In both of the above examples, the caller holding (0.7) can beat (bet)/(pot) = 50% of the bettor's value range. Initial findings validate the bold imo.

Sorry for so many posts in a row. Just trying to keep everything organized.

So the proper response here for player A is to call at minimum defense frequency because player B's bluffs have 0% showdown equity.

player A:

(0.4) = 0% call
(0.5) = 0% call
(0.6) = 100% call
(0.7) = 100% call

I decided to check the bold statement by figuring out player B's profit on the bluff:

1pot + 1pot = 2 pots

(2 * 0.5625) = 1.125 pots of ev coming back to player B on the bluff.

Notice that if player B misses value with (.65), then the result is that player A's ev goes up with (.7) all the way to (1 pot of profit/hand). Since (.7) is 1/4 of player A's range, and the other hands are 0ev, then player A's total ev/hand is 0.25 pots.

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Going to the other example, we can compare the evs of possible exploitation, but a tourney is starting.

Then player A can also call with (.6) and (.5) for a healthy profit, averaging +1/3 pot per hand.

Let's say the player in position has a very weak showdown value, it almost never wins. Your A or K high wins something like 0.11% of the time! There is still showdown value and small EV!

If you bluff this combo, and your opponent calls 'at minimum defense frequency', your bluff has 0 EV ? So you lost value by bluffing this combo ? Does it make sense ?

So you would bet this combo if you think your bluff has some EV and the opponent slightly 'overfold' according to the concepts of Min defense frequency.

If you are the bluff catcher, does the fact your opponent has the option to bluff or check this combo means you should defend less ?

Maybe it does. Because you want him to be indifferent to bluffing. You don't want him to be specifically 0 EV. In fact, as soon as he bets balanced, he has more EV in the pot I am pretty sure.

So if he decides he start bluffing a combo that has like 0.01% pot value in check down, well I guess that means you could let his bluffs be the equivalent profitable, so he is still indifferent to bluffing.

I think the difficulty is more in identifying in practice how we do it. It's easier when we just talk about the theory of the game.

But notice it's only a matter of call frequency. For sure you would never make a fold that would be a +EV call. If you do this, he is not indifferent to bluffing.

Sincerely I don't understand how you really apply all those maths in game.

I think a good example of this concept would be like, it's pretty common when you play HUNL, an opponent rarely showdown a worst hand. When you showdown Ahigh, King high or bottom pair it's never good. Well I guess you could call down this player a little less ? since by bluffing more he's kind of renouncing to his equity he would already have by showing down.

All real river situations can be described using the (0,1) game. By studying these situations away from the table we’re better prepared to handle the decisions at game speed.

I Think Q2 is too low because he may bluffs some better Qhigh. You said yourself he could bluff QT. So if KQ is a break even, Q9 is logically negative and every lower Q is slightly more negative.

Then I know that's another topic, but I'm not too sure with how you assess your opponent's range. And I may be wrong. I get confused because it's a tourney with ICM considerations too bc I play cash game.

I think the 567 is bet a lot with air, especially 8x, nuts, most pairs, 9x, overcards with little showdown value. But honestly I'm not too sure because I think the board somehow is good for OOP, who has more 89, but much less overpairs.... I think most of the checking back range is want to showdown range.

So I think the J turn could be a good leading card for OOP because the only showdown hand to have a J would be AJ KJ and they even bet flop sometimes.
And when IP does not bet J turn, and the J turn would be a bad card for his range, I would deduce he unlikely has J.

Am I right to think we can deduce something from an opponent betting or checking a card that is good or bad for him ? I think this is more related to individual tendencies.

The river being a J, and you not leading at it vs a range that apparently wants to showdown, IP should think you don't have the J, so I think both ranges are capped. a smaller sizing anyways leaves most nuts out of his range and seem to target a pair & Ahigh bluff catcher.

so he would be representing a range of weak pairs & showdown value that wants to get some thin value and some give ups who reopen. I'm not sure what I'm saying.

But given the original question of : what about his showdown value. Well does he bluffs with KQ KT K9 K8 ? if yes, you have to be careful calling too weak bluff catchers . But I would not think players bluff those.

This is very a very important point that I'd like to get back to. I've written before about the most important hands in the opponent's range, this is how it breaks down:

assume you're playing against a good opponent on the river:

step 1) estimate the bottom of the opponent's value range.
step 2) estimate the top of the opponent's bluffing range.

these are the two most important hands in the opponents range, by identifying these hands, we put ourselves in a better position to effect a proper counter strategy.

Some do, but let's look at how this mistake (imo) would change proper counter strategy:

a) if the mistake is compounded by bluffing way too much, then the out of position players bluffcatchers gain significant ev, as well as some potential value check raising hands that gain significant ev.

b) if the mistake is compounded by missing value bets, then the out of position player's bluffcatchers gain even more ev.

the list goes on but you get the idea. To see where the ev comes from, think about the loss of ev that the in position player experiences when he bluffs the wrong hands, thus missing river showdown value.