I've been reading the new J Little book 100 Essential Tips to Master No-Limit Hold'em. In Chapter 52 he has an example hand which I found quite interesting. Of course, the aspect of the hand that I found so interesting, Little doesn't even mention and seems to be completely unaware of. But I thought it would make an interesting thread.

The hand is as follows:40 bbs deep the button raises (probably to 2bb) and only the big blind calls. The flop comes AQ7. The big blind checks, the button bets 25% pot, and the big blind calls.

The turn is the A. The big blind checks, the button bets the size of the pot, and the big blind who never check-raises, calls 25.5 percent of the time and folds 74.5% of the time. And it's this calling/folding frequencies that I found so interesting (and it's solver output so it must be correct).

Now according to poker game theory, when there's no raising range, and the aggressor bets the size of the pot, since the pot is offering the aggressor 1-to-1 the defender should call at that ratio which would be 50 percent of the time in this case. But here the defender is only calling 25.5 percent of the time. (If there were some raises in his range he wouldn't need to defend at this high a frequency. You can see this in the first example of the next chapter.) So what is going on?

I think I know the answer but would be interested in hearing what others have to say. For those interested this example appears on page 198 in the kindle version of the book.

Mason

I would assume it has something to do with shallow SPR and your turn/river frequencies and equity realization dictating to you to overfold on the turn. 20bb solves look even weirder.

I suspect that there may be more than one reason contributing to what the solver determines is correct strategy and this might be one of the reasons. But it's not the one I'm thinking about (and, of course, I don't have to be correct).

Mason

Probably a combination of two things:

1) BTNs bluffs can check behind and realize all their equity. So you're not trying to make their bluffs worth $0, you're trying to make it indifferent between betting and checking.

2) BB being pacified on all streets. I'm assuming if turn was all in then BB would defend closer to MDF.

So basically the solver can't find enough hands at +0ev @ 40bbs to satisfy MDF against a pot size bet on the turn?

It's a possibility, though you'd think if that were the case BTN would simply bluff more often. I'd have to run a pacified sim and dig into the details to be sure

As I said on twitter to Mason. The most likely outcome is that the sim is somehow messed up.

As for how/why I think I'd need to see the inputs.

Here are my results which are very close to what we'd expect to see. Apologies for the low quality phone screenshot.

There are a few factors that I think are leading to this result:

1) Range advantage: The BB's range is so capped on this particular board that it cannot meet MDF without giving up additional EV.

2) Preflop discount: Related to #1, the BB can call preflop with hands that actually have -EV against the button's range. Since folding preflop means losing 1 BB, the BB should be calling with all hands that lose less than 1 BB.

So the solver game tree will have negative EV in some spots for the BB. As long as the EV of the full game tree is greater than -1 BB the solver's solution is preferable to folding preflop.

3) Bluffs have + EV for button: The MDF is calculated assuming a perfectly polarized range. In reality the button's bluffs will have some chance of winning. This skews the equation somewhat. Slightly + EV bluffs for the button will prefer checking to realize their equity, provided the EV of checking (however small) is greater that the EV of bluffing. So to achieve indifference the BB only needs to make the EV of the button betting less than the EV of button checking, not 0.

Somehow, this doesn't surprise me. When I first saw this result, and again Little doesn't even mention it in the text, I thought that perhaps Little entered into the solver by mistake a far larger bet than the size of the pot, and that would certainly produce the kind of solver output he has in the book. And the above output from JustLuck sure seems to confirm that something was wrong.

But for the purpose of this discussion, let's assume that the solver result in Little's book is correct (and again I now highly doubt that it is). What would cause the solver to determine that you should call far less than Game Theory indicates that you should call, and I do think that this can come up in other situations. (See below for a contrived example).

First, let's define what a bluff catcher is. A bluff catcher is a hand that will lose to all the value bets which the aggressor makes but will also bet all the bluffs that the aggressor makes. So, in this example we see, for instance, that any hand which contains an ace or a queen, will beat all bluffs, so these are certainly bluff catchers since the range has no check-raises.

But when the aggressor bets the pot, the game theory calling frequency is 50 percent of the defender's hands (when there is no raising range). But what happens if the defenders uses up all his legitimate bluff catchers before he hits the 50 percent mark. That is, he'll be left with hands that will be in the top 50 percent of his range yet will not beat all of the aggressors bluffs, and thus, many of these hands should probably now be folded.

Here's a contrived example. Suppose the board is as before, AQ7 and the turn is again the A. Except that this time the range of the defender is all ace-deuce hands and all trey-deuce hands. Also notice that since there two aces on board, the defender can only have 8 combinations of ace-deuce while he'll have 16 combinations of trey-deuce. So, in this contrived example, two-thirds of his range will lose to all bluffs and the defender should only call with the top one-third (33.3 percent) of his range the ace-deuce hands, and not with the top 50 percent of his range since he doesn't have enough legitimate bluff catchers, and he'll also have no bluff-raises since he has no value raises.

