Interesting spot at Red Rock yesterday between two villains. The button straddles for $5. Folds around to MP, who has let his stack dwindle down to $55. MP calls the $5. Folds to the cutoff who raises to $25. BTN and the blinds fold and it's back to the shortstacked MP with some dead money in the pot.
Two players, one left to act: (Pot $38) - Effective stack $50.
MP tanks for a few seconds and then CU--who may have thought he took down the pot preflop--tables Q
Q
. The hand is still live. MP looks at CU's QQ and folds.
I think this was a big mistake. Let's look at it. And let's ignore the fact that most of us wouldn't be sitting at $1/$2 with $55, never mind open/limping the straddle with it.
Now, MP's limping range equity is around 23% against QQ, and he would need around 42% equity to just ship in his remaining $50 preflop with essentially zero fold equity, so that would be a spew.
However, with perfect information on our opponent's hand, we should be taking a flop for $20 and then either continuing when we're ahead of QQ, or bluffing our remaining $30 into a $58 pot if the flop comes A or K high.
Obviously, CU isn't going to fold very often in the latter case, but will he fold say...25% of the time?
I'm not entirely sure how to approach this question properly, but that's never stopped me before, so let's gooooo.
As I mentioned, Equilab gives MP's limping range 23% equity vs QQ if the hand goes to the river. Does that mean that MP would be winning roughly 23% of the time if the hand was somehow stopped on the flop? I would think so. Certainly, I could be wrong, but let's run with it...say MP is beating QQ on the flop 23% of the time. Therefore he's folding 77% of the time and losing his $20 investment.
Not so fast, because MP is bluffing A or K high flops. How often do those show up? According to a
Poker News article, the flop comes A or K high 41% of the time when you're holding QQ. (For now, let's not count blockers if MP is holding an A or K, because then he's betting for value, and that instance will fall into another section further below.)
So, instead of folding 77% of the time, MP is only folding 45% of the time and then losing his $20 investment on the flop. The rest of the time, he's either bluffing or value betting. So when he folds it's 45% times a $20 loss which equals
(-$9.00) in EV.
With 45% of instances covered by folding, that leaves us 55% of instances to go. If MP is ahead on the flop 23% of the time, that means he's bluffing 32% of the time, as these two figures add up to the remaining 55%.
Now, sometimes MP is going to suck out after bluffing on the flop and getting snapped off, and sometimes QQ is going to suck out when it's behind and MP was value betting on the flop. Let's just take those two instances and say that they roughly cancel each other out, and give them a 0 EV overall.
Granted, QQ is much better than MPs range, but it's also harder for pairs to suck out when they're behind on the flop, as they often only have two outs to a set or some unlikely four straight or four flush outs (or counterfeiting board pairings with overpairs vs lower two pairs.) But we'll call it even and move on.
We'll break down the bluffing next. MP bluffs 32% of the time, during which he gets snapped off 3/4 of the time and gets a fold 1/4 of the time. That means he gets snapped off in 24% of total instances and loses his $50 for a
(-$12.00) EV loss. He gets a fold 8% of total instances and wins $88 for +
$7.04 EV gain.
Finally, MP actually wins the full $118 pot 23% of the time for a
+$27.14 EV gain.
Eh, not so fast...because QQ is going to be either folding or checking down some of the time, particularly when an A or K flops. We'll give MP his full $118 win 3/4 of the time and only the $58 in the pot on the flop 1/4 of the time...well no, that's not right either, because even on bad runouts, MP might still be able to fish another $10 or $15 out of QQ on the turn, and then QQ will be almost obligated to call off another $15 or $20 into a roughly $100 pot on the river.
So, let's say out of the 23% of value spots, MP wins the full $118 in 20% of total instances for a
+$23.60 EV gain, and wins $58 in 3% of total instances for a
+$1.74 EV gain.
The final tally, then, is:
(-$9.00) +
(-$12.00) +
$7.04 +
+$23.60 +
+$1.74 =
+$11.38 in EV.
Now, some of you may have noticed that MP appears to have lost EV overall from bluffing, so what happens if MP
never bluffs?
In that case, he folds 77% of the time and loses his $20 for a
(-$15.40) in EV.
For the remaining 23% of value, it's the same as the top equation:
+$23.60 +
+$1.74.
This gives MP a total EV of
+$9.94, less than if he had used a bluffing range. So what happened here? If my math is right, MP loses less money bluffing than he does folding. I know that folding is supposed to be 0 EV, but here we're accounting for the $20 MP put in on the flop...so it's the EV of call/fold, call/bluff, or call/value bet.
So does that mean MP should always bluff? Probably not, because QQ may never fold on a flop that doesn't have an A or K, so that brings the overall fold equity way below 25%.
I'll mention here that I left out MP semi-bluffing flopped draws on non-A or K high flops. If anyone can think of some math to work that out, then cheers to you, it would be great to see it. I'll venture again that for all types of bluffs we'll have to assume very low fold equity on non-A or K high flops.
What interests me most about this hand is thinking about how I would play this if we both had $300 stacks, instead of the dinky $55 effective.
I would think that the bluffs would have to be bigger and across multiple streets to maximize EV. Could we ever bluff a low scary runout? What about check/shipping an A or K high flop for maximum fold equity? Is it too much of a risk, or do we risk actually ruining our fold equity with a suspiciously huge bet on the flop, and if so, then what about shoving the flop for value?
I think I'm going to cross post this one over at LLSNL. Might be fun.
Last edited by suitedjustice; 10-15-2020 at 03:57 PM.