Im not sure if we're on the same page regarding 'ex showdown equity'. This is completely aside from the hand in question, but I wanted to clear this up because I think there is a huge misconception about what GTO defense is when facing a bet and you've got me thinking about it now. I know we talked about this before, but I want to bring it up again because although I felt like I understood what you were getting at, I had to build a model to conceptualize it. Also if this isn't what ex-showdown is referring to then I don't think I get it.
(This is not really the most realistic example, it is a model to illustrate a point)
We get to the river in a spot where villains range is KK, and 77 exactly on a T92,3,2 board
Our range on the river is 86, A9s
Villain bets KK all in for pot ( needs to bluff 77 half the time), we are facing 1:1 on a call. The game theory guys will chime in and say, 'you gotta call 50% of your range or else you're exploitable!'. This is wrong under the assumption that villain played his range in the most profitable way on earlier streets and should not have done anything differently to arrive at this particular river with this particular range. It just so happens that on this particular board, we will not be defending 50% of our range. It would be ridiculous to call 8 high combos when the strongest possible bluff in villains range is 77: therefore to determine what to defend vs a bet, we completely disregard our 8x combos and focus on A9s. We call A9s 50% of the time to make villain indifferent to bluffing, which means we are going to fold our 16 combos of A6, and 1.5 combos of A9. That has us calling 8% of the time - and this is our maximally exploitative response to villains maximally exploitative response - which is the equilibrium.
Takeaways:
Playing unexploitable means making your opponent indifferent to bluffing with all of his BLUFFS, not simply ATC. If villain started playing ATC on earlier streets to give himself ATC in this spot, he would by definition be exploitable and we would be adjusting to maximally exploit the holes in his strategy on earlier streets, which would make this spot non-existent outside of equilibrium play. Therefore in GTO play there will be spots that arise where you appear to be vulnerable to being bluffed by ATC, but because previous lines leading up to this spot do not give villain ATC, you are not exploitable when folding >50% of your range facing a PSB (in this specific river spot).
Rewording for additional clarification:
We are attempting to make 77 indifferent to bluffing. In doing so, we are going to fold A9s sometimes.
Villain has 6 value combos, and 6 bluff combos. If we overfold A9 here villain can bluff all his 7x combos and exploit us. So we have to defend A9 sometimes. But not always, because if we call A9 100% of the time, villain can just stop betting 77, and value town us with his KK which increases is EV. That is why we defend a fractional amount of A9. for For anyone trying to follow along but needs a visual aid to see it (I certainly did when first exploring the problem) plug this into cREV and you'll see that when you call the right frequency with A9s, the total EV of villains strategy is lower than if you were to fold always or never and he exploits it.
I am open to any further insight on this, as I recognize the possibility that I may have it wrong. Any additional comments on that would be appreciated.
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