Seat 1: Big Blind ($23.79 in chips)
Seat 2: UTG ($41.54 in chips)
Seat 3: UTGplus1 ($74.20 in chips)
Seat 4: Donzo ($51.80 in chips)
Seat 6: Small Blind ($74.36 in chips)
Small Blind: posts small blind $0.25
Big Blind: posts big blind $0.50
*** HOLE CARDS ***
Dealt to Donzo [4s 3h]
UTG: folds
UTGplus1: folds
Donzo: raises $1.50 to $1.50
Small Blind: folds
Big Blind: calls $1
*** FLOP *** [2h 2c 6c]
Big Blind: checks
Donzo: bets $2
The action folds around to us and we're holding 43 on the button. We make a standard raise hoping to steal, but also prepared to play the hand in position. The big blind calls and the flop is pretty dry with 226 and two clubs. The big blind checks and we bet $2 because A) we have 4 high and can't win this hand unless we improve, B) we have a gut shot straight draw, and C) that flop misses almost all of the big blind's calling range.
Big Blind: raises $5 to $5
The villain check-raises and we're left with a decision. The most obvious play is to fold, as we are guaranteed to be way behind no matter what the villain has and we could simply write it off as a continuation bet that didn't work. Calling is a legitimate option, as it indicates a lot of strength, implying that we have either a 2, a 6, an overpair, or at worst a flush draw. If we call, it's with the intent to bluff the turn (assuming the villain checks). Finally, we can re-raise in order to get the villain to fold out all of their bluffs that were bluffing us on the flop, as well as the occasional underpair to the 6. Whereas it might be appropriate to call here with just ace high for value, as that would be ahead of much of the villain's bluffing range, our 43 can only function as a bluff if we don't improve.
We unfortunately don't have any stats or other reads on the villain and therefore can't say how often he/she check-raises. Consequently, our decision has to be based on how the board connects with the villain's range and how it connects with our perceived range.
Our Perceived Range
Using Hold Em Manager 2 (which I'll discuss in a future post), I can isolate my stats for just the session in which I was playing. For the amount of time I'd sat at the table, I'd been raising on the button 50% of the time. If the villain had been paying attention, he/she is likely to notice that and consequently assume that I'm opening an extremely wide range of hands on the button. Furthermore, because the board is so dry the villain has to expect us to be betting with most of our bluffs. As a result, our post-bet perceived range on the flop is huge, which means we must expect the villain's check-raise to be a bluff a substantial percentage of the time.
Villain's Continuance Range
For the sake of argument, let's say that the villain was calling preflop with ATC (any two cards). Let's also say that the villain check-raises his/her entire range on the flop. If that's the case, and we're attempting to discern the viability of a raise, we need to determine the villain's continuance range, or the hands with which they would continue in the hand in the face of a re-raise from us. To do that, we can use Equilab to separate all the combinations of hands that fit that criteria. If we say that the villain continues with quads, a boat, any 2, any 6, any overpair, or a flush draw, that makes up about 30% of all possible hands. In other words, unless the villain has the capacity for a 4bet flop bluff, which is a very aggressive and rarely-seen play at these stakes, they are folding 70% of their hands to a raise.
However, remember that those numbers are based on the assumption that the villain called with ATC and is playing all the "continuing" hands for a check-raise followed by a call. In reality, the villain's continuance range is almost certainly lower than 30% because even after accounting for a smaller total number of original hands from which to select, if you look at the image above you'll see that there are a lot more combinations of hands with a 2 than we would expect all but the loosest villain to play, e.g. T2, 72, etc. Furthermore, the villain is less likely to call a re-raise with a weak 6 than with a 2, overpair, or flush draw. Finally, the villain would probably just check-call with some of their bigger hands (trip 2s, a boat, etc), especially knowing that I'm bluffing a significant portion of the time, as they wouldn't want me to fold all my air.
Armed with the above information, we need to determine the profitability of a raise. After the villain's check-raise, the pot stands at $10.25. We raise to $10.50, which is the equivalent of betting $8.50 (because our $2 is already out there) into $10.25 (the pot including our $2 and the villain's $5 raise). If we do a little basic math, we'll see that we need our bluff to work 45.33% of the time to be profitable:
($8.50) / ($8.50+$10.25)
($8.50) / ($18.75) = 45.33%
If our evaluation of the villain's continuance range is even remotely in the ballpark, and they're folding over 45.33% of their hands, then it's a profitable raise. The villain does fold, and we take down the hand with 4 high.
Donzo: raises $8.50 to $10.50
Big Blind: folds
To phrase the idea of bet profitability more simply, think of a hand in which there's a $100 pot on the river and the action is on us holding nothing. If we bet $100 (the size of the pot), we need our bluff to work 50% of the time to break even, as 50% of the time the villain will call and we'll lose our $100 bet and 50% of the time the villain will fold and we'll win the $100 in the pot.
Estimating a villain's continuance range can be a difficult task in the moment, but in fairly clear cut situations like the one I've outlined here, having a healthy grasp of the likelihood of a fold is crucial to maximizing profit.
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