the EV for the hand as a whole you need to consider all preflop scenarios (limping, raising, folding, reraising, and for what amount), hand ranges, and the cost of the blinds. Even then, due to the inherent value of position, you won't get a symmetrical result. The small blind has a positional advantage that will force the BB to fold more often.

We'll you'd have to solve it completely so it doesn't really matter which end you start from. There's really only two ways to solve for GTO play:
1. solve for a bastardized version of poker
2. solve using ingenious formulas and a supercomputer that doesn't exist yet.

Theoretically, one way to solve a game is to map out the complete gametree and work backwards to find the best branch at every decision under every possible scenario. However, if you did this for hu nl at decent stack sizes, there would be more decision nodes than atoms in the universe.