A situation in which I need some help to solve

You need to give more details about the players ranges.

Using PS charts, the buttons folding range is pretty much empty (supposing he is facing a check-raise). All the buttons hands have at least a gutshot except 77 which he strictly speaking should rebluff with (at least a few of them).

/Johan

Well, given that the button doesn't fold, filter out the hands in MP2s opening range that has 55%+ equity vs a button 3-betting range (if that is what you require to check-raise for value). This gives the check-raise range for MP2:

• AA-JJ, 99
• All queens
• KTs
• [Not TT, not AJ and not KJs]

That's 57 combos. Take 6 more to bluff with. I'd pick some AK. (There is nothing worse left.)

Kind of bizarre

/Johan

And then the button reraises for value (again using the 55% rule) with sets only. Thats 9 combos. He needs one bluff. One might argue that either player should "delay" action, but since both can be convinced that the other is coming along, I see no big reason to slowplay.

Conclusion: The MP3s check-raising range is per above, and the buttons folding range is exactly 5 of the 6 possible 77 holdings ♠_biggrin:

Disclaimer: I have surely screwed up somewhere in thought process and calculations. The calculations are based on PS standard ranges. The 55% is picked out of thin air. The assumption of "no slowplay" is a simplification, but not an unreasonable one with these ranges and this board ♦

/Johan

If you have no folding range on this board you're not 3 betting wide enough, but since you're using the ORCs for a guide that's unfortunately the case here.

While the button may not have much of a flop folding range he will have both a turn and river folding range so not bluffing at all is obviously wrong.

If you have no reads the best thing to do is turn hands into bluffs and try to at least make your range as unexploitable as you can but this board if it goes to Showdown will tell you a lot about the button and you can adjust your ranges from there (Especially if you see 55 or something at SD)

Originally posted by Boomer2k10
While the button may not have much of a flop folding range he will have both a turn and river folding range so not bluffing at all is obviously wrong.

If that was directed to me; the MP2 does bluff in my scenario, with 6 AK

/Johan

Is this a FR hand?
If yes, then i would have A6s, A7s to xr bluff with, sometimes i also raise A5s or 87s, wich is a perfect bluffing hand.
BU should fold some low pocket pairs imo, but i can't find out the exact ranges.
Since the flop hit BU's range, we shouldn't xr too much.

Well if it wasnt obvious Im looking for a GTO solution, so you thats why I didnt tell anything about the ranges. I think even if you give a massively wide 3bet range (like 66+,A8s+,KTs+,QTs+,JTs,T9s,ATo+,KTo+,QTo+,JTo) for the button he'll be still overcalling the flop due to the potsize and the fact that how hard he hits this board.
Not to mention that if someone 3bets this wide, or wider he has to overfold on boards that are the opposite of this (i.e. that he completely misses), because otherwise he would be making -EV calls simply due to the lack of pot equity.

So basically MP2 doesnt have enough immediate fold equity on the flop, and has to calculate turn and river fold equity in advance. But to do that, he needs to know what how many combos he is going to value bet on the turn and the river, which is quite plastic (cause some turn and river cards cause equity shifts compared to flop vbet ranges).
And even establishing a v-bet range only for the flop is quite difficult, because BU-s line will be call flop raise turn around 50-60% of the time, which makes the flop x/r a lot more expensive so I dont think that value x/ring with just 50% equity is a good idea. I think you need 56-58% at least, and this almost means like 2pair plus. Almost because the AA is not x/r-ed but KK is (due to the GS outs). But that means on a turn card that isnt a T you have to b/f that KK that sounds like WTF.

So I came up with 2 pair + and KK for a value range,and added 3 combos of 87s with a donk end plus backdoor flush draw, but this means MP2 is underbluffing, and button is overcalling the flop and if this is the best they can do it means that there is no GTO solution for the flop.

And just to make things more complicated, I think that card removal may play an important role here, because the both ranges are so close to each other that they both hold a significant part of eachothers outs.For ex you think that you have four T outs, now the chance that button holds a T himself in his calling range is 42%.

