Top two pair very deep vs nit - uh ohz

[foobar37]

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Interesting spot. Not as close as people are suggesting, though. It's a pretty clear check behind to me. Worth doing a bit of math to demonstrate that, I think.

We'll assume a best case scenario where villain does not have a flush. If we include flush possibilities, the math skews even further to a check, since he'll never be folding a flush, so if we can make a case for checking behind without a flush in his range, it's even more clearly a check if it is.

For a 16/9's opening range, things are pretty narrow by the time he gets to the river like this. AJ (8 combos), AQ (8), AK (4), 66 (3), 77 (3), KK (1), and AA (1), pretty much. I don't think villain's range is weighted strongly towards one end or the other beyond what the available hand combos provide.


Hero has a 64% equity against that range, so a check behind is clearly profitable. The question is whether betting is more profitable.

If we're betting for value, we need to know how often villain needs to call with hands we beat to make up for the times when he has a hand that beats us. There are 8 combos that beat us (sets) and 16 combos of hands that we beat (AQ/AJ), so villain has to call more than 50% of the time with AJ/AQ for a river bet to be profitable. I think that's a pretty optimistic percentage for any kind of reasonable bet.

Shoving as a bluff is interesting. In that case, we assume villain will fold AJ, AQ and AK, so we're left with figuring out how often he needs to fold a set for a shove to be profitable. When villain has AJ or AQ, shoving has the same EV as checking, so we can ignore those hands for this part. The pot is 216 and Hero has 445 behind, so we're betting 445 to win 216. If 'n' is the fraction of the time that villain folds a set, the EV of a shove is (we gain 1/2 the pot when villain folds AK, so we add that into the equation also):

((216n - (1-n)445) * 8) + (1/2 * 216 * 4) = EV

solving for zero, we get

n = 64.4%

So villain needs to fold a set more than 64% of the time to a river shove for our bluff to +EV. I find that extremely unlikely to be true on this board, as it would require him to call with AA and KK only.

Add in even a slight chance of a flush for villain, and the case becomes even stronger. A river check behind is the right play.

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You did a lot of nice math here, but I think you missed two key points.

1) Certain hands need to be discounted a lot more than others. If his stats were 16/13 I would give him credit for raising small pairs UTG, but at 16/9 I expect he limps 66 or 77 most of the time. There is very little chance he would fire close to pot on the flop with KK into three people, discounting KK. AxQh and AxJh are much more likely than the other AQ and AJ hands. Etc.

2) You can pick your bet size. We have zero fear of being check-raised as a bluff, so there is certainly some bet which will get AQ and AJ to call >50% of the time (or whatever the actual percentage cutoff is when you do the discounting). If nothing else, min-betting should have higher EV than checking.

Overall though a nice job.