[5NL] A9s in CO, in depth

This is a hand that we were talking about in the IRC. We determined our two best options were either C/F or B/F 1/3 pot OTR, and I said I'd do an in depth analysis breaking down our opponent's range to figure out which was the better option. So, here it is!

Hero (CO): $5.22 (104.4 bb)
BTN: $4.86 (97.2 bb) {26/5, 0/2 fold to CB, 0 3b, 20 hands}
SB: $5 (100 bb)
BB: $5 (100 bb)
UTG: $5 (100 bb)
MP: $5.12 (102.4 bb) {29/10, 0 3b, 21 hands}

Preflop: Hero is CO with A 9
2 folds, Hero raises to $0.15, BTN calls $0.15, SB folds, BB calls $0.10

Flop: ($0.47) T A T (3 players)
BB checks, Hero bets $0.33, BTN calls $0.33, BB folds

Turn: ($1.13) 7 (2 players)
Hero bets $0.54, BTN calls $0.54

River: ($2.21) 3 (2 players)
Hero ??

I've made the assumption that our opponent will raise AT, AA at some point earlier in the hand, and probably raise 77 OTT. He's a fairly passive player so I've assumed that he won't raise his trips OTF and OTT the vast majority of the time. He'll probably call these two bets with his gutters and JJ-KK sometimes, so I discounted two thirds of his JJ-KK. So his range should look something like this to the river:

Board: AdThTs7c3h
Equity Win Tie
Hero 64.50% 62.72% 1.78% { As9s }
BU 35.50% 33.73% 1.78% { TT, KdKh, KdKc, QdQh, QdQc, JdJh, JdJc, AJs+, A9s-A2s, KTs+, QTs+, JTs, T8s+, AJo+, A9o-A2o, KTo+, QTo+, JTo, T9o }

As for breaking down his range and how he will play it, these are my assumptions:

- When we check he'll bet AJ+, his Tx, his TT, and check back the rest
- When we bet 1/3 pot he'll fold his few JJ-KK and his BW gutters, call all his Ax, raise a few of his Tx (say KT, QT), and of course TT


EV FOR CHECKING

We have 97.32% vs his check back range, and he's checking back 66.3% of his range so

(0.663)(0.9732)(2.21)=$1.43


EV FOR BET/FOLDING 1/3 POT

When we bet he's raising 10%, calling 58% (against which we have 56% equity), and folding 32%.

(0.1)(-0.74)=-0.074
(0.58)(0.56)(2.95)-(0.58)(0.44)(-0.74)=0.769
(0.32)(2.21)=0.707
Total: $1.40

So... based on my assumptions and assuming I didn't make any mistakes it's pretty damn close!


edit: I'm interested in what betting bigger does to our EV so I'm gonna do that real quick, betting 2/3 pot without changing his range breakdown even though it probably does change a bit..

(0.1)(-1.47)=-0.147
(0.58)(0.56)(3.68)-(0.48)(0.44)(-1.47)=0.885
(0.32)(2.21)=0.707
Total: $1.44

Though his calling range probably actually does narrow a bit... I bet if we even took out A2 it'd be an inferior option to 1/3 pot.

Chances are that he will not raise AT or TT at any point before the river. If he does, then it's a minority of the time. He's a passive player to begin with, and you're taking the betting lead, so he doesn't have any reason to be aggressive with those hands. This is along the same lines as trips.

When it comes to discounting KK-JJ, I feel like JJ makes it to the river more often than KK because of pre-flop 3-betting.

If you want to go deeper, then look at how you should play your entire range.

First I'll redo the original task with a more accurate range..

So now his range before any action OTR looks like this (I've taken out KK completely, left 1 combo of QQ, and given him 4 of JJ and one of 77)


Board: AdThTs7c3h
Equity Win Tie
BU 37.57% 35.84% 1.73% { TT, QdQh, JdJh, JdJs, JdJc, JsJc, 7d7s, A2s+, KTs+, QTs+, JTs, T8s+, A2o+, KTo+, QTo+, JTo, T9o }
Hero 62.43% 60.69% 1.73% { As9s }


- When we check he'll bet AT+, his Tx, his TT, 77, and check back the rest
- When we bet 1/3 pot he'll fold his few JJ-QQ and his BW gutters, call all his Ax, raise a few of his Tx (say KT, QT), and of course TT, 77


EV FOR C/F
He's checking back 64% of his range and we have 97.3% equity vs his checking range.

(0.64)(0.973)(2.21)=$1.38


EV FOR B/F 1/3 POT
He's calling 57% of his range, raising 13%, and folding 30%.

(0.13)(-0.74)=-0.0962
(0.57)(0.56)(2.95)-(0.57)(0.44)(-0.74)=0.756
(0.30)(2.21)=0.663
Total: $1.32

Okay now to try and figure out how we should be playing our entire range. Oh boy.

So I guess we should first figure out what my range is in this spot and go from there.. Pretty much never going to be betting this sizing 2 streets vs. this player as a bluff so my range is going to look something like:

{ AA, TT, A5s+, KTs, QTs, JTs, T9s, A8o+, KTo, QTo, JTo }

Our opponent's unexploitable folding frequency and how he deviates from the unexploitable strategy will tell us how we want to be playing our range. The thing that complicates things is I'm going to want to be betting different sizes with different hands, and frankly I don't know how to handle this mathematically. So to simplify things lets say I have 2 sizings here, 1/3 pot and 2/3 pot.

BETTING 1/3 POT
When I bet 1/3 pot our opponent's unexploitable folding freq is

0.74/(0.74+2.21)=33.5%

Which means his unexploitable calling freq is 66.5%. Any deviation from these frequencies is exploitable.

From our above post we know when we bet 1/3 pot he's calling 57%, folding 30%, and raising 13%. I'm not quite sure how to factor in his raising frequency here. Does it matter? He's folding less than his unexploitable freq which means we can exploit him by bluffing less (probably never vs. this opponent as he's never going to adjust).

Our Ax only starts to have enough equity vs. his calling range when we hit A8.
If he folded more than 33.5% could we turn A5s-A7s into a bluff here profitably?
Obviously we should bet our AA, TT, and trips.

So our B/F range is A8-A9, AJ-AK
Our B/C range is AA, TT, AT, KT
Our C/F range is A5s-A7s

Problem: We've already figured out that checking A9 is more profitable than betting it, so presumably A8 will be also.

Amended C/F range is A5s-A7s, A8-A9


BETTING 2/3 POT
When I bet 2/3 pot our opponent's unexploitable folding freq is:

2.21/(1.47+2.21)=60%

Which means his unexploitable calling freq is 40% and any deviation from these strategies is exploitable. I figure when we bet 2/3 pot he will fold some Ax (say A2-A5) and probably only call his QT, KT, but continue to raise AT, TT, 77 so his new action frequencies are:

Fold: 44%
Call: 52%
Raise: 4%

So he's folding a lot less than his unexploitable calling freq which means we should never be bluffing..

So our B/F range is AK, T9s, JT-KT
B/C range is AA, TT, AT
C/F range is A5s-A7s, A8-A9, AJ-AQ




Now obviously we don't just have these two sizings and we aren't just using these sizings all the time for our entire range. We can increase the value of our range by choosing to B/F 1/3 pot with like AJ-AQ... Also I'm not sure how to account for how our blockers affect his range frequencies when we're not dealing with hero's individual hand but his entire range.

So that's a wall of text. My best shot. I suppose practice makes perfect and this is the type of thing I should be doing over and over again until it's completely internalized.