My post from last night flubbed a simple equation badly enough that I felt the need to scrap it and start over from scratch. I’ve left the old post up for posterity’s sake and so people can see what not to do. Here’s what I should have said:
Blinds are $.50/$1 and effective stacks are $100. Hero opens to $3 with AQs in the CO. An aggressive player on the BTN 3-bets to $10, blinds fold, and action is on Hero. I believe that 4-bet-folding here is a mistake. I also believe that when possible, we should prefer 4-bet-calling to flat-calling if for no other reason than because it is more effective at deterring future 3-bets. The question is “How often does BTN have to be 3-betting before 4-bet-calling becomes better than just flat-calling?”
There are better and more complicated answers to this problem than the one I will find here. You will get a better answer if you can estimate a 5-betting range for Villain and then perform the same calculations that I do here. You will get an even better answer if you come up with an entire 4-betting range for yourself, including both hands that fold to a 5-bet and those that call. As an example, though, we will solve for the minimum 3-betting range such that 4-bet-calling AQs is unexploitable.
Finding the 3B% at which 4-bet-calling is unexploitable requires finding the point at which this play is break-even against a worst-case scenario 5-betting range that contains every hand that is tied with or ahead of AQs: {22+,AQs+,AQo+}. This range constitutes 8.3% of hands and has 56.7% equity against Hero’s AQs. We need to write an equation expressing Hero’s EV as a function of Villain’s Fold % (%F) and set that equation equal to 0.
0= %F * $14.5 + (1-%F) * (.433 * $104.50 + .567 * -$97)
9.75=$14.5%F + $9.75F
40% ~ %F
With a little rounding, we see that Villain needs to fold 40% of the time to make this a break-even 4-bet-call. We know that the 60% of his range that he does not fold constitutes 8.3% of all hands, so now we need to ask, “What is Villain’s 3-betting range such that 8.3 constitutes 60% of it?” Dividing 8.3 by .6 gives us 13.8%, so if Villain’s range for 3-betting BTN vs CO is greater than 13.8%, then you are guaranteed to have a profitable 4-bet-call with AQs.
In practice, his range will probably not even need to be this wide, as this assumes that he 5-bets only and exactly the hands that are not losing to AQs. If small/medium pairs are not in his 3-betting range, which is true of many players, then your fold equity increases. If he occasionally 5-bets hands that AQs dominates, then your equity goes way up. So 13.8% would be a very safe number, but realistically if I feel like I am getting 3-bet more than 10% of the time, I am ready to 4-bet-call it off with AQs.
As I finished this last paragraph, I realized that I didn’t account for the card removal effect of Hero holding an A and Q. I think this is otherwise correct, though, and the point was really to see the method more so than the result. If you care about that level of specificity… well, you’re on your own.
Great post. I’ve already used this model to find appropriate 4-bet-call situations with other hands against various player types.