Originally posted by zulusierra
Is this a FR hand?

It doesnt matter unless you open a different range from FR MP2 than 6max MP2.

Originally posted by kavboj84
Well if it wasnt obvious Im looking for a GTO solution, so you thats why I didnt tell anything about the ranges. I think even if you give a massively wide 3bet range (like 66+,A8s+,KTs+,QTs+,JTs,T9s,ATo+,KTo+,QTo+,JTo) for the button he'll be still overcalling the flop due to the potsize and the fact that how hard he hits this board.

With that range, supposing the button reraises right away with what he perceives as value, he will fold a few hands (depending on what he perceives as value) from the 66 and 77 pairs, the rest of them will be bluff raises.

Originally posted by kavboj84
So I came up with 2 pair + and KK for a value range,and added 3 combos of 87s with a donk end plus backdoor flush draw, but this means MP2 is underbluffing, and button is overcalling the flop and if this is the best they can do it means that there is no GTO solution for the flop.

Sure there's a GTO solution. It's just beyond the "I give you 10:1 to call, therefore it's 10:1 that I bluff" principle that really applies only for the very last bet in a hand.

//
I had a similar, but much easier problem hand posted some time ago. I had QT in the BB, called a raise, the flop came A8x (don't recall details). I folded - so what. With that board, I simply had to fold close to 26% (or whatever) of the range I called with preflop. Yet, the preflop call was right - as my fold. Nothing very wrong with my preflop calling range either. Note that a 26% folding frequency on a particular flop renders you seemingly exploitable to a c-bet. (Yeah, like if the c-bet isn't coming anyway...)
//

The answer lies in Game Theory (GT), what it says and, in particular, what it does not say. GT says that given any single hand, there exists a way to play it, based on the board and the opponents moves such that the expected outcome is zero or better for all players employing the strategy. (No one can profitably deviate from it if all players present employ the strategy.)

This says that for each holding in your range, there is a GTO way of playing it. But this does not say explicitly that you should have a certain percentage of bluffs and value-bets for each street in a range. GTO does not operate with ranges at all. GTO applied to poker results in ranges, that's very different.

Once you know how each and every hand should be played (this knowledge is what the Nash theorem ascertains the existence of), then you can a posteriori just list the strategies for the individual hands and simply count the number of checks/bets/raises for the board in question. There is nothing that guarantees a certain amount of remaining bluffs, value-bets and rebluffs for each and every street. From this you build your unexploitable ranges, not vice versa.

I believe that it is impossible to find the solution starting in the other end, namely "guesstimating" how much value/bluff should remain for the later streets. It's only guesses, but it is the best we can do.

I think you may be seemingly exploitable in some situations when playing GTO (or close to it), but you aren't. Note that in the example of the original post, both players are pretty much showdown bound, so bluffs on either part will have extremely little effect (except for in the real world, where people don't play GTO.) Likewise, in my example above, my opponent can't c-bet much more than his regular 98% either.

Sorry about my rant.

Sure there's a GTO solution.

yea and whats that ?

With that range, supposing the button reraises right away

I think it isnt worth 3betting the flop for the button, since even a call-flop raise turn line is hard to balance, so if you have a b3b range on the flop how do you blance your call* lines with that weak range whats left ? Also after a x/r except A,K and T turns I dont see MP2 checking at all and even on those cards he will bet sometimes.

I was editing when you posted. See previous post.

Originally posted by kavboj84

Sure there's a GTO solution.

yea and whats that ?

We know there is a solution for each individual hand by the theorem of Nash, but nobody knows exactly what it looks like. (But see post above.) I believe that there exists algorithms for how to find it, but there aren't supercomputers enough for it.

Originally posted by kavboj84

With that range, supposing the button reraises right away

I think it isnt worth 3betting the flop for the button, since even a call-flop raise turn line is hard to balance, so if you have a b3b range on the flop how do you blance your call* lines with that weak range whats left ? Also after a x/r except A,K and T turns I dont see MP2 checking at all and even on those cards he will bet sometimes.

