About two weeks ago I played a seemingly simply hand against my buddy Hank. I found myself in a 20/40 game with him and Pete on my last day as a Bay Area poker player (the following day would find me loading a U-Haul with the help of Bravos, and the next day driving to Orange County). Anyway, the game was playing pretty fast and loose, with at least Hank opening up his opening ranges pretty substantially. I wasn't actually doing this and neither was Pete, I don't think, but that's not really important. What is important is that at the time of this hand I wasn't yet sure just how much bonus aggression Hank was putting in. First I'll just give you what happened in the hand, then hopefully something interesting will fall out of me writing about it:
Hank opens UTG + 1. I 3-bet QQ 2 spots later in the highjack. All fold and Hank 4-bets it. I opt just to call (no more cap since we are heads up) and we see a flop.
Hank bets dark and I call
752-3 with a flush draw
Hank bets and I call
Hank bets and I call
"So what" was kind of my original thought on this hand. But there is actually a fair amount going on, and I may have missed some value with my hand. Here was my thinking on each street.
Obviously I have to 3-bet with QQ the first time around. When Hank 4-bets I have a decision to make. Hank and I's dynamic is that basically neither of us ever folds to the other, so if I 5-bet I'm setting myself up to get crushed by the hands that beat me, especially AA. He can either just call and check/raise any flop, or 6 bet me right away at which point I'll basically have to call down almost regardless of what board cards come. If the pot swells to 13 small bets preflop I'll be getting 18:5 on a pure call down, which is a price I won't be able to pass up. Also I figure to be about even money or maybe even behind his capping range, so there's not even immediate value in putting in a 5th bet. This is assuming that his range is something like TT+ and AK (against which I'm a slight favorite), with maybe a slight discounting of TT and AK (which would obviously put me behind pretty quickly, since those are 2 of the 3 hands I beat). More on the discounting later.
This is one of those hands where the board cards basically don't matter. There's always a small chance your opponent left the reservation preflop (but with Hank this chance is exceedingly small), but even if 77 had somehow crept into his range it's hard to flop a set and it's just something you have to live with (and if it did, then 88 and 99 are in there also, both of which I have dead to rights). So for all intents and purposes Hank and I are playing with our preflop holdings. I chose not to raise here for basically the same reason I chose not to raise again preflop. I feared that doing so would encourage Hank to play well against my hand. It's not just a matter of me winning one extra bet when ahead and losing just one when behind. If Hank has AA (or maybe even KK) I'm going to suffer a pretty heavy penalty for raising the flop. He'll either 3-bet me right away, or worse yet check/raise me on the turn. With JJ or TT, he'll likely just call me down. Again in a perfect world I could perhaps get away from my hand at some point, but I have queens against a good player and friend who is kinda horsing around on a 7-high board. With TT or something we could start the search for a hero fold. But I have too much hand and am committed to showing down.
After thinking about the hand this is the street I wanted to mulligan. So far in the hand I have under-repped my holding pretty substantially. Hank could easily think I was horsing around preflop, and all I've done since the 3-bet is call two small bets and gotten to see 4 of the 5 board cards hit the felt. In short, he can't narrow my range nearly as much as I've narrowed his (which may or may not be a mistake), and I should be free and clear of the hammer of AA. Specifically if I raise this turn, I believe Hank is going to have a hard time 3-betting me with AA for fear that I got out of line preflop and have some ridiculous straight, two pair, or set type hand. But at the time I didn't properly weight that possibility and was assuming Hank was pretty confident that I was playing close to correctly preflop. It was only after the session in discussions over email that I got the sense Hank was really taking some preflop liberties and therefore would probably assume me capable of the same sort of stuff. So basically a raise here probably gets called down by all the overpairs left in his range (against which I'm about 50/50), but puts extreme pressure on AK, forcing it to either call a bet at roughly break even odds or fold (he'd bet getting a little more than 7:1 to call the raise, and AK has 6 outs of the 47 he can't see).
