What do you mean by this? Do you mean to say that 99 should be played a certain way here because of our whole range? If so, why would you take a line that is less +EV when compared to another line with a specific hand? Your goal should be to play every hand in a way that maximizes the expectation of playing that hand.

What do you mean by this? Do you mean to say that 99 should be played a certain way here because of our whole range? If so, why would you take a line that is less +EV when compared to another line with a specific hand? Your goal should be to play every hand in a way that maximizes the expectation of playing that hand.

Well u put a set of conditions (wich i mostly agree with) where 99 has more EV as a bet then as a check. However for instance for a while i played a strategy where i didnt 3bet vs EP opens on any position (not saying this is good) vs this strategy 99 is pbb better as a check then as a bet, since villian has QQ+ and can float some more aswell beceause of this.
And probably more important: intuintively 99 seems like a hand in the middle of our range, on a board that slightly favours our opponent. And therefore seems to me like a very good checking candidate

Well yes, 99 works better as a bet than a check given the conditions under which I set out to analyse the hand (and which in my experience are representative of a lot of opponents you would encounter). Of course you can't expand this to how you should play versus all opponents - like you said if villain didn't have an IP 3betting range this analysis wouldn't hold up.

I strongly disagree with:

99 seems like a hand in the middle of our range, on a board that
slightly favours our opponent. And therefore seems to me like a very
good checking candidate

This is a general statement but IMO making a gameplan via such general approaches leads to leaks. The rule of thumb of playing the middle of your range passively isn't there because playing all hands which are in the middle of your range passively nets the most EV, rather it's there because it is easy to remember and is generally an okay, but not perfect approach. If you can definitely show that line A nets higher EV than line B in this spot when it comes to 99, would you still check 99 in spite of this based on it being a hand that is in the middle of your range?

Hey, lots of questions here so I'll reply in parts.

How I calculate the % realized: yes, basically your approach, although I think your estimate of 25% is a little optimistic - I don't think we stack our opponent quite as often as you think, plus we don't have 100% equity when we do set over set him. These little things add up when it comes to playing potentially huge pots. My 15% top-end estimate was I believe a bit on the conservative side - reason being I wanted to see if we can definitively show that betting 99 is better even given pessimistic/careful assumptions. But - who knows what the actual % is. The only thing I would add is that you have to make sure to be logically consistent. For example, if you think you get that much when you set over set your opponent, you also have to make yourself lose that much in the other part of the tree where your opponent has top set and you set-under-set yourself.

To answer your other questions:

When you get to wider ranges or more wet boards, yes it becomes much harder, but it's by no means impossible. A spot like that would have to be approached differently though, as you would need to have several turns/rivers (or at least several types - for example flush completing and non flush completing on a two-tone board). The process is often a pain in the butt, and sometimes halfway you realize that the spot is so complex that you can't really get CREV to spit out a meaningful number. IMO this is fine because the process itself is useful, even if you failed at your goal. So the short answer to your question is: yes it gets harder in those spots, yes sometimes I try to solve, mostly I fail, but always I'd learn something in the process.

Equity realization - this is a really tough one. The 65% was basically an educated guess (that in my opinion was on the extreme end so would provide a useful boundary). It was based on experience with toying around with CREV and also seeing what other players/people I talk to estimate in this spot. As you allude to - it's pretty unfortunate to have to essentially guess when taking an otherwise technical approach to the analysis, but I've yet to find a way around it. I would certainly NOT use HEM to figure out what that number is, though. As you point out, you are never going to get a meaningful sample for this spot. The only way to do it is to generalize it and filter for spots like this one, but that defeats the purpose.

Thinking about it more the first thing I would look at is how often we have to c/f. That shouldn't be the hardest thing to figure out, and the equity versus villains check back range can be estimated fairly accurately as well - knowing those two variables you could get closer to a more supported estimate. All in all though I would approach the problem from that end, not a data-driven one.

Thank you!