Yeah, you are right, and I wasn't suggesting a strategy, I just wanted to mention how bizarrely strong the buttons fairly wide range is on this board. (I threw it into Combonator.)

My thoughts about this particular hand (now after thinking and writing plenty) is that balance is of minor importance this time. The hand is pretty much doomed to go to showdown. Maximize value with the nuts, never ever fold (because the pot is big), unless it's between 1 AM and 2 AM on a rainy Monday or something like that. River rebluff only when drunk on new years eve. I hope my above post will clarify why I believe this, whether it's right or wrong.

/Johan

I know what Nash equilibrium means, but I am not sure that it exists in every situation. And since you have no proof its only an assumption for you as well that it exists.
A lot of times poker seems chaotic to me(especially viewing poker in multi street/multi way perspective), where one player makes an adaptation which results a temporary higher EV line until the other player makes an adjustment as well.

For example imagine a hand (this one is also close to it) where the agressor who bets the flop c-bets the turn 100%. Now the caller must call here 1SB + 2SB = 3SB instead of 1 to see a SD,so it isnt worth playing those hands that you would call flop and x/f turn. But the agressor knowing that the other player never folds the turn should not bluff the turn. So he must start checking the turn but then the other player must call the flop with hands he would x/f the turn with, and then again the agressor has fold equity on the turn.

GTO says that there must be a balance point somewhere but you should get to that point by constant leveling as well and this case I just cant see a stable point, instead of that I see a constant imbalance, which is the unmistakable earmark of chaos.

Multi way pot bluffs, same story. It is very rare that you can make a profitable bluff in 4-5 way pot due to the pot odds, therefore when you bet it must be value. But then people start overfolding and exploit you, but then you can start bluffing, and they can stop overfolding and the endless cycle starts again.

The Nash equilibrium exists for a certain category of games. And, yes, I do have a proof. I can email it to you. It's the original proof by Nash. Or, Google "nash non-cooperative games" and you will find a pdf download.

Poker is a game to which the Nash theorem applies and it means that there is an unexploitable move for each and every situation.

GTO says that there must be a balance point somewhere but you should get to that point by constant leveling as well and this case I just cant see a stable point, instead of that I see a constant imbalance, which is the unmistakable earmark of chaos.

GT says there is a balance point, but doesn't tell you how to get there at all. If you knew the GTO solution you could just play it and don't give a damned about what the opponent does in his leveling, or what happened last hand. You care about your hand, the board, and what your opponent has done this hand so far.

I perfectly understand your skepticism. I "demanded proof" just like you, but nobody around here could really provide a proof, so I dug it up and read it up myself. Nash equilibria exists.

Here is an analogy from reasonably understandable mathematics. Imagine a continuous map from the closed unit disk in the plane to itself. There is a theorem called Brouwers theorem that states "Every continuous map from the closed unit disk to itself has a fixed point."

See here: Brouwer fixed-point theorem

[It's a harmless link to Wikipedia, if it's removed, just go there manually and search.]

Now, the Nash theorem says mathematically something very similar. In fact, its proof uses the Brouwer fixed-point theorem. There is even an example in the Nash publication about poker.

It would be pretty interesting to see the GTO solution for just one nontrivial hand, like this one. But I'm sure that there are no real surprises to be found. Approximating partial solutions like Boomer does in his videos is probably the best we can do with reasonable effort. A full analysis of even a pretty trivial one-street poker game is extremely cumbersome.

/Johan

it means that there is no GTO solution for the flop

Maybe i misunderstood the definition of GTO, but since the number of actions and combos are limited, there must be a GTO solution for every situation.

"GTO says that there must be a balance point somewhere but you should get to that point by constant leveling as well"

"It is very rare that you can make a profitable bluff in 4-5 way pot due to the pot odds, therefore when you bet it must be value. But then people start overfolding and exploit you, but then you can start bluffing, and they can stop overfolding and the endless cycle starts again."

