It’s always a function of value hands. The more value hands the more bluffs. In fact I chose 3/4 pot then I need 28% give or take (I want to overbluff tighter players and underbluff looser players) of my value range. The card removal issues are secondary concern when I have too many bluffing hands.

So I c-bet 80% of time and fold to small turn probe 12% of the time and medium about 40%. Basically the exploit you should be looking for is someone who bets medium strength hands consistenly on the flop -- like second pair, 3rd pair and pocket pairs under the top card. If they do this, then basically they have to overfold to the probe, because they have no calling range.

Here's where PIO deviates from traditional poker logic. The ratio creates a situation where if they have a hand worse than X, they've lost the pot whether they call or fold. This means that if they are in this region, they'll never make a mistake on the node (because the decision doesn't matter). However, we are guaranteed an extra percentage of the pot, when they are in this region. The math works out that we get an extra bluffing percentage of the pot. So if the ratio is 3:1 to value bluffs we are guaranteed an extra 25% of the pot, when they are in this bluff catching region (no matter they're decision).

In practice, if they always called, we could make an extra 50%* Value Region of the pot, by never bluffing and if they always folding we could make an extra 100% * Bluffing Region > 25% by always bluffing. So the number that PIO produces is a lower bound on our actual max profit. Because we want to maximize profit, we should pay attention to calling/folding ranges and their sizes.

The mixes you saw are related to calling values. Because poker is actually just a giant sum: ValueofCall(2c2d vs Player2Range) + ValueofCall(Player1Range vs QhTs) + ValueofCall(AdKd vs Playing2Range) + ... = ValueofCall(Player 1 vs Player2). If we happen to only bluffing certain regions due to card removal, then Value Calling of 2c2d could be less than 0 and Value of calling AdKd could be greater than 0. So for example our sum of bluffing-catchers could be -5 x .5 + 0 x .5 + 5 x .5 = 0, where .5 is our calling frequency. They key thing to realize is that I could improve my value by calling: -5 x 0 +0 x .5 + 5 x 1 = 5. That's clearly sub-optimal, because now I just change my strategy and I make more money. PIO can do this math down to the combo, so comes up with wickedly complicated bluffing strategy to prevent subtle exploitation in the sums.

In practice, I think the matchup level analysis should dominate your thinking -- with the understanding that you need to bluff to keep people from changing to always fold strategies.

I ain't gonna lie you lost me on the math. The Solver is showing a bluffing region here and green is checking region. You mentioned calling region below, so not sure if we are on the same page or not.

I am still trying to balance my combos to 2 value : 1 bluff on the river per Janda's books. If the ATo/QTo is 12 combos, I still don't understand why we can't bluff all 6 combos of QTo and check all 6 combos of ATo. Frequency wise it is still balanced for that region with 6 bluffing combos, rather then 3 combos of each. I just prefer to bluff the less sdv hands and hopefully ones that don't block opponent's opponents folding range. I can see the issue with QT blocking opponents folding range on the J9675 board, but in my mind you just have to pull the trigger because blockers only count for like 10% of overall range anyways. I'll likely have to read your message 10x over to fully understand it and try and use your formula myself to fully grasp it. On the surface I understand, just not very clear even though it looks like you explained it clearly.

I'll read it over a few times and see if I can understand a little bit better by Monday.

I reread my post I wasn't very clear. Basically in the hand if I have Q9, you have 8 combos of bluffs so Q9 is always a fold on the river, so I can improve my expectation by just folding all hands with a Q. If I have A9 now, it's an easier call, because you are bluffing 4 extra combos. PIO takes this type of insight and runs many iterations until the range all hands are indifferent. Otherwise you can gain expectation by pure calling regions that don't block bluffs or alternatively you lose less money by folding QJ, because you face less bluffs, so we aren't indifferent anymore between calling and folding.

this is how i understand it ,say villain has KJ and AJ as bluff catchers and there both 0ev, say the solver is bluffing AT and KT and 50% frequency say you start bluffing AT pure you become exploitable because KJ becomes a plus ev call and AJ becomes minus ev. So solver will now just call KJ and fold AJ and increase his ev because rather than having 2 0ev calls he has a winning call and a fold which is 0ev, so his strategy over the 2 hands now has ev rather than being 0ev.

Thank you that was a very good explanation.

The idea is and this is dictated heavily by the choice of betsizing in the sim, that I want to bet small and have a reraise range to hurt his slim value raises. I played a hand yesterday where my opponent raised the block on T976 with QQ. I really want to cut down the value of this sort of exploitation, so I need to bet 3-bet sometimes.

Players are all over the place in this spot. Some people fold frequently, others basically rarely fold. Basically, I think the key to remember is that everybody plays this spot a little bit differently, so you need to adapt to the player here. GTO gives you some ideas about how to play solidly against an unknown, but the exploit level here is still reasonably high.

I wish it was complicated by my database says that is fiendishly difficult for my hands to lose money as defends in the big blind.