What’s Your Play? Second Nuts on Four Flush River Results

Thanks to those who commented on this week’s “What’s Your Play?”. I think this may have been the liveliest discussion yet, and I’d encourage everyone who hasn’t to read those comments, because I think those conversations are at least as valuable as what I’m going to say here.

I’ll begin with a small mea culpa: not only did I apparently screw up the pot size, but I think I also misremembered the board cards, because I’m pretty sure a straight flush was not possible. Thankfully I don’t think either of these mistakes affected the analysis much, but I do apologize for them.

Plan For a Raise

This is a big pot, and it may be about to get a lot bigger. Making a mistake when the seventh and largest bet goes into the pot would be extremely expensive, whether it’s a bad call or a bad fold. Even moreso than usual, then, you need to have a plan for how you will respond to a raise before you bet.

I think folding would be correct. A lot of you, Nate most insistently, made the same basic point that it’s a tough spot for Villain to check-raise as a bluff because the Ah is a healthy part of Hero’s range.

Moreover, it’s just generally difficult for people to pull the trigger on a 180BB river check-raise bluff.

Finally, as Rant 2112 says, “We know from his line that he has a value hand on the river 99% of the time. He’s not going to turn it into a bluff. He’s just going to show it down.” By “value hand” I take him to mean a hand with showdown value, ie a hand that could beat most of my bluffs. In that case, most players, live players especially, will prefer to call rather than check-raise bluff. People like to see showdowns, especially when it means risking a lot less money than a huge bluff into a nutty range.

That said, we shouldn’t read too much into the “primarily a live player” read. That was just a hunch, I didn’t have a lot to back it up. Regardless, whoever he is, it’s not too likely that he’s sick enough to check-raise bluff here with any great frequency, so I’m all for folding if raised.

How Often Do We Lose?

Given that we’re going to fold to a raise, any time that he does surprise us with a check-raise bluff, it’s going to result in a very expensive mistake. We have to determine just how confident we are that he won’t bluff and then allow for an appropriate risk of being wrong as part of our argument against betting. Here I think the risk is small, in the absence of extenuating circumstances which we’ll get to in a moment.

So let’s say that 5% of his raises will be bluffs, and the other 95% of the time we’ll be beat. Depending on how much we bet, that those bluffs would cost us $150-$200 in equity that we’ll need to weigh against the upsides of betting.

Of course he’s never folding a better hand, so next we should try to estimate how likely he is to hold the Ah and assume that we always lose the pot when that happens. There are three ways he could get to the river with the Ah: 1) slowplaying the flopped nuts; 2) calling down with AhX where X made a pair on either the flop or turn (we can put AhAx in this category as well); 3) calling down with AhX and no pair.

Given the depth of the stacks, I don’t see a lot of reason for (1). He’d need to raise at some point in order to get stacks in by the river, and I’d expect him to want to build the pot before the board pairs or a fourth heart comes, so I don’t think there’s a high risk of this. If he has a flush to the Ace, he almost certainly made it on the river.

Most of the scenarios for (2) would require him to call a 3-bet with an offsuit rag Ace deep and out of position against someone he probably assumes to be aggressive and one of the tougher players at the table. I don’t think we’re going to see A3 or A5 ever, and even AT would be a pretty bad call. AhQx would make a lot of sense. It slightly contradicts my read on his pre-flop range, but that wasn’t a strong read, so I think we can safely count three combos of AhQx and none of the others.

We also need to consider AhAx. That hand is more unlikely based on the preflop action, both because he may have made a larger opening raise and because he my have 4-bet it. Then again, this deep he may just choose not to have a 4-betting range pre-flop, so let’s count one of the three combos of this hand.

As with (2), the possibilities for (3) are constrained by the fact that Villain would have had to call the 3-bet from out of position with an offsuit Ace. Only AK and AJ are at all plausible, and as with AA/AQ both may have been played differently pre-flop. Unlike with AA and AQ, though, it’s less clear that he would take this line post-flop. Although I don’t think it would be awful, I wouldn’t call it standard either. Both AhKx and AhJx would be good candidates for check-raising the flop or turn. Generously, we’ll give him three combos of those.

The cost of betting then, is getting called by seven combos of hands with the Ah, plus a small risk of a bluff which amounts to much less than one combo (.35 to be precise, if we go with our 5% guess).

How Often Does He Call With Worse?

We estimated that he’ll raise with somewhere between seven and eight combos of hands, so if he calls with just eight combos of bluff-catchers, we have a profitable bet. These are harder to count, but we can get an idea by looking at both his and Hero’s range for getting to the river.

Villain probably doesn’t expect Hero to turn many hands into a bluff on this river that he thought were good enough to bet for value on the turn. If Hero expected to be good on the turn, then he’s probably still good on the river unless Villain holds a high heart, in which case he probably isn’t folding. That means Villain can only hope to beat triple barrel bluffs.

