Probably boring / basic but I'm lousy deep and I don't think I'm alone.

PokerStars Zoom No-Limit Hold'em, $1.00 BB (9 handed)

Button ($144.13)
Hero (SB) ($275.55) (Reg hero playing 17/13, 3bet 4%)
BB ($284.58) (Bad reg with laggy zoom stats)
UTG ($282.24)
UTG+1 ($80)
MP1 ($50)
MP2 ($334.56) (Unknown with reggy stats)
MP3 ($231.50)
CO ($149.69)

Preflop: Hero is SB with A, A
3 folds, MP2 bets $3, 3 folds, Hero seeks advice.

There are a lot of parameters that you could / should take into consideration to choose the appropriate bet size.

In this specific spot, I think that because effective stacks are deep, you are likely to 3 bet with a depolarized range OOP right? There is no reason to 3 bet bluff much, as vilain is likely to call very wide because of high implied odds when IP.

Considering you are likely to 3 bet mainly for value (depolarized) then I think you should make make it on the higher side, like 4x.

MMQC, If I was going to depolarize and weight to value (which isnt an approach I really like), then 4x to $12 seems like a bad size... any deep players left to act can call in position for less than 5% of effective stacks against a reasonably narrow hero range. If I'm not balanced then I'd rather go to something like 9x and give villains a chance to make a real mistake. But I would expect I want to be balanced here anyways.

min 3b to induce

There are few things here.
So you continue to have a polarized range here even deep? You don't even adjust? When deep, we have less FE, so why bluff, at least why bluff as often as when 100bb deep? I hope you at least adjust your range?

Sure I'll adjust my range - for example there are some starting hands I'm going to like better than others when 300bb deep. But I'm still going to seek some balance. Contrary to your initial post, imho there are still very good reasons to have bluffs in my 3bet range.

OK, just for the purpose of the discussion (I think there are few things to develop)

you still want to have 3 bet bluffs in your range, so I guess that you will include a lot suited aces and even some suited connectors in your 3 bet bluffing range?

Also, by definition, bluffing means getting folds from better hands. IF vilain adjust properly to deep stack play, he should call very wide here, so I am wondering what better hand can fold? Yes, if you 3 bet TJs you can get a fold from KJo, KTo type of hand that have you dominated.

If vilain don't adjust correctly to deepstack play and still fold as much, then I think we can continue to 3 bet bluff him all day long.

In position it would be a whole different story tought. We can still continue to 3 bet bluff a lot and can make it 9$ or even 8$ if vilain folds everytime. (make our bluffs cheap)

Also, I don't understand why you would