As a side note, just because you only have 45% equity against his calling range doesn't mean shoving isn't the best option. If you have 26% equity against his shoving range if you check, shoving here will be better than c/f even if you're only ahead 33% when called.
Ok, so at first I was confused because I was pretty sure you needed to have more than 50% equity on the river for value betting to be correct. After doing some quick numbers it looked like Nick was correct but I was still very skeptical. I've been out running errands all day and every so often I would think about this at it was driving me pretty crazy, damn you Nick... ha just kidding, thanks for the comment that really made me think. Also thanks to TJ for your comment a definitely look back through your HH's every so often to look for spots like this.
After doing some more work, what I've realized is that Nick is correct in some scenarios where shoving is going to be better than c/f'ing even when we have less than 50% equity. But those situations come up when you are afraid of c/f'ing the best hand some of the time which I'm not really afraid of in this spot, although admittedly I'm not 100% sure what Padirk will do with the 6 combos of AK he gets to the river with if I check.
Anyway here is what I did. I basically broke down villains range for getting to the river. He gets to the river by my best estimation with 26 combos.
6 combos of AK
3 combos of KQs
3 combos of KK
6 combos of QQ
3 combos of JJ
3 combos of TT
and 2 combos of KJs.
So we are ahead of 15 of those combos and lose to 11 of them. Lets assume he calls with all the hands we beat except QQ. So then we get 6 combos of folds, 9 combos of called by worse, and 11 combos of called by better. This is assuming about as wide of a calling range as we can give him(probably too wide for this particular player)
EV= ((6/26)*570) + ((9/26)*570+397) + ((11/26)*-397) = $298
Ok, and then the EV of checking assuming that he checks back all worse(never bluffs/never value bets worse) is
EV= ((15/26)*570) = $329
This is about as simple as we can make this calculation, so as you can see c/f'ing is about $29 better than shoving. And both are way better than c/c'ing. If you are wondering what the EV of c/c'ing is with our above assumptions is
EV= ((15/26)*570) + ((11/26)*-397) = $161.
The really tricky part of all of this is figuring out what he's going to do realistically with those 26 combos on the river. My best guess is that he folds all of his QQ, KQs and maybe hero folds 2 of the 6 combos of AK. So now we are at 11 combos that fold to our shove, 4 that call and lose, and still the 11 that call and win.
EV= ((11/26)*570) + ((4/26)*570+397) + ((11/26)*-397) = $222
Now we have to guesstimate what he is going to do with AK when we check. This is the most important part of everything because this will decide whether to shove, c/f, or even (gasp!) c/c.
My best guess is that he shoves at most 1/2 or 3 of the AK combos he gets to the river with, and never turns a hand into a bluff. So now the EV of c/f'ing the river is
EV= ((12/26)*570) = $263
And even with him shoving 3 combos of AK that we beat c/c'ing the river is
EV= ((12/26)*570) + ((11/26)*-397) + ((3/26)*570 + 397) = $207.
So in general it appears that check/folding the river is always going to be the best play vs this player given the above tendencies. With all of my number crunching I don't think I ran into a scenario where shoving was the most + EV play however. So I'm still not sure what Nick is referring to. Maybe Nick you could post a scenario? My brain is a bit too fried to go through the scenario you described above with regards to the 26% and 33% equity.
The only scenario I've come up with where shoving is better than c/c'ing is when c/c'ing then becomes the best option. And that comes up when I think the player will hero fold 2 of his combos of AK when I shove into him, but valueshove the 6 of his combos of AK when checked too.
Those EV's look like this
Shove = ((11/26)*(570))+((4/26)*(570+397))+((11/26)*(-397)) = $222
C/F = ((9/26)*(570)) = $197
C/C = ((12/26)*570)+((11/26)*-397)+((4/26)*(570+397)) = $253
Ok. phewwwwwwwwwww that was a lot of work. I'm out. If anyone finds and discrepancies or problems leave a comment. Later.
Nice post Nelson. I wrote down a few things in generalities to clear up some stuff in my own head, and your observation about the relationship between shoving, c/c and c/f is not an accident.
ReplyDeleteHere's a few abbreviations I'll use to simplify my equations for shove EV, c/c EV and c/f EV.
