Edit: As several commenters pointed out, I screwed this up in multiple ways. I’m getting ready to start up a session, but I’ll be making a new post with a corrected equation soon. In the meantime, you can disregard this post unless you’re curious to see whether you can spot the errors (there are hints in the comments section). In my defense, I was addled with fever at the time that I posted this.
One of my students and I were recently discussing a common situation: You open with AQs in the CO and get 3-bet by an aggressive player on the button. Assuming 100 BB stacks, I don’t like any line that results in you folding pre-flop. I think your options are either to 4-bet-call if you think his shoving range will be wide enough or just to call the 3-bet if you think you’d have to fold to a 4-bet.
He then asked me what would be the cut-off, in terms of the BTN’s 3-betting range, for 4-bet-calling vs. just calling the 3-bet and taking a flop. I wasn’t prepared to give an answer off the top of my head, and as I started working it, I figured it would be an interesting blog post.
The answer will change a bit depending on the sizes of the bets, but the method will remain the same. I’m going to use the actual sizes from the example we were discussion. So, Hero opens for $3, BTN re-raises to $10, and Hero is contemplating a 4-bet to $24.
Basically we’re assuming here that BTN will always 5-bet or fold if we 4-bet. We’re also assuming that we have a good guess at his 3-betting range but don’t know what his 5-betting range will be. Thus, the real question is how wide does his 3-betting range need to be before 4-bet-calling AQ is unexploitable?
To solve this, we can pretend we’re playing our hand face up, with the constraints that he must either shove or fold and we will always call. Thus, he will shove AK, AQ, and all pocket pairs but fold everything else. Our equity gets a bit better the more pocket pairs are in his shoving range, but to make the math easier, I’m just going to give him a shoving range of {TT+,AQs+,AQo+}, against which AQs has about 38% equity.
We want to find the point at which 4-bet-calling is break-even, so we need to write an EV equation for this play and set it equal to 0. The EV of 4-bet-calling AQs will be equal to
0= %Fold * $13 + (1-%Fold) * (.38 * $201.50 + .62 * -$97)
(1-F) (-$16.43) = -13F
F = .56
If he never 5-bet shoves a hand that beats AQs, then he would need to fold 56% of the time to make a 4-bet profitable. The shoving range we gave him represents 4.7% of hands, so if he 3-bets at least 11% of the time in this spot, we can profitably 4-bet-call AQs no matter what his true 5-betting range is.
In theory you’d also want to compare the EV of 4-betting to calling, though it’s considerably more difficult to estimate the EV of calling. Against an aggressive 3-better, though, I want to 4-bet pretty much any hand that I can profitably 4-bet just because it’s so annoying to have someone sitting on my left 3-betting me. The more I can do to discourage that, the better, and generally a strategy that involves a lot of 4-betting is a better deterrent to aggressive 3-betting than is a flat-calling strategy.
Cliff’s Notes for the Math-Phobic: What I did here was find the point where, even against a worst-case scenario 5-betting range, it would be profitable to 4-bet-call AQs as a semi-bluff. If his 3-betting range is at least 11%, then there is nothing he can do to make it unprofitable for us to 4-bet-call AQs.
AQ’s equity does get better, but still the 3-bet range that the opponent would need to have goes up considerably I’m sure if you include 22-99.
In reality though, I think the cutoff for 3-betting range is actually much lower than what you have here, both because 1) many 3-betting ranges don’t some of the hands you have listed as value hands, in particular JJ and TT, and 2) You haven’t considered the non-value hands that might get 5-bet shoved. Hands like A5s that have some good card removal value, and which AQs has completely crushed, are popular choices for 5-bet shoving.
Also shouldn’t the numbers be: win $14.50 if villain folds to 4b, win $104.50 if vill shoves and hero wins, lose $97 if vill shoves and hero loses?
hehe yeah, how do you win 201,5$ when you get it in and win? 🙂
Fixed (I hope). Nice catch as usual.
Using the above example, how often should you 4 bet-fold?
That’s a much trickier question to answer, I’m afraid- so much so that I’m not even going to try to summarize how to do it right now. Sorry 🙁
The short answer is that the above example does not provide all of the information needed to solve that.
How often you should 4-bet/fold is a little more involved, but how often you should be folding to 3-bets (at least in order to be unexploitable) is easy to solve – it has to be enough so that the opponent can’t 3-bet you profitably with ATC. If he’s raising to 10 and you’re opening to 3, he’s risking 10 to win 4.5, so he’s going to have to be successful 70ish percent of the time, which means you should be 4-betting 30ish percent (assuming as above that your only defense against the 3bet is a 4bet).
This also abstracts from the possibility of either of the blinds 4-betting. At lower stakes games, this is somewhat rare as a bluff, and you’ll only see it from top 2-3% hands, but at mid stakes games it seems to be more common. Clearly if you have the blinds helping you out in defending against 3bets to some degree then you don’t have to it as much yourself and tone down your 4-betting while still remaining unexploitable in this respect.
Does 4-bet-bluffing from the blinds really “help” you in this regard? They make it less profitable for the BTN to 3-bet but only in a way that still makes your open raise less profitable. This is particularly apparent in cases where you have a hand like AQs that is good enough to 4-bet-call but that you end up mucking to a cold 4-bet from the blinds.
If there’s more 4-betting from the blinds it would make you want to open less, sure. But I’m sure not so much that you wouldn’t want to open AQs. But you’re right that “helps” probably isn’t the right word in that it doesn’t add to you EV. But it does mean that you don’t have to 4bet as much to deter the button’s 3bets.