I really have to disagree with you. I think you are a bit too concerned about extracting max value with our trips instead of maximizing the ev of our strategy.

We don't want to sacrifice the EV of our trips by betting small in order to possibly benefit some other part of our range when betting large with our trips would net us more EV with them.

why wouldnt we?
I think if the EV we gain with our non trips hands is bigger than the EV we lose with our trips and therefore the ev of our entire strategy increases by leading small there we would be more than happy to "sacrifice" the ev of our trips.

I think the main problem of leading polarized here is the same one any strategy with leading ranges has: it's extremly difficult to be balanced and unexploitable. I'm almost positive that fellow rio coaches sauce and jeans both said that they have never met a player with a leading strategy they didnt feel thay can exploit pretty well.
I would say leading the entire range for very small sizing doesnt counts as a leading strat in the sense they meant it since you dont split your range and basically only say you realized that range strenghts shifted and therefore you put some money in the pot in a spot where opp is very unlikely to bet himself.

Btw i also think that you don't even really sacrifice any ev by betting smaller since you can just go ahead and make overbets on rivers in order to get in the same amount of money in the pot as you would by betting around 3/4 twice.

So as a conclusion i would say leading small with the entire range is supperior to leading big and polarized for reasons

1. Many hands in our range strongly benefit from the bet (in that spot a hand like 77 denies any 2 random overcards their equity and you do get quite a bit of value with your Ax)

2. By overbetting rivers polarized you can minimalize or even counter the effect of not getting enough value with your trips
   => adding points 1 and 2 makes it pretty clear that the ev of our strategy increased a lot
   and most important of all:

3. You prevent yourself from splitting your range and therefore playing a strategy that might be exploitable or very tough to balance (since you give off quite a lot of info any time you split your range)

A core tenet of game theory is that the goal of our strategy is to take the highest EV line possible with each hand. Any time we take a sub-optimal line with a particular hand it is a theoretical mistake. This would apply to intentionally taking a lesser EV line with our trips in order to benefit or "protect" some other part of our range.

I think if the EV we gain with our non trips hands is bigger than the EV we lose with our trips and therefore the ev of our entire strategy increases by leading small there we would be more than happy to "sacrifice" the ev of our trips.

I would argue that if we are sacrificing EV on our strongest value hands we are probably not actually maximizing the value of our strategy.

Btw i also think that you don't even really sacrifice any ev by betting smaller since you can just go ahead and make overbets on rivers in order to get in the same amount of money in the pot as you would by betting around 3/4 twice.

I agree there are situations when we don't sacrifice EV by betting our trips small (of course these board pairing spots vary in board texture, opponent strategy etc.). In this case it would not be a theoretical mistake to bet our trips small. One situation that allows us to sometimes bet the majority of our range small is when EV bet trips small = EV bet trips large and we use a mixed a strategy with our trips and balance both sizes.

3.You prevent yourself from splitting your range and therefore playing a strategy that might be exploitable or very tough to balance (since you give off quite a lot of info any time you split your range)

Yes these spots can be difficult spots to model and balance. Just because a strategy is more difficult to implement does not make it inferior.

The above reply did a good job of describing most of my preferences for the small lead with range over the polarized lead. That said, there are certain board textures (primarily the ones where trips will maintain near-nut status quite well on almost all rivers) where I think the polarized range will yield higher ev for the strategy. I haven't worked it out yet for nothing short of laziness and lack of time, but my intuition says that those should be the spots where larger sizing prevails while boards that can change more easily on rivers will prefer the small lead with my range and a variety of river sizes.

"We don't want to sacrifice the EV of our trips by betting small in order to possibly benefit some other part of our range"

the frequency in which we turn trips is much lower than the frequency in which we do not turn trips, therefore the benefit to other parts of our range with the smaller sizing occurs much more often than does the detrimental effect of losing value with trips due to the smaller sizing. Whether or not one outweighs the other will depend on the exact makeup of the flop x/c range. So, you cannot make the sweeping statement that we don't want to do something unless you can back that up by also stating that our range has enough trips in it beyond some threshold that makes the larger sizing more +EV for the entire range,

The "sweeping statement" is backed up by a core concept of game theory, that the goal of theoretically optimal poker is to play every hand in the highest EV line possible. You maximize the value of your strategy by playing every hand in the highest EV line possible, not by shifting EV around between different subsets of your range. Therefore, if you are taking a suboptimal line with a particular hand, you are making a mistake.

If the argument is that we are actually maximizing the value of our strategy by intentionally taking lesser EV lines with some hands, I would respectfully disagree. However I would be very interested to hear an argument from a game theoretical perspective that it is ever correct to intentionally take a suboptimal line with a particular hand.

Trying to choose one or maybe two bet sizes for my range and trying to maximize the EV of each hand in my range are not compatible strategies. If i wanted to maximize the EV of every hand in my range I'd be playing a strategy that's far more mixed and has far more bet sizes and lines than I can possibly keep track of as a human. I'm making some compromises in an effort to make my strategy more easily employable.

