around 2:1, optimal bluffing frequency would be .3358, if it is the question

Optimal river frequencies on sizing is to push the percentage of the villain range to the point he would be indifferent on calling/folding the river (EV(call)=EV(fold)), right ? You can make a 1/4 pot bet and your bluffing range would be optimally 16%, if it is your intention and how your perceived range look like. 

On my first thought I consider a much wider bluffing range, but you don't have many bluffs to put a pot size bet on the river I think. What ranges you would continue bluffing ? Value is kinda std, counted 36 combos (JJ, 66, AT+).

So are you saying when deciding how often to bluff river, instead of just saying ok I have 30 value combos I need 15 bluffs we should take into account that since its so likely villain has Ax (for example he has AQss) we now have 17 perceived value combo's so should only have 8.5 bluffs?

I would agree with that, I think it makes a lot sense that we should be bluffing this river much less considering how strong villains range is, and all draws missed. 

Is it still correct to use a GTO value:bluff ratio OTR if we assume villain is going to be calling way more than 50%? If villain is calling an exploitatively high frequency should we not bluff at an exploitatively low freq? 

Don't 56cc + 67cc make the better 6x bluffs?

I can see how when you're deciding to represent the 666 then it doesn't matter that you have few outs to improve, but I don't see how 5h6h doesn't make you both too cbet happy and too river jam heavy as well.

It seems kind of like this hand was a river decision that got made on the flop, but I don't think even c-betting does all that much for your range on this flop because:

 1) You end up running some really weird bluffs with your hand on lots of run-outs.  I mean, what would you think if you saw somebody c-bet into two players with 5h6h and keep bluffing on some of the most common run-outs for this board, i.e. flushes and straights?  I would think, why didn't they just do that with KQ type stuff?

 2)  You could just use better c-betting hands (the 6x combo draw stuff) to bluff the run-outs that miss the draws, and this seems to randomize much better for big river bluffs.  

My initial instinct here is something like 80:20 value to bluff.

This is somewhat supported by some very rough math.  If he blocks ~50% of our aces we adjust by cutting our bluffs in half.  This makes for 16-17% bluffs.  Adjusting for card removal effects from our bluffing range, which will be less drastic,  makes me pretty comfortable with a 80:20 split.

Agreed.  Now lets assume that villain's range is 90% Ax+ OTR.  My understanding of optimal river frequencies is that they're purpose is to push a large portion of villains range to near indifference. 





I'm actually kind of interested to see what Janda would say about this, because making assumptions on Villain's range seems kind of open to debate and who the optimal strategy is actually playing against.  Sauce, for example, has said a few times that when he doesn't know what to do, he plays the strategy that does the best against all possible strategies.  If this is the starting point for constructing your strategy, then you wouldn't really make assumptions on people's ranges in spots and you would strictly strive for balance among your own ranges no matter what they could be doing.  
On the other hand, Mathematics of Poker defines an optimal strategy pair as two players who are maximally exploiting each other.  If that's the case, then you're playing against the nemesis,  which seems kind of the same thing since the nemesis gets to pick from the universe of all possible strategies when deciding how to exploit you, but I don't think it's quite the same.  It might be, though.

Yeah, I've thought about this more and I think it's really important to recognize that you're assuming that you know Villain's strategy--this will give you much different answers than a river model based upon the AKQ game, which is what the "optimal" bluffing frequency is.  In fact, in Mathematics of Poker the authors actually make the point that when we hold the K we remove the K from our opponent's range and therefore we cannot ever bet it because our opponent folds the Q and continues with the K.  This seems kind of similar but on a bit more complex of a scale.  

I want to say that it doesn't actually make sense to talk about this hand in the guise of optimal frequencies because you're starting with an assumption on his range, but it still might make sense theoretically from a non-exploitative perspective.  Regardless, this is probably a really important exploitative thing to master. 

It seems to me as if some points are getting mixed up here. :) I´ll try to explain what I mean:

1) GTO actually DOES take Villain´s range(s) into consideration, namely it assumes THE optimal one (which still IS a "known" or at least assumed strategy). What GTO does NOT do is to consider anything else but the optimal strategy for his opponent.

It´s like the old "rock-paper-scissors"-idea: a GTO player expects his opponent to choose his options perfectly randomized (1/3 rock, 1/3 paper, 1/3 scissors) - so he does the same (and remains unbeatable). Still he makes an assumption about the range of his opponent!