This street is pretty straight forward in a vacuum. The way I've played the hand so far basically makes raising kind of pointless. I'm gonna get looked up 100% of the time by AA and KK, any non-pair hand will almost certainly fold, and against the preflop range I originally assigned Hank there aren't enough pairs that I beat for the raise to show a profit. I realized afterward that without explicitly thinking about it I had heavily discounted TT from Hank's range, assuming that a cap and dark lead was at least not something he'd do 100% of the time with that holding.
So now for a little bit of math, which was the point anyway. First, we'll assume Hank's range really is TT+ and AK and figure out what happens if I take my line, raise the flop, or raise the turn. Also, I won't consider 5 betting preflop. First, the hand combos:
AA - 6
KK - 6
QQ - 1
JJ - 6
TT - 6
AK - 16
For government work, it's close enough to say that against AK on the flop I am a 3:1 favorite, and that an under pair is a 9:1 dog against an over pair. Also, I'm going to leave the one QQ combo out of the discussion and assume that Hank barrels off with his entire range if he's never raised (something he confirmed via email later).
My triple call line pretty straightforwardly wins 2.5 bets 100% of the time when ahead and loses 2.5 bets 100% of the time when behind. What's my equity on the flop? Well PokerCruncher tells me 59.3%. We could figure that out ourselves just to see if my government work is roughly correct:
12 combos of AA-KK I win 10% of the time.
12 combos of JJ-TT I win 90% of the time.
16 combos of AK I win 75% of the time.
1.2 wins, 10.8 wins, and 12 wins against 10.8 losses, 1.2 losses, and 3 losses
24 wins, 15 losses, which comes out to 61.5%. Obviously I'm off a little bit, the main reason for which is that AK actually has ~26% equity, not 25%, and leaving out the QQ also helps my cause slightly. The point is that I win about 60% of the time, showing a net profit of:
2.5 * .6 - 2.5*.4 = 2.5 * .2 = .5 big bets
So a triple call down is profitable. Great. Now let's try a flop raise, with the assumption that I don't ever fold and that Hank check/raises the turn with AA and folds the river with AK. Folding the river with AK is obviously bad for me, and Hank probably wouldn't do it 100% of the time. However, he might also put in more action with KK which I'm just gonna say cancels out.
So for this line, I win 3 full bets against TT and JJ, but lose 4 bets against AA, 3 against KK, and collect only 2 from AK. Actually when you say it that way it's pretty obvious this line is worse with the assumptions that I've made. It's also becoming kind of unwieldy to figure out what happens in the cases where one of us improves. What if Hanks spikes of set of tens on the river, do I have the presence of mind to check? I should. What if a king falls on the turn and Hank donks? What if he doesn't? What about the 1 in 200 times he set over sets me? I set over set him? Is he really calling the turn with AK? All of the set flopping kind of cancels out and actually works in my favor since I have position, but how we'd handle aces and kings falling off the deck is just more work than I want to do here.
Now we'll try a turn raise, which is easy enough to figure out completely since I know the turn card was a blank. To further simplify it I'll assume it was a 4th rainbow card so I don't have to deal with the one combo in Hank's 40 hand range that just picked up a flush draw. Our assumptions are that Hank won't ever 3 bet me on the turn and if he does I have a mis-read preflop and can just fold, and that he'll actually fold AK on the turn. Basically the problem boils down again to the handling of AK, since I'm breaking even against all his over pairs (assuming he never folds them). But getting him to fold AK on the turn and not see the river is a coup, since it steals away 1/8th of his equity and he was getting odds to draw against my specific hand (assuming he collects a bet on the river). Sorry if that was a little dis-satisfying of a conclusion, but I've been at this for a while now and have some other things to do today. In short, I think I should raise the turn because it forces Hank to make a tough decision with a hand that makes up a huge percentage of his range. Limit Hold 'Em is all about making your opponents make mistakes, and a turn raise was a chance to do so in this hand.