I do think that when comparing the two strategies suggested in this thread there are clear EV losses and gains to different parts of our range. We can compare the two strategies (knowing that they are likely both valid and near-best) and pick the one with higher EV. I'm unlikely to try and accomplish more than this, even though the game theoretical solution will definitely net higher EV than either choice.

I think plenty of better hands can and should fold here (66, 3x, for example) but an early street play also doesn't really need to be so strictly defined into bluff/value as it does on the river. I think once we start betting this hand it often plays well as a bluff on some more common runouts where I'll struggle to have effective bluff combos.

Despite saying that I'm not too confident this is the highest EV line with my hand in retrospect - I think it's probably best to check the flop mostly. I think it's a pretty useful hand to bluff with if I decide to use overbet sizing on turn, which is something I'd be more likely to do on this board more recently, but at the time of this video I don't think I was using that line on this board texture. The idea is kinda similar though but I think it's just truer to the intent of the line if I overbet the turn.

why over betting the turn strategy?

28:40 - This depends on his sizing, but given the .5pot flop bet is the only action we've faced so far I'm pretty confident that's not the case against most < pot bets.

33:50 - I thought he was paired reasonably often given the low flop (which lends to merged betting ranges) and the failed bet on a good turn bluff card. He's unlikely to bet here when checked to and could certainly still end up having a raising range so I was happy not bothering to try for any x/r and just hoping to get a call from weak pairs.

44:15 - Yeah, mostly just for board coverage vs someone who is going to flat me fairly often. I like having some small suited hands too or just using blockers occasionally like you implied. It can become troublesome to never have < A high in a variety of postflop situations so it's good to have some hands like this that are no better than marginal calls vs 3bets in the first place.

Yeah it's hard to say what his calling decisions will be like since his value range appears to be squarely Qx of every possible rank. Something with a heart Jx or Ax would be nice since that will remove his highest quality Qx hands and something that has no low cards would be nice to improve the chance that he's pure bluffing. I could see Ah5/Jh5/AhT/JhT perhaps being the best ones.

18 - This is a spot I'm leading my range for this size; I don't have any checks here.

28 - Why do you think a straight would be worse than a pair? There's definitely some chance he'd bet a chop for small sizing and I do a good job of never blocking bluffs when I have an ace with a straight card.

To your last question - it's not something that needs to be done at a high frequency at all. It's rarely the best play with bottom pair to start putting money in so it's definitely a small percentage of my range. I often lean towards betting with higher equity if I'm going to bet a marginal hand so using a gutshot or a backdoor nut flush draw is natural. This also makes bluffing more effective if that ends up being our preferred late street option on some runouts.

The ace is a card where if I don't create a leading range my opponent will choose to check an extremely high frequency. He's doing this because he knows my range took the lead here and I have a large % of hands that would like to put money in the pot on this street. I'm countering this by putting in a small amount of money with that whole range, which allows me to maintain balance easily while gaining EV for my strategy when compared to a high frequency of check/check on the turn.

Yep, good observation. I think you're right that we should be leaning towards a lead on all board pairs vs his cbet strategy - there's a lot to be gained on this street with trips+

dont we want to lead when the board pair is not in his range? if he cbet depolar, the board pair is in his range and ours

being on his range and being a significant part of his range are different.

when we x/c the air distribution across ranges is much heavier on IP who has a high cbet frequency. it is hard to define for which range the top card pairing is better but it doesn't should make us never lead. against this type of cbet strategy most pairing turns aren't pivots in range vs range equity so I should be leading with some frequency and A3 seems a good candidate.

I ran a simulation on this and was surprised to find that Td isn't a bet here. It had about 65% equity vs my range getting to the river and if he chose a half pot river sizing that would narrow my continuing range to one that beats his hand.

I had slightly overestimated in game how light i'd need to defend vs his flop sizing (it appears a lot of ace highs are folds for example). It appears at first glance that something like the Qd is a reasonable cutoff for him to value bet IP here.

I watched some sauce vids and he plays these hands so I just assumed it was good

Seriously though, 32o isn't hugely useful but i felt that if I was going to fold 10-15% of hands I should pick the ones that offer the least benefit in rare big pots. It's nice to be able to have some hands that my opponent can't have (namely, a straight on A45xy runouts) which seems more beneficial than having another shitty Tx hand. They both offer similar value in making arbitrary two pairs and full houses so the main advantage of Tx is having an overcard on certain boards which is hard for me to evaluate. Essentially, my intuition said that keeping the hands pre that can make straights made sense to me. And seeing a good player like Ben choose hands in this manner does help get the gears moving!

I certainly wouldn't say I prefer not having a club - I have higher equity with a club and I remove a card that he's likely to peel to a raise by having a club. I just think it's important to bluff sometimes with no clubs and of course gutters are a nice choice for some equity as well. I'm not going to have 100% frequency with non club gutters and I think I'd like to mix in some big and small gutters here (calling whenever I don't raise).