Mason

Yeah I didn't initially pay much attention to the exact numbers, just the fact that the BB was overfolding. After reading Mason's latest post, I wondered if maybe the data was based on a 2X pot bet.

I ran a quick sim on GTO+, and in fact the BB was only folding 70.8% vs a 2X pot turn bet. Granted I allowed for a few different sizes, so the ranges were split in some ways and the results will not be exact. I also only ran the sim to .1% accuracy. JL's data looks like a misprint to me.
Screenshot is BB's response vs 2X pot Button turn bet.

There is a PLO video on youtube by Chris Wehner that touches on a flop spot where the solver overfolds significantly more than what it "should".



The time stamp where he gets into it is around 6:30. Situation there is you open, gets three bet, and you are now oop in a small SPR pot, and the flop comes down 666. In this PLO example, the solver folds to a 1/3rd bet 80% of the time. Why? It wants to fix its distribution for the future streets, and it has such a range disadvantage that the only way it can fix this is to fold. It's worth a watch.

When I read Mason's post, I felt like there was a lot of symmetry between the examples (low SPR, with significant range advantage for the opponent and future betting rounds).

I think the villain in the original post is overfolding to improve the range vs range distribution on the river.

Or, after reading this thread, it's probably just a mistake - but the result from the Wehner video is still very interesting and enlightening because it shows where a solver would be overfolding based on simple pot odds analysis for a very distinct reason.

I think the terms "range advantage" and "range disadvantage" are not good terms. Suppose on the river you have the nuts 40 percent of the time and absolutely nothing 60 percent of the time. That is, if your hand is not the nuts it will lose to your opponent's bluffs (when you check it). Do you have a range advantage or a range disadvantage?

Mason

Typically "range advantage" in the sense that I've seen it mostly used refers to range vs. range equity. So if you get a given flop, and the button will win the hand 60% of the time if both players continued with their entire range through to the river, then they are said to have a significant range advantage. At least that's what I meant in my post above when I referred to range advantage. I'm guessing that's what the video is referring to as well.

So for your example Mason you would have a range disadvantage, but you would have a strong nut advantage that may be more significant in that example.

Link, in case it didn't work with yhoutub tags.

PLO Range Advantage

Polarization advantage, range disadvantage.

How would you prefer to see it describe Mason? This language has gone a long ways to help describe these situations and I'd be curious to see better descriptive language used.

Regardless, I'd like to see you comment on the content of the video - and the situation as I find it interesting to see such a dramatic folding range to such a small bet.



What are you going to call that?

This doesn’t look right to me. If we defend by pure calling and never raising, it usually necessitates defending more than MDF. Otherwise BTN can start bluffing all his air, knowing that they are breaking even from fold equity alone (and are outright profitable once you consider equity when called).

Of course this is offset by the fact that checking also has some EV. And that BB might also be overfolding river after turn goes check check. But this usually happens in lines where BB has lots of air, eg after flop goes check check. Not so much in check call lines where BB has folded most of his air on the flop.

you only post the final part 20K3gbOQ1ZE

that's why it didn't work

here it is, you can quote this post to see it in action

First, let me say that I'm not very good at PLO.

I finally got a chance to watch this video and I suspect that what is happening is that the solver is saying that the player in the big blind just doesn't have many bluff catchers. That is, looking at the defender's range (with the KK88) on a 666 flop, once you get past his very strong hands there are very few hands left which can be considered bluff catchers. So, even though GTO says you should normally call much more than is being done here, that's because (I think) that in most situations, there will be enough hands to beat the opponents bluffs (which won't be many considering the small bet size). However, if your so called bluff catchers can't beat all your opponents bluffs, this changes things.

One thing that I do disagree with are his comments about range disadvantage. In many situations, I suspect the majority of them, you can be at a range disadvantage yet still have plenty of bluff catchers. In those situations, I also suspect that the solver results will be much more in line with general GTO poker theory. That is there will be many more hands to call with, especially if the bet on the flop is small.

Another interesting thing about this hand is why does the solver bet small? In this spot, I suspect that the solver is recognizing that the opponent, the player in the big blind in this case, won't have enough bluff catchers to call. So, there's no need to bet big since the bettor will often pick up the pot no matter what the bet size. In a more normal situation, I suspect the solver would suggest a larger bet to get the defender to fold some of his bluff catchers.

Another spot in the video where I disagree a little is where he says we don't want to give up a percent-and-a-half of the pot and that's why we don't just check-call the turn, While this is obvious true from an expectation perspective, some players may want to do this to reduce variance, and this, of course, depends on the size of your bankroll.

Another thing that I would have mentioned in the video, is that when you can (and should sometimes) raise, you don't need to call as often as GTO would suggest if you could only call.

To finish, even though some of my comments may seem negative, I thought that this video was actually well done and that Chris Wehner presented the information in a clear and concise manner.

Mason