I posted this stuff in my journal, but I thought the readers of the NLHE section might appreciate seeing it too. If it's not appropriate to double post, mods can feel free to just close this.

K high flops, Early Position vs BB call -- Part 1, IP strategy

First, let's take a look at IPs equity on our random selection of flops:

(Note that the last bar is just average of all the other bars, and I accidentally included one queen high flop and one 7 high flop in there somehow...)

In general, IP is doing pretty darn well on these flops. Not too much of a surprise, since OOP is going to call with a fair amount of suited low cards and doesn't have as many nutted hands like KK, AK, AA. I believe Kings are also simply more frequent in IP range than OOP, so IP is more likely to have top pair than OOP.

Because of our equity advantage, we expect to be able to bet pretty frequently on these flops. So, let's take a look at just how often we bet on each flop, broken down by hand type:

There are a couple of interesting points that stand out to me.
(1) If the board is dry and disconnected we bet a lot.
(2) If the board has is connected, we want to polarize more. My suspicion is that these boards are good to check raise on if we bet small, since OOP has a lot of bluffs that can improve to the nuts. We'll take a look at that in the OOP response.

It's also important to note that on most of these boards, we're betting around 60%, so we're checking pretty frequently, even though we have a good equity advantage. So, let's try to figure out which hands we're checking.

Here's how often we're betting our strong hands:

First thing to note, we're basically always better AA or better on these flops, which means we have to check back a good number of kings to balance our checking range. Of course, the KQJ board is a bit of an outlier. The problem is that our opponent has some very nutted hands there, and a hand like a pair of kings really isn't very strong. Even two pair is not amazing here.

For a random check at which Kings we're checking back, here are some of the Kings from the K73 board:

This is a general trend on most of the boards -- we check more with our weaker kings. We have a slight preference toward betting the backdoor flushdraw too, so K8s bets a lot more often than K8c.

Finally, here's how often we're betting our weak hands:

(Obvious note that two_overs doesn't really make sense on K high boards... but it shows up because my code displays it when it exists... and I accidentally included that 7 high flop.)

Weak draws are backdoor flushes and gut shots. These are filtered to not have a made hand to go with them.

An interesting note is that it looks like we're giving up with a lot of our air on the "scary" flops that have multiple face cards.

IP Live play

Hand One:

The first thing to note is that this board is quite disconnected, which tends to lend itself to a frequent bet strategy. Because of that, we'll likely want to restrict our strategy to either betting small or checking. (If you haven't thought about this before, think about betting volume -- we can bet the same total amount of money with our range by either betting big rarely or by betting small frequently). The reason I tend to like the small bet on disconnected boards like this is that it's a lot harder to play against. To meet the MDF, villain will have to defend with something like 75% of their range.

Now, for our specific hand. If we had 76d, I think this would be an autobet. As-is, I think it should still be a bet the vast majority of the time. Our hand has pretty good equity on this dry board, but really needs protection. We benefit a lot from getting a hand like JTo to fold (which I suspect it will).

And it worked out as expected. The one interesting thing I noticed in PIO is that villain actually underdefends compared to MDF here, making up for that by frequent check raising.

Hand Two:

(Yes, I know this says hand 6... it kept giving me either the K73 flop or one of the mistaken 7 high flops and I just sped through them. Back to thinking.)

This board is very connected, there are several straights possible, and since it's all broadways our opponent has a ton of two pairs. We have the advantage in sets though, because our opponent will frequently 3bet KK, QQ, and JJ. We both have most of the nutted AT hands, but our opponent sometimes 3 bets ATs so we have an advantage there.

Along with our nut advantage, we also have a pretty good equity advantage (something like 56% here). In order to take advantage of our nut advantage, I suspect we mostly want to polarize here, and get the most money out of our nutted hands.

Given that, I think check with AQ is pretty standard.

The turn is the 4c and villain checks again.

Now we're in a relatively interesting spot. This card is a total blank, so we have to analyze how we're doing against villains xx range. I think we're well ahead of villains xx range, however, I think weak kings make up enough of the xxc range that we're not really making much money by betting. Additionally, the hands that our are value targets are things like QT, Q9, JA, JT and J9. I suspect the majority of these hands will call one bet whether we make it here or on the river, and if we're checked to on the river, I think these make up a much bigger portion of OOPs bluff catchers than they do now.

We have no one to deny equity from, because anything with a straight draw probably also has a pair. So, I decide to check back.

On the river we face the 3h and a full pot donk:

Clearly our hand is call or fold because we're a pure bluff catcher at this point. We need to defend about half of our range against the full pot bet. I suspect we're in the top half, so I go ahead and sigh-call.

And we lose:

K high flops, Early Position vs BB call -- Part 2, OOP strategy

As we've already seen, these boards are quite good for IP. Here, we do a little bit of analysis of what the OOP player does to respond to various c-betting strategies.

Let's examine how much PIO likes to xr the flop. First, I should note that I force OOP check on these, so we still have our full range always.

Here's our xr vs a small bet:

(Again, the outlier is the accidental 764 board I put in.) Our check-raise percent is mostly hovering around 15%. Note that "nuts" is two pair plus, so "strong" in this scenario for OOP is basically just "top pair" "medium" is mostly any other pair and "weak" is a hand with no pairs.

We can compare this to our strategy against a big bet:

Naturally, we check raise a lot less, because IP is polarizing more when they bet this big size. Where do we lose the raises? Let's take a look at what each category of hand is doing more closely.

We xr our nutted hands a bit less (perhaps about 2/3 as much in aggregate), but our strong hands see a big change. We're xr our kings 3,4,or 5 times as much depending on the board (I'll leave it to you to analyze each board individually if you're curious). We see the exact same pattern with our weak hands, unsurprisingly. The amount we can bet our weak hands is directly correlated to how often we're betting our kings.

More Live Play, OOP

Now for some live play OOP.

In our first hand we face a full pot bet on the KQ8 board.

This is unsurprising because we expect IP to polarize more on these two high-card flops. I think we can actually just fold pocket 3s here pretty much always. We definitely have better hands to bluff catch with, especially since we have many hands that can improve where as 3's are mostly just done.

Okay, looks like our second hand is a little more interesting. We have top two, and our opponent checks back.

That's not too surprising, as we expect villain to be checking about 40% of the time here, and in particularly to be checking a pair of Kings about 40% of the time.

As we saw, our opponent is much more heavily weighted toward weaker kings at this point. Additionally, we just block a lot of continues from villain.

I think we definitely want to bet this turn at a decent frequency because we can get value from both of the flush draws, any pair and potentially straight draws with a blocker. I suspect we want the small size given that villain is weighted toward weaker pairs at this point.

Villain folds, we win two hands in a row this time!

It looks like we gave up some small amount of EV as compared to betting bigger here in equilibrium. I still feel like the small size does a good job of targeting the weaker paired hands, although we do block a lot of those. So, I'm not sure what is best here or why we prefer the larger sizing. I'd love to hear your thoughts!