Of course if I’m 3-betting K2s my actual 3-betting range is quite wide, but I also wouldn’t 3-bet it if I expected Villain to realize quite how wide I actually was. My best guess is that he might expect my 3-bet range to look something like {TT+,A2s+,K9s+,Q9s+,J9s+,T8s+,97s+,86s+,76s,ATo+,KQo}. Even so, that leaves about 87 combos (according to my rough count that, historically, is likely to be inaccurate) that he expects me to 3-bet pre-flop that can’t beat a pair of Tens on the river, which is about the point where I think he’d expect me to take my showdown value rather than bluff.

Most if not all of those hands bet the flop. The turn is a pretty good barreling card, so even if he assumes that I’d barrel only half of my air range, that would still give me more than 40 combos of air that get to the river this way.

The one thing we haven’t accounted for yet is the fact that my barreling range ought to be weighted towards Ah, Kh, and other hands with hearts in them that improve on this river. So if I were actually betting half of my hands with trivial showdown value on the turn, much less than half of that range would still be air on the river. Let’s say then that I have 20 combos of air after seeing the river.

I’ll have about that many combos of Ah and Kh, depending on exactly how weighted towards Ax and Kx he expects my 3-betting range to be. The range that I gave above is pretty biased in that direction. This is a good bluffing card, and I’ve got plenty of air in my range, so if he perceives me to be aggressive, it’s a good bluff-catching spot, especially if he isn’t sure whether I’d bet the Kh for value (and I wasn’t sure myself!).

The catch is that his best bluffcatchers are hands containing the Kh, which of course Hero’s hand blocks but Villain doesn’t know that. So his bluffcatching range may be narrower than usual for that reason.

Ignoring the fact that we have the Kh, his plausible bluffcatchers are AxQh (3), KxQh (4), KhQx (3), AA (3), KK (6), QQ (3), TT (3), 55 (3), 33 (3), JJ (6), 9h9x (4), 8h8x (4), QhJh, KhJh. That’s 47 combos, of which 7 are blocked in this instance and some, like the big pairs, discounted based on the preflop action. We’d only need him to call 20-25% of his bluffcatching candidates for us to show a profit. If he only continued to a river bet with Ah or Kh hands, he’d fold to a bluff roughly 75% of the time, which is about twice as often as we’d need for a bluff of $1500 to be profitable.

If he bluffcatches with anywhere near the right frequency, and if he rarely check-raises bluffs, then this is a bet.

Bet Sizing

There was some interesting discussion on bet sizing as well, with some people advocating an overbet to polarize our range and deter a check-raise bluff and others advocating a small bet to keep his calling range wide. I incline towards the second camp myself, because I think that when we bet big, we should actually be polarized.

This is because the first hands in his bluff-catching range should be those containing Kh. The larger our bet, the smaller his bluff-catching range ought to be, and so the greater the ratio of Kh to other combos. In other words, when we have the Kh in our hand, he should call disproportionately more often the smaller our bet.

However, as swallsjr points out, “We want to size our bet on the river to maximize the chances of getting called by worse, without inducing a check raise by worse.” That means we don’t want to bet so small that he excludes the Ah from our range, because then we don’t know what could happen. I think somewhere in the neighborhood of 1000 to 1500 is what I should have bet.

Results and Lessons

Remember I mentioned extenuating circumstances? Obviously I couldn’t have thought through things so thoroughly at the table, but I still went into the tank for a while when he checked to me. At some point, I began to worry that the longer I tanked, he would bluffcatch less and check-raise bluff more, correctly putting me on a thin value bet.

That may have been giving him too much credit, but eventually I just gave up and turned my cards over. He mucked, and I collected the pot. There are a few lessons to take from this:

1. Do the work ahead of time. You may be looking at the length of this post and wondering how one could possibly do this sort of analysis at the table. You couldn’t, for lack of both time and tools. That’s what makes it important to do this kind of analysis away from the table, so that your instincts and rough approximations are better in the heat of the moment. No matter how much you try to balance your tanking range, you simply can’t avoid giving away some information if you have to think for several minutes about a complex situation.

2. Small edges matter in big pots. Don’t look at this, throw up your arms, and say, “It’s all so complicated, it’s a close decision anyway, just check it back and collect what’s already a nice pot.” If a $1500 gets called worse 60% of the time and raised 40% of the time (and very rarely as a bluff), then you lose nearly $150  $300 by checking back. That’s 6 12BBs, at least an hour’s two hours’ worth of work at a live cash table if you have a solid edge in the game.

Thanks for playing, and good on ya if you made it this far!

Edit: Corrected math error in the second-to-last paragraph.