He won´t adjust though, even if his opponent (for whatever reason) played rock only. So, the GTO player remains unbeatable, but he forfeits profit (by not choosing paper 100%). Poker is not RPS though, so a good balanced strategy doesn´t mean 0EV, but regularly +EV, because opponents make enough mistakes that are "exploited" even by a GTO-strategy.

2) What Nick aimed at (if I understood correctly), was not a certain assumption about a certain Villain´s range that we want to exploit, but simply the question what the best counterstrategy against an 90%+Ax-likely range would be. So, we should assume that the "optimal" strategy for Villain would consist of 90%+ Ax on the river. This has nothing to do with exploitive play, it´s simply the question what the optimal strategy in a certain spot would be. And there are spots (due to board-runouts etc.) where even optimal ranges result in "unbalanced" ranges.

So, back to the question - which actually is an interesting one! Say, Villain had 100% Ax, then it would be easy, we could have a fixed number of combos to valuebet with (any hand with >50% against Villain´s calling-range), and could perfectly balance it with bluffs. But - now Villain does not have 100% Ax, but only 90%, rest of his hands don´t contain an A. How do we get to the amount of valuecombos now? Say, our valuerange would be AJ only. Purely from combo-counting we would get 2xA * 3xJ = 6 AJ-combos. Now we would add 3 bluffs and everything was fine.

If we gave Villain a 100%-Ax-range though, our value-range would shrink to 3 combos - and we could only bluff 1.5 combos. If we still sticked to our 3 bluffs, Villain could call 100% and print money. But we wouldn´t do it, because we´re good. :)

But what if Villain has 90% Ax? How many combos do we count now?! I´d say the correct approach would be to say "in 10% we have 6 valuecombos and in 90% we have 3, so overall we have 3.3 valuecombos", and then add the appropriate amounts of bluffs. But - did anybody ever made that or even thought about that? Or is it a purely academic approach because it rarely will happen that Villain has 90% of something? What if it were 30%? Before Nick brought this up, I still had counted 6 combos, instead of saying "it´s 70% six combos and 30% three combos = 5 combos overall".

Very interesting.

@Nick: Did I actually understand your question correctly?

Nick, I just read this thread and you have a very interesting question.

Seems like this comment I made there (A Deeper Look Into Blockers) might help answer your question :

this makes understanding blockers even more useful :
if we are facing a river bet against a "GTO player" (at least someone trying to make us indifferent to calling or folding, and at least in a perfectly polarized vs bluffcatcher situation), then 

- whenever we have a blocker to the bluffing combos (like in the JdJx first hand), we know that the bettor is unknowingly unbalanced towards value hands, and we have an easy fold, because we cannot be indifferent to calling or folding : calling has a negative expectation instead of the constructed zero EV.

- whenever we have a blocker to the value combos, we know that the bettor is unknowingly unbalanced towards bluffing hands, and we have a mandatory call, because our expectation is always positive ! In this case, the bettor still has a positive expectation, but lower than the constructed 1 pot EV.

Sounds like a huge implication of blocker usage.

So here you know the GTO river:bluff ratio to use OTR in a PvBC situation; you're then asking how / if knowing the blocker effect of Villain's range should influence your range and bet sizing construction.

From my comment above, it becomes obvious that Villain has an easy call if he suspects you're aiming at making him indifferent without taking his range into account.

Knowing with reasonable certainty that Villain's range blocks a big chunk of our value combos should have us adjust accordingly our bluffs combos to have Villain remain indifferent to calling or folding.

Villain doesn't even need to block "a big chunk" of our value combos : even a few ones would give him an easy call.

Running the numbers in a spreadsheet makes this very obvious.

cool, thanks robert.  so it looks like are we concluding that the "true optimal" bluffing fqcy here should be significantly less than 33% for 1PSB.

bear in mind that I'm much less qualified than BigFiszh and yourself, obv.

It surprising that I had like an "aha" moment while thinking about implications of blockers on GTO river play and that you have these questions at the same moment; it even seems "new thinking material" to BigFiszh ("back to the question - which actually is an interesting one! ")

Anyways, we need to be crunching some numbers, I guess, (like BigFiszh has started) :

if we know the value betting range and the corresponding GTO bluffing range (33.58% as per ScienceBitch1), it would be easy to measure the impact of Villains blockers on both EV, I think, and then adjust accordingly.

By the way, some similar situations are studied in Felipe's video and I want to overcome my laziness and make the calculations in these examples : GTO with and without blockers.

the optimal play is to check fold flop

ill have to check that felipe vid out