OOP range: 20% nuts & 80% bluffcatchers
IP range: 33% nuts & 67% no-showdownvalue-hands
OOP nuts splits the pot with IP nuts
lets say that the pot is 10 chips, there's one round of betting and OOP is forced to check, IP is allowed to bet any amount or checkback (no-showdownvalue-hands always lose when checkedback)

IP bets 12,5 chips with all of his nut hands, and bluffs this same size 17,6% of the time when he has air.

OOP then calls with his bluffcatchers 30,6% of the time.

How did you solve it?

I compared the EVs of different bet sizings for our value hands, and chose a bet size with the highest EV, which is at around 1,236x the pot.

I tried it using crev. But I couldnt get to a pinpoint result (I knew the bet size was around 125% pot tho). Did you use some special software?

I just used calculator.

Calculate how much money you're making with nuts on average, when you bet Pot/ 1,25xPot/ 1,5xPot, and you'll see that on average you're making most with 1,25x bet size.

Cool problem. Can you better explain how you got the result? There must be a way to do this without guess and check

You can Crev but you wont get the exact solution because the software isn't perfect.

do a sim

Mind explaining how you got the 17.6 bluff %? I'm prly doing something wrong, but I'm getting 27.23% bluff with air (meaning checking 72.7%).

I just used solver in excel and set up Hero's Value Bet/Bluff ratio based on bet size. Standard (x+1)/(2x+1) where x equals bet size in relation to pot. Once I had these inputs set, I created a total expected value formula basically just the weighted avg of the 7 different scenarios we could have (Hero Nuts Villain Nuts, Hero Nuts Villain bluff catcher calls, Hero Nuts Villain bluff catcher folds, Hero Air check, Hero Air Villain Nuts, Hero Air Villain Bluffcatcher calls, and Hero Air Villain Bluffcatcher folds). Then just max our EV by changing x and I also got 1.236X as our optimal bet size.

All my formulas were linked but I basically set our valuebet/(value bet +bluff) percent to make villain indifferent to folding or calling when he had bluffcatchers. With our 1.23X bet size our value bet/bluff ratio was roughly 64:36. Given that we have the nuts 33% of the time, this gives us a P(bluff given we have air) of 27.2%. Different than the 17.6% you mentioned. Also, how did you get the OOP bluffcatcher call % at 30.6? I was under the assumption it would be indifferent for him to call/fold and his percent wouldnt change our expected value.

Appreciate the replies in advance, I'm a noob here so I am probably overlooking something obvious.

Your math is good.

I should've said 27.2% there, my bad. I'm not sure where I got the 17,6% from, I should've double checked.

OOP's calling % changes the EV of IP's bluffs. OOP has to make sure, to make IP's bluffs indifferent, just like IP has to make sure that OOP's bluffcatchers are indifferent when he bets.

Therefore, if OOP doesn't want to be exploitable, he has to make IP's bluffs exactly 0EV.

And he does that by folding 1,25/2,25 = 55,6% of hands. Which means he has to call 44,4% of the time. 20% of that are nuts hands, and the rest (24,4%) are bluffcatchers.

Therefore, when he has a bluffcatcher, he calls 24,44% of the 80% of the time he has it, which is 30,6%.

Ahhh makes sense, thank you.