So basically my question is about how many 3bet bluffs we should have for every combination of value.

I've seen different numbers in different places. I've seen people suggest 1 bluff for every 2 value hands, as well as 1 bluff for every 4 value hands. On the more aggressive side, Janda's "Applications" advocates a 2:3 value-to-bluff ratio assuming all your value raises are "true value" that you will 5-bet shove. Building off of Janda's work, I've also seen it recommended that we should have a 1:1 bluff to value ratio if we're going to have a 4-bet flatting range.

I know this is a complicated topic, but in general, how often should we be looking to 3bet bluff (when using a polarized range) so that we are bluffing enough but not overdoing it? Seems like there's a lot of disagreement on what the most effective ratio is.

I think that there are (infinite) preflop strategies that could satisfy the conditions of a Nash Equilibrium. This is because many hands will earn exactly 0ev preflop depending on how one constructs preflop ranges. For example:

100bb no limit holdem with four strong players.

cutoff raises 3x, I 3 bet on the button to 10x, blinds fold, cutoff folds, calls, or 4 bets.

Now, this is obviously a bad game, but we've found ourselves in this position and we gotta play it as best we can.

I'd use this range on the button in that spot:

77+, ATs+, A2s-A5s, AJo+, KJs+, KQo, QTs+, JTs, T9s, 98s.

The worst of that range, or A2s and 98s, most certainly derive most of their evs through preflop fold equity, but they also make big hands that can win big pots; they're not pure bluffs. Thus there is no need to differentiate between bluff and value. There is only +ev and -ev.

Notice that my opponent in the cutoff has a decision to make with a big chunk of his opening range. If he doesn't continue often, then my preflop ev goes up, effectively protecting my strong hands and getting more bluff value with the bottom of my range. If he does continue often, then I'm getting more postflop value with my strong hands, which in effect turns more of my 3 betting range into value. Think about how A2s performs vs someone that calls a lot in that spot.

Also, notice that with strong players in the blinds, we should only call on the button rarely imo; this is because the threat of a squeeze is too great with something like A6s or 33.

This turns some hands that are very profitable as calls in normal games into either folds or 3 bets in a tougher environment.

I'm not one for value:bluff ratios pre-flop, particularly in a multi-player game. In 6-max, MP can't 3-bet as many bluff combos as BTN can, partly because MP still has 4 other players yet to declare an interest in the pot (as well as the UTG raiser), while BTN only needs 3 players to fold (inc. UTG) for his bluffs to insta-profit.
In addition, some of the lighter 3-bets can/should call 4-bets some of the time. Unless your range is incredibly polarized (like AA and 72o), then the terms "value bet" or "bluff" don't even make much sense pre-flop. You can 3-bet a hand like AK and simultaneously get called by worse (like AJs) while folding out better (22), so it's both a value-raise and a bluff. (It's mostly a 3-bet for equity protection imo. You just want villain to fold, unless he'll stack off with something you dominate). Very few hands (big pairs) actually want to get action when you 3-bet. Everything else has at least a small component of "bluff/protection".
In short, I wouldn't try using ratios. I'd just try and build ranges that simply make the most money by 3-betting (instead of doing something else) in each situation. In some spots you can 3-bet light very often. In others, you can't, so your 3b range should be much stronger.

Arty is right that the vast majority of our 3 betting hands want the opponent to fold. This isn’t because we’re bluffing though. It’s because when our opponent has a chance to win part of the pot, it makes our 3 bet less profitable than when our opponent folds. So from left to right we can see a progression of best possible results to worst possible results:

Opponent makes a -ev call>opponent folds>opponent makes +ev call>opponent makes +ev 4bet.

Why is it better for us if opponent makes -ev call than if he folds? I don't think that's right.

Because that means we’re winning the pot plus part of his call.

Ah, ok. I got confused for a moment. Thought we were talking about our bluff range's EV.

Makes sense. I was actually surprised that you of all posters didn't understand. Compliment intended.

Stop thinking in terms of value and bluffs. Start thinking in terms of raw equity (AA) and board coverage (76s). Combining those two things is how you maximize your preflop ev. Also keep in mind that making the nuts a lot has diminishing returns.

You ask how many we should have, but I suspect you mean how many we should have to be play GTO?... In most games you shouldn't be using GTO.

fwiw Janda thinks the preflop section of Applications aged badly and suggests people skip it

Wait really? CRAP! I literally just spent the last few days taking detailed notes on it :/

Sorry

Problem is the terms "value bet" and "bluff" really don't work that well. I think it took most players (myself included) a lot, lot longer to realize this stuff than we should have. A lot of stuff in poker seems kind of obvious once you begin understanding it, but it's confusing as hell when you're trying to get it sorted out, especially when nearly everyone else is confused/wrong too and poker advice tends to be an echo chamber.

Example: When does 3-betting a pocket pair become for "Value" pre-flop? JJ? TT? 99? 88? Let's say you say TT is a 3-bet "for value" and that's the weakest value 3-bet. Well, what's 99 then? What so special about TT to make it a value 3-bet but 99 is not? Also, nothing magically happens when you go from having 49% to 51% equity against the calling range, so it's not like it's a "value bet" just because you're now the favorite. Besides, not all equity is created equally as the GetMeOffCompletely hinted at before (who is a very good poster btw and I almost always agree with what he says).

For preflop ranges you are much better off going off what PokerSnowie recommends than what I recommended in Applications.

All of my extra book copies are at my parents house approx 2000 miles away but if you haven't bought the new book and assuming I can find them PM me and I'll try to send you a copy whenever I visit next time (might be several months but meh free book)

Poker be confusing

Equities run so close preflop that there really isn't a such thing as a bluff or value raise. Most of the hands in the bottom of your range are 0ev or worse. Preflop solves using Monkersolver have essentially solved preflop for the most part.