[2-4]: How do you play this 2p on the Turn?

It's far less of a disaster OOP.

You tried to deny villain's equity OOP on a dynamic board and they ended up denying some of yours. C'est le poker.

Forgot to link the thread I referenced above.

Regardless of what anyone else does or doesn't understand, I know I never understood why range advantage should lead to betting more at all. So I'm happy taking this advice.

This may be a stretch, but your example in this thread reminds me of the AKQ game, which I think makes your point about polarization in the simplest possible setting. (The way this game works is that two players are dealt one card each from a three-card deck of Ace, King, Queen, then the first player can bet or check; if he checks the second player must check behind; if he bets the second player can call or fold; if the second player doesn't fold, the game goes to showdown and the higher card wins.) The GTO strategy for this game is

The reason I think this makes the same point is that a bet is profitable if and only if your range is polar from your opponent's perspective. Which is the case exactly if your opponent has the King; then your range is {A,Q}, which is polar. If your opponent has the A, your range is {K,Q}, which is not polar, and a bet is pointless (opponent always calls), and if opponent has the Q, your range is {A,K}, which is also not polar, and a bet is also pointless (opponent always folds). Consequently the only card that doesn't bet is the K because the K guarantees that your opponent does not have the K and so your range is never polar.

After tabooing the concept of range advantage, here's how I'd still justify the general tendency of "lead more as BB when the Flop has low cards":

1) With a weak hand, I can bet as a bluff (because opponent is more likely to both (a) have missed and (b) believe I have hit than if Flop is AT2)
2) With a strong hand, I might prefer betting for value over check-raising because opponent is less likely to bet, which means I may not be able to check-raise

This seems sort of reasonable, although 1) is stronger than 2).

Responding to this in case my earlier points weren't clear. With the line and bet sizing scheme OP took in this hand, I'm not convinced she didn't fold the best hand here.

If she did fold the best hand, it's a disaster. If she didn't have the best hand, she was forced to fold away her equity in a spot where she didn't need to 3B and had a chance to improve if she just called, or at least could have blocked the river and folded if V raised again, which seems very unlikely if he didn't have the straight he was apparently repping here.

Either way, it just seems like such an unnecessary line to take if we happen to get to the turn the way she did. If we want to have some 3B's with 2P when a straight draw comes in, I'd think we should either bet small to induce, with a plan to call if V raises, or bet large enough that we don't have to wonder if we induced V to spaz-raise.

The line of wide defend pre, 40% pot donk, 1/3 pot barrel, min-click 3B, fold to a jam can't possibly be better than...well, almost any other line.

If the solver likes to defend with this hand, and likes to donk flop, and barrel small on the turn, okay, fine, I'll buy the line as far as that last decision node. But clicking it back against a guy with the description given seems like we're inviting him to do something spazzy, then surrendering when he does something spazzy.

So he's a fish we haven't studied, other than noticing he plays odd with weird ranges, and will randomly bluff or get aggro in spots that don't make a ton of sense.

Maybe he had it. Maybe he didn't. If we're not sure (and I don't see how we can be), then I don't see how this line would be correct, at least not if we're trying to exploit V's tendencies.

We could have folded pre. We could have check-called the flop, or barreled turn for a size that isn't all but guaranteed to induce a raise from a spaz-tard V. We could have called his raise and not 3B. We could have done a lot of things. We didn't need to do anything hero did here.

I don't know how much we need to argue about this. I wouldn't have done it in game, and even if we're going for a merge, my personal heuristics especially don't like two pair where the pairs are specifically the top and bottom two cards for the more aggressive lines. And if we don't give OP the benefit of the doubt, then I'd say it's likely a newish player overplayed a middle-of-range-ish hand on a heavy board, which is a standard leak for newish players.

And but also EV tends to run way closer OOP against aggressive players than people think, and there's tons of mixing. I can go through each item in your "Why didn't we just x/y/z" litany and say what could go wrong with each of them. That's being OOP against aggressive players on dynamic boards for ya. You're constantly choosing between losing value, under realizing your own equity, and letting them over-realize theirs.

Also the 3b wasn't a clickback; it was 56% pot. And I think you're vastly over-estimating how often turn 4bs happen just because sizes are smaller than you personally would use and villain is capable of stochastic aggression.

yea, 3bet bad. That seems to be what both the GTO-adjusted-for-V-tendencies-strategy and the let's-think-through-equities-without-anchoring-strategy suggest. It's also the main thing I won't do in a similar situation next time.

I haven't thought about it very thoroughly, but I think it's clear that some version of range advantage (probably better defined in terms of median hand-versus-range equity or percentage of combos with >50% hand-versus-range equity) has some positive (almost certainly causal) correlation with betting frequency. The regressions in that thread do show positive slope, even with low confidence.

To use the 99/1 case, the player with the polarized range will bet more total combos if the distribution were more favorable: for each losing combo that switches to a winning combo, more losing combos (and total combos) get bet. It's probably better said that the shape of the range-versus-range graph is really what matters and we're trying to project it onto single axes with these concepts like polarity and "range advantage". Obviously it's not "bet iff you have 'range advantage'", but all else equal it should be causing more betting.

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i think this is more so due to how some here interact with people

here i am thanking you and apologizing for the mixup and explaining why i interpreted what you said to mean something different from what you intended

and sub oddly chose to view that as me doubling down on insulting you guys and then throwing in a stream of fresh insults

this isn't really a divisive issue, it's a pretty typical hh discussion, just the parties involved are taking things as personal insults and attacking posters not ideas - nowhere in this thread have i insulted you

and, you don't need to agree with my viewpoints, that's the entire point of having open discussion of hands here, to allow thoughts to freely flow - discussion and disagreement should be encouraged

yes, many different opinions (on almost every street!) is what I meant.

All I know—and it’s not much—is that non-ace high connectors & gappers, suited or not, play HORRIBLY OOP HU for a preflop raise in SSNLHE, even with this GOAT rake and nothing will ever convince me otherwise. You could throw every one away in the blinds and never limp to begin with first three spots in a nine-handed 1-3 or 2-5 with 4-9+2 rake and at worst it’s < -1BB/100 EV under laboratory conditions.