Because by definition a polarized bet means we don't bet hands in the middle of our range. Which means our bet happens at less frequency. Frequency and bet sizing are directly correlated.

I still don't see that we have got to an answer for "how polarised" we would be though. Working that out needs us to assume with what cards villain is checking, if he checks a range bet spot we are out of GTO space and into the unknown.

You said "we are checking back more than we are at equilibrium earlier". I still don't know what you meant by this, as there is no equilibrium that involves a villain check.

X % of your range is "air" . you want to be able to bet as often as possible with the "air" part of your range , while balancing it with value and making villain indifferent between calling and folding. when you bet polarized you basically bet the "nuts" for value and add some bluffs. Since villain is always only bluffcatching and pretty much never beating a valuebet, the pot odds he gets and the construction of your betting range decide if he is indifferent or not. if you bet 25% pot with a polarized range that means you get to bluff only 20% of the time you bet. if you bluff more often villain has a +ev call. but if you bet pot you get to bluff 33% of the time... the more often you get to make a blanced bet, the more money you eventually make. that is why you want to bet big when using a polarized range. Obviously you need the "value" hands to make these big bets. because if your valuebets lose at showdown too often that doesnt work because your bluffs always lose when they get called. so stronger your hands are the bigger you can bet. the bigger you bet the more often you get to use bluffs because villains odds are worse.

Thanks for the basic lesson, but I feel you've missed the point somewhat.

All of the above applies is in situations where Villain has a condensed/capped range, that's why "villain is always only bluffcatching", all the logic follows from there. Villain having a condensed range is an additional assumption we are making, which wasn't stated in the exercise (we were just told they check sometimes in this situation that's a range bet).

If Villain was checking 100% of the time here, they are making a mistake by not betting in a situation where they have range advantage (so they aren't realising as much value as they could), but they are not capped when they check and therefore still have the same range advantage they had as the preflop 3 better.

bro you asked this question right? literally you did. and now you complain about it being answered? i seriously dont get you.

He said we should bet big, so I asked him to clarify if he's saying that purely because we are polarised.

I know the logic between betting big and being polarised.

What don't you get? I'm explaining to you why I don't think it's clear that we would be polarised here.

Your point of contention is that SB range cbets. In my initial post I said for all intents and purposes SB should be range betting but no at equilibrium it is not a range cbet. It is around a 75% frequency bet

It's redundant to say we should bet big because we are polarized. By definition, big bets are polar bets.

Betting polarized and betting big are synonymous. Saying statements like "we bet big because we are polarized" is like saying we bet big because we bet big. It's a nonsensical statement.

Also it doesn't matter if this was a range cbet spot or not and Villain checks. We would still check back more often because IP benefits more from seeing a turn card.

I know you did this to try to avoid the debate around whether it's actually a range bet or not, but it changes the question imo.

Yes, that's why I asked you to clarify, I was actually asking why we are polarised but more thinking out loud, I should have clarified more.

There is no "more often" if it's a range bet and he checks. There is no node for checking in your GTO sim in a range bet spot. If you force a check you need to assume a range. I agree that we would check back a lot though.

We are polarized because we are betting into an uncapped range since we don't know what his checks mean. You don't bet the middle of your range into uncapped ranges - you simply check back.

Isn't betting a weaker capped range into a stronger uncapped one generally a bad idea? That's why good donk betting spots are rare. That's leading me to think we check back everything.

We aren't really capped - the only hand we can't have that SB can have is AA.

So we would always bet sets. And since we have some value hands we need some bluffs. We would want to unblock the weakest parts of his range so we would not want any broadway cards in our hand when we bluff.

That's the nuts though, it's a pretty important hand to be missing from our range. When we bet it really ****ing sucks to get check raised and we are 200bbs deep.

It's 3 combos out of like 120-130 combos. A <3% nut discrepancy doesn't make enough difference in our strategy to start range checking back OTF.

You're happy getting 66 and 44 in here when we get check raised?

I'm talking about overall strategy not some unicorn situation.

We don't change our strategy drastically based on small % discrepancies in ranges.

A 3 better having AA is hardly a unicorn situation, it's not like the nuts is 85d or something.

The reason the board is a range bet is because his range is much stronger than ours, and the reason him checking is a mistake is because he allows us to play a smaller pot and fails to capitalise on this. When we bet we are rejecting the offer to play a smaller pot, it seems to me that we should be very careful about doing this. I know I always feel a bit lost when I get raised in a situation where I crush villains range, because I end up second guessing myself.

Him having AA AND us having a set is a unicorn situation.

XR frequencies go down at higher SPRs. Which makes your hypothetical situation even less likely.

Which hands are you betting for value and then happy getting in?

Well just because we bet big OTF has nothing to do with getting stacks in.

You will almost never get XRed in this spot because of board texture and SPR so I wouldn't worry about that part of the game tree.

I'd comfortably bet my AK/sets/K high FDs in this spot no problem.

You can't just dismiss it as rare, if we don't have hands we bet and are happy getting in on the flop/on blank turns then we don't have an unexploitable strategy.

So according to your logic we can't bet the flop unless we can have the absolute nut hands in our range.

You realize there are spots in poker where you literally have a 0% XR range? And 0% 3bet range OTF?

So we can have a 0% 3 bet range and a 0% x/r range, but we definitely don't have a 0% bet range here?

Obviously there are a ton of spots where we can't have the nuts and can bet, but also Villain can't have/is very unlikely to have the nuts, this isn't one of those. Villain shouldn't be 3 betting that wide here deep either.

I'm clearly not saying you can't bet if you never have the nuts, but you have avoided saying you would be happy getting 66 in on the flop/on a blank turn. I don't see how an unexploitable strategy can have a betting range that folds 100% to raises/turn bets on blanks - if I'm wrong then please just explain it to me, as i said I don't have solver experience.

If we take the extreme - villain is checking 100% of his range and we check back. If we extand this to all streets - villain checks all streets and we check back all streets we will lose as villains range is stronger.

So to make this situation +EV we have to bet on at least one street.
So as an conclusion I dont think checking back flops 100% in this situation if villain overchecks is the right play.

I think you would likely make more money if every time you called a 3 bet you were allowed to check down, than if the hands played normally, so you get to realise your equity. You don't win money calling 3 bets, you just aim to win back some of your original raise to make it better than folding.

Your incorrect assumptions is that SB would only XR AA here. That would never be true. And since that isn't true - we can easily get 44/66 in by the river.