I tought GTO is a static strategy, it doesn't take your opponents actions into consideration, and if you make adjustments you are not playing GTO anymore.
I don't know if it applies to 4-5 way pots, but maybe it doesn't want to make profit with every single action, it simply have to make those bluffs to remain unexploitable. (?)

BTW, i doubt that a human being will ever play a hand by GTO, not even HU. The solution you came up with must be good enough IRL.

@zulusierra. There is a GTO solution for every form of poker for any number of players except for some contrived variants for which the proof and applicability of the Nash theorem breaks down.

An exception could be a NL Hold'em variant where any bet amount (not only fractions of chips (dollars, cents), i.e. rational numbers, but also irrational real numbers with infinite decimal expansion) is allowed, except that you aren't allowed to go all in. In such a game, there is for every strategy a strictly better one. Hence no GTO exists.

@ zulusierra

IDK what you mean by static strategy, but GTO play does take the actions of the opponents into account.It can be viewed in two ways, either your opponent is a GTO player and your reference is that, or it can be a nemesis, who automatically and maximally exploits your deviations from GTO play. And also I dont understand what you wanted with adjustments, ofc its true, but that doesnt answer anything,cause the question was if there is a GTO strategy.

For example:

you open from MP2 and everyone calls except the SB.On the flop you give your opponents 11.5:1 for calling, that means 8% bluffs, and since you risk 1 into 10.5 means you need to make all of them fold in 9% of the time. But this means each single one of your opponent needs to fold 54% of his range, because 0.54^4 = 0.09 , which is simply not going to happen at 11.5:1. Even if you hold AA they have a profitable call against it with almost any2 (not to mention implied odds), and even if not any 2 certainly not only with the top 46% percent of their ranges.And this means you shouldnt bluff 8% but way below that, if you should bluff at all.

@YohanN7: Thanks for the tip,I will examine this paper when I will have some time, sounds interesting, but I think I need a few hours to go through it.

Until then, Im still waiting for the best solution of this example.

Well, im just asking rather than stating anything about GTO.
From what i have read so far, i think there is only one GTO solution for every situation, the same against every type of opponent.
It uses many variables like number of combos, bet sizes, possible upcoming cards, one street and multi street alfa, etc, except the opponents tendencies.
This way, after the mathematical operations you will have only one solution, at least in FL.
Thats why i said its "static".
Maybe im using the wrong english phrases, sorry for that, shame on me...
If its true what you have said, im obv. wrong.

For the "hopeless" bluff situations:
I agree with you that there are situations when its ok not to bluff at all.
I think its ok IRL, because nobody will exploit you, but don't know if there is a situation where GTO says that.
Maybe when the number of combos to bluff with becomes 0.4, for example?

well since the lowest number of combos can be only 1, and its not hard to realize that as you add more opponents to the pot the number of bluff combos converge to 0, after a while you should get to a number that is below 1, i.e. a point where you have no bluffs in your range anymore, even if all players int the pot play GTO. But if this is so, then the consequence of this is that there is no fix point (which is the BE point in poker), and the GTO strategy collapses into a constant leveling war.

GTO is entirely static. Of course, it takes into account all the opponents moves the present hand.

For a conceptual understanding of GTO per se, it's best to simply forget about ranges and percentages of ranges with which we should bluff. Just view GTO as a HUGE Nash table. It simply tells you what to do in every conceivable situation. From that table, you can then build your ranges. Then you can look into the ranges to find how much you are actually bluffing with respect to each board and constellation of opponents.

When you play GTO, it will, by definition, never collapse into a constant leveling war, and GTO provably exists so that trying to find counterexamples isn't going to work (because they simply cannot exist).

Starting in the other end, namely "we have this and that many opponents on a XYZ board" will probably not lead to the true GTO answer. You will have to do too much guessing. The "N:1 odds = N:1 bluffs" is true (with freak exceptions) only for the last bet vs one opponent. On earlier streets vs multiple opponents, it's at best a rule of thumb.

Until then, Im still waiting for the best solution of this example.

You may have to wait for a very long time. Boomer quote: Calculation of GTO for FL HUHU would take 120 years using the full computer power of Google