You are right, I believe this is one of the reasons why poker is so hard to slove by computers. We are actually not playing our range vs his range, but each hand from our range against his range (minus bloker effects). See Bayesian Games in some GTO text.

This happens very often when balancing preflop ranges: say you 7bet jam and you have AA for value and 54s as a bluff and 6-4 is your "theoretically correct" value-bluff ratio (I completely made up the numbers). Now if he holds AT for example, he knows your range is only 3 value hands and 4 bluffs and you are not balanced. The solution is to bluff with Ax hands, so when he holds Ax, he blocks your AA and Ax bluffs at the same time so that your ratio doesnt change. Its actually a bit more complicated, usually you cannot construct your range so that the ratio doesnt change for every hand he holds, but you can do it so that when he holds a high equity hand, you ratio shifts to value and when he holds a weak hand, he knows you have a bit more bluffs but he still cant call because he has too little equity. Its hard to construct such range without a software...

So on the river, its the same idea. Your range is different from his point of view for every hand he holds and you need to account for that. If both ranges are reasonably wide, you can use the standart 1-a ratios, but as the ranges get narrower, this effect is more and more significant. I studied this a lot with Combonator and GTO range builder, you can find some interesting spots if you concentrate on this. Models like 1-0 game or nuts/bluff vs bluffcatcher toy game doesnt account for blocker effects at all so the 1-a solution is only applicable when the bloker effects are neglible.

Thanks both! Luke,

"Models like 1-0 game or nuts/bluff vs bluffcatcher toy game doesnt account for blocker effects at all so the 1-a solution is only applicable when the bloker effects are neglible.

Yeah, that is pretty much what I intuitively understood. Could you give me a quick example of how a narrow range could be different when holding a value hand and when holding a bluff so that I can have a starting point? I found no much depth on neither Applications of no limit nor HU no limit expert.

Try spots where flush is possible. Like board is 5c6cJc2dKs, your value hands will be AcXc and so when he holds some bluffcatcher like AcJd, he knows that you have less combos in your value range. Remember that you are balancing against the weakest bluffcatcher, but sometimes even the weakest bluffcatcher blocks some part of the value range. In this case, you should bluff hands with Ac in them, so when you bet you have the key card in both your value and bluffing hands.

So it's not only the hands we block but the hands villain blocks... It's seems pretty fucking complicated! I will try more easily:

In the hand you mentioned, I see in Flopzilla that against a range of cold call in BB vs BTN he has 20 flush combos on the river, but if we hold the Ac he has only 14! What does it mean? If we have 33 flush combos OTR we should normally bluff with 16 (with a PSB), but when we hold the Ac, how many more combos could we bluff? And if we don't hold it, how many less combos could we bluff? How do you get these two numbers?

In my opinion its more about what he blocks than about what we block, but its really both:)
I dont understand the question: if you hold the Ac, it doesnt affect how you play other hands...you just decide if you bet this hand or not.

Try this, construct your range of 33 value hands and add 16 bluffs not using the Ac. Then see the situation from the bluffcatchers point of view. Give him a hand with Ac. Now he sees your range as 14 value combos and 16 bluffs, so he can easilly call. This would be a problem - if he has enough bluffcatchers that contain a nuts blocker, you would be very exploitable to bet that seemingly balanced 33/16 range. So now go back to those 16 bluffs and choose them so they include Ac. Now when HE has the Ac, he knows that you have less nuts but also less bluffs, because he blocks both, so your ratio of value-bluff is OK...thats the idea.

You do not need to do this in every spot. It is important mainly when your valuehands are all of the "same kind" like when all of them are AcXc or QcXc, you need to bluff hands of the QcXs or AcXd type. And its even more complicated, because you dont want to bluff hands with too much showdown value at the same time.

Maybe the basic idea is: nut blocker is important not just because he cant have the nuts, but also because he cant have the nuts blocker:) Its really easy to see with preflop ranges: if we have range of AA,KK+bluffs, its important for those bluffs to be Ax. So when he holds a good blocking hand like AK which blocks half of our nuts, he blocks all of our bluffs also.

"but it is true that if villain calls with top pair or better, most of his hands will be top pairs, so the calling frequency will not be the same: it will be lower when we are valuebetting and higher when we are bluffing. So in fact we should add less bluffs, right?"

Your assumption is correct.

To figure it out, you need to look at the hand from villain's perspective:

Consider that villain holds KTs on the river. (The T is irrelevant and thus a good side-card for the example given). Now villain has one K in his hand and there is one K on the board. Two kings remain and your AK combos are now reduced to 8 combos. (Two kings and four aces remain, 2 x 4 = 8).

Holding KTs, villain knows that you value bet the following range:

AA for 6 combos.
AK for 8 combos.
88 for 3 combos.
A3s for 2 combos.

Total value combos are 19. Given a PSB on the river roughly 10 bluffs are needed to make Kx indifferent to calling. Had you bluffed only considering your own range and not the blocker effects of villains holdings, you would have bluffed too much, giving Kx a profitable call.

Wow thanks!

You have given me more than enough material to work with. I will come back here when I have practiced a little bit on what you just gave me.

I appreciate a lot!

So, FriendlyCritter,

do you think it's better in any given spot to try to figure out what his worst bluffcatching hand is and then build your bluffing range from his perspective of your value range as you did in your example (bluffing 10 and not 12 combos)? At least in the spots where it has a remarkable effect, as in the example of Kxx.