Mathematical Bet Sizing
I played a hand in the $33 Bounty Builder this week that made me wonder what the correct sizing would be on the river. UTG+1 (68BB) opens to 2.29bb and I (98.5BB) 3-bet with AA to 7bb, which is about 3x his open. As stacks get shorter you can definitely lower your 3-bet size, but here I wanted to maximise value in position against a player who I thought was playing sub-optimally. Creating a bigger pot in position is never a bad thing, especially when your opponent will make more mistakes by being out of position. Also, the bigger I make it pre-flop, the easier it will be to get stacks in by the river and pick up that bounty.
The flop comes 3d 2s 7c (16.85BB)
Villain checks and I bet 6BB, which is roughly 10% of villain's stack.
Pro tip: Bet 10-12% of your opponents stack on the flop if you are looking to bet turn and jam river.
The turn is the Qh so the board now reads 3d 2s 7c Qh. This is a great card for me if I was bluffing, but when I have a value hand it can make my opponent think twice about calling with 88-JJ when my value range is QQ+,AQ.
The pot is now 28.85BB and Villain has about 55BB behind. In order to jam the river for value I need to bet enough on the turn.
Pro tip: Take away the pot size from the effective stack size, and then divide this number by 3. This will always leave you with a pot sized jam on the river.
55BB - 28.85BB = 26.15BB
26.15BB / 3 = 8.71BB
Now the Qh will either make Villain fold more often on the Turn or the River. If he has 88-JJ then he might call one more bet on the Turn and then fold on the river. For this reason I decide to size up my Turn bet to 12BB, which in turn (no pun intended) also makes the pot bigger than the effective stack making it easier for Villain to call on the river with his strong holdings like AQ and KK.
The river is the 2d and the board now reads 3d 2s 7c Qh 2s and the pot is 52.85BB. Villain has about 43BB at this stage.
He checks and I jam and he folds.
It's pretty difficult to know what bluffs I would have in this situation so I'm wondering if I should bet the river smaller especially as he's having to call off for his tournament life and will most likely be a little tighter. If I bet 1/3 pot or 1/2 pot there's a greater chance that he'll call. Just because the plan on the river was to go bet/bet/jam, it doesn't mean we can't change our mind on each street.
I also ran this hand in PIOSolver to see what size is believes is best on the river. I gave it the options of 25/50/75% on all three streets and it believes that if we bet 25% on the flop we should bet 75% on the turn and all-in on the river. If we go for a 50% sizing on the flop, we should bet 50% again on the turn and all-in again on the river so it looks like going bet/bet/jam in this spot is the highest EV line, but that doesn't take into account that our opponent is unlikely to call the triple barrel for his tournament life with some of the hands that PIO suggests he should, Interestingly, PIO suggests tripling it off here with most of our AK combos as our bluffs.
I've done a lot of work with PIOSolver recently and while I think its suggestions are great in theory, in practice it's important to also consider tournament life and opponent tendencies since nobody is playing 100% optimally yet.
Good luck at the tables.