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I just ramble - don't look for answers here - I'm just really posing questions to myself.
It's true that there's nothing wrong with letting lags value bet for you, but this hand is a good example of how there is more to people than stats typically reveal - a poker tracker or HEM would based on this hand rate this villain as very likely to bet, making it likely that you would check to him and let him do the betting - but actually he's doing a pretty poor job in this hand of value betting for you because his flop bet is so small.
If the flop had been more dry I'd have been perfectly happy with the sequence of bets as I think he'll bet this way with a large part of his range - and fold a large part of that range to any aggression. But as has been pointed out the flop is pretty wet - you basically have to bet a decent amount to protect your pair ($0.25) on the flop when first to act.
In the present sequence of events you can definitely call him down on every street - you have plenty of equity for that as he'll play a lot of weaker drawing hands and lower pairs this way. However, if you did do the betting on the flop he'd fold a good portion of his trash and you'd have to keep in mind that you are continuing to the turn against a more narrow range.
I'm not sure I'm completely agreeing with your hand range analysis. You go from "PF = smaller pocket pair" to "he must have 99 or TT" - which imo is not a smaller pocket pair. I think point is not so much that he'd reraise JJ preflop - but rather that he wouldn't have limped 88+ but rather raised them UTG when first in.
To let him hit any flop with some equity I'd be tempted to call his limp calling range: 22-77, A5s-A9s, AT, AJo, KTs, KJ, KQo-JTo, T9s-65s, T8s-97s.
(This leaves his PFR range UTG at: 88+, AQ+, AJs, KQs-JTs)
I'm not saying this range is particularly better than what you are thinking about, rather that the correct decision varies hugely based on what the actual range for the opponent is - but let's just not forget here that he chose first of all to LIMP.
Let's say you bet $0.25 against that range and he will fold:
A5s-A9s (not clubs, not pairs), AT, AJo, QJo-JTo.
He will continue with
22-77, A5s-A9s (clubs), A7s, A8s, KTs, KJ, KQo, T9s-65s, T8s-97s.
Question now is - what part of his range will he raise rather than call? Sets? Two pair? Flush draws? OESDs? (I had him fold the gutshot). For the sake of having something to calculate I'll assume he raises OESDs and FDs. This leaves his calling range:
22-77, A8s, A7s, KTs, KJ, KQo, 98s, 87s, 76s, T8s, 97s.
Of these we are ahead of: 22-66, A8s, A7s, 98s, 76s, T8s, 97s (48 combos)
Of these we are behind: 77, KTs, KJ, KQo, 87s (25 combos)
Now my question is - if we lead out on the flop with $0.25 and he calls - how large a part of the range that we are ahead of will oblige us with a bet if we decide to check/call? I'm pretty sure the whole range we're behind will be happy to bet into us if we check to the villain - but what about the part of the range that we're ahead of?
I'm just not sure even a lagtard will start shooting bullets with mediocre hands if we make a sizable flop bet that eliminates the worst trash hands from his range.
Ok, let's go the other way and check then - see what he bets:
22-77, A5s-A9s, AT, AJo, KTs, KJ, KQo-JTo, T9s-65s, T8s-97s.
That's all of his pre-flop limp-calling range.
We are ahead of: 22-66, A9s-A5s, AT, AJo, QJo, JTo, T9s, 98s, 76s, 65s, T8s, 97s (~114 combos)
We are behind: 77, KTs, KJ, KQo, 87 (25 combos)
So the principle of letting him bet seems sound. However, in this situation he has a lot of straight and flush draws (since ok - I cheated and eliminated them before by saying he raised them) which has much better equity against our queens than a lower pair.
Draws: Ac9c, Ac6c, Ac5c, AcTc, T9s, 65s (12 combos)
So, by betting we fold out 50ish combos (fold equity - about 35-40% of hands), we're raised by 12 combos (unhelpfully defining his hand as something we're ahead of but where we might not be able to play a big pot anyway), and we make it less likely that hands we are ahead of will bet into us if we take a check/call line on the turn/river.
I think the general principle of c-betting this flop is good if the opponent is playing a narrow range of hands that contains a proportionally big amount of possible straight and flush draws. The hand ranges I defined for his limp calling range happens to have A7s and A8s as examples where the clubs on board are blockers for the flush draw - and not have other suited hands in the range because they are more likely to have been raised rather than limped like AJs and KQs. Also the range I defined is so wide that the drawing hands are less than 10% and you could justify in this hand simply check-calling from the flop to take maximum advantage of the villain's tendency to bet with nothing/anything.
I think the normal principle of pricing out draws is somewhat nullified by the opponent continuing with a very wide range - if we were to put him to the test we might force him to only continue with hands with equity or a draw - and suddenly draws become a bigger part of his range and we need to be more conscious of pricing him out. This is something we need to be aware of in the action we take that narrows our villain's range (the flop bet in this case) and needs to be taken into account when we decide if the action is profitable.
I don't think I'm really reaching a conclusion other than: It really depends on what the opponent plays and how he plays it.
But I think this is the kind of analysis that is needed to ascertain which is the correct decision.
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