Quote:
Originally Posted by bigburge10
Also, and surely someone will correct me if IĀ’m wrong, but you would only call at MDF across the hands that hold enough equity to call. For example, if your range held 10 combos, 2 of which had 0% equity, and the other 8 held enough equity to call pot sized vet, you would call with only 4 of the 8 hands. Calling more than 4 leaves you over calling, and less than 4 as overfolding. The 2 0% are simply folded. Of course, in this model, a bluffcatching range canĀ’t actually hold a 0% equity hand.
This is correct, but allow me to try to better explain it. When we reach an equilibrium on the river in the nuts or air scenario the goal for the polarized better is to make the other player essentially have 0ev with his bluff catchers. If a hand has 0 equity, then it isn't a bluff catcher and isn't really counted.
Let me provide an example.
IP range = AA, KK, QQ and 99 88
OOP range = JJ, TT, and 66,55
Board = 2 2 2 2 3
pot = 100
stack = 100
On the river IP will bet all 18 combos of value (AA, KK, QQ) and bet 9 combos of bluffs. If blocker effects don't matter then it doesn't matter how the 9 combos are distributed. IP could bet six combos of 99 and three combos of 88 or 75% of each (3/4)*6 = 4.5, so 4.5 + 4.5 = 9 also. The reason we arrive at 9 is because 9/(9+18) = 33% which in this case is the pot odds laid to the OOP player (pot/(pot + pot + pot). In this situation 66 and 55 are pure folds for OOP as they have 0 equity even vs the bluffs, so the defense frequency or "mdf" which would be pot/(pot+pot) = 50%. This means 50% of the combos of JJ/TT would be called or only 6 combos out of a total of 24 (so only a 25% call rate).
1. That is the equilibrium situation and would have IP winning 95 chips
2. Non-equilibrium where OOP folds everything IP wins 100 chips (because he always bets and there are 100 chips in the pot
3. Non-equilibrium where OOP calls everything IP wins >100 chips (because he only bets his value hands) this ends up being 109.45 chips
4. Non-equilibrium where IP over bluffs and always bets 88/77 OOP always calls JJ/TT and always folds 66/55, this would increase OOP EV because IP is unbalanced... JJ/TT gain ev here.
5. Non-equilibrium where IP never bluffs and only ever value bets AA/KK/QQ OOP gains EV because now when IP bets OOP can fold 100% and when IP checks behind his JJ/TT get to realize all that equity when in scenario (1.) JJ/TT is indifferent to calling/folding due to the odds laid.
You can attempt to calculate IP/OOP EV in the various scenarios as practice. (2.) is the easiest. I've included the EV for (3.) in case you would want to practice the calculation and reference it, although I cheated and used PIO :P