FR = folding range
CRB = calling range better than our hand
CRW = calling range worse than our hand
CBR = check back range
SRB = shoving range better than our hand
SRW = shoving range worse than our hand
Using that, we have these equations:
Shove EV = FR*(Current Pot Size) + CRB*(-1)*(Max Pot Size) + CRW*(Max Pot Size)
C/F EV = CBR*(Current Pot Size)
C/C EV = CBR*(Current Pot Size) + SRB*(-1)*(Max Pot Size) + SRW*(Max Pot Size)
You'll notice a few things about this set of equations. First of all, C/C EV depends on C/F EV, more specifically, C/C EV = C/F EV + A term that values our profit/loss from calling his bet when he chooses to bet.
Now, let's assume that CRB = SRB, in other words, hands better than aces are always calling our shove and always shoving if we check. This isn't a very strong assumption for this hand but maybe it is a fairly strong one in other cases.
We'll notice two things about our equations now:
First, if villain's CBR is wider than his FR, this implies that we err towards c/f over shove, because he is going to check back hands worse than ours which won't call a jam, thus saving us money against hands we lose to without losing value against hands we beat.
Second, if villain's CRW is wider than his SRW, this implies that we err towards shoving over c/c, because he is going to call with hands worse than aces but not bet them himself when we check.
Note that since FR + CRB + CRW = His total range and CBR + SRB + SRW = Total range, thus with our assumption, FR + CRW = CBR + SRW. This shows that the two observations above are correlated, such that when CBR > FR, CRW > SRW and vice versa. Since the equations are related in such a way, we yield that when we prefer c/f to shove, we also prefer shove to c/c, and when we prefer c/c to shove, we prefer shove to c/f.
I know this doesn't further discussion much but it does show an empirical reason for why it works out this way.
Kevin
Very nice breakdown. I agree against this particular opponent shoving can easily be inferior to c/f, just wanted to throw out the disclaimer than shoving shouldn't be dismissed just because we're ahead <50% when called.
ReplyDeleteTo tweak your scenario a little bit creating a situation where shove > c/c and c/f:
He'll call AK+ vs your shove, he'll fold KQs/QQ. If you check he'll shove all hands that beat you as well as 4 combos of AK, he'll check behind with 2 combos AK as well as KQs/QQ.
EV(shove) = (9/26)*570 + (6/26)*(570+397) + (11/26)*(-397) = $253
EV(c/f) = (11/26)*570 = $241
EV(c/c) = (11/26)*570 + (4/26)*(570+397) - (11/26)*397 = $222
In this scenario we're beat 35% of the time when our shove is called and we're ahead only 27% of the time vs his river betting range. It's pretty intuitive c/f > c/c, but what might be less intuitive is that shoving is better than both.
I totally agree it's a pretty rare spot. Some criterias that usually need to be fulfilled:
1. Villain's range is mostly made up by made hands (check)
2. Villain won't/can't bluffshove over our bet (check)
3. Villain is more likely to call a bet than he is to bet himself (probably true as well here, right?)
4. Villain will bet worse hands for value and/or bluff with a decent frequency, but not necessarily enough for c/c > c/f (this is probably where this opponent's tendencies make c/f > shove)
If a bigger part of villain's range is made up by missed draws, the situation changes somewhat. Checking to induce will become more of a valid option, but as long as you think your opponent's betting range will put you in a close spot that doesn't matter. However, if you think villain might check behind better hands, that will screw up the situation completely and should make you more inclined to c/c or c/f.
Usually the spots where shoving will be correct even though we have <50% equity when called will come up against tough players with good bluffing frequencies and/or the ability to v-bet thin. This is basically math's way of saying "position is everything". When we're OOP, a good opponent can always force us to put money in as underdogs. In those situations our aim is to find a way to put in the money being as small underdogs as possible.
I remember getting involved in a discussion about this on 2p2 a long time ago: http://forumserver.twoplustwo.com/19/high-stakes-pl-nl/whats-my-play-here-top-pair-25-50-hu-v-krantz-351832/
nice thread i never read that before, ty for the link
ReplyDeletegood thread nelson and cheers for the abbreviations above.
ReplyDeletemy head hurts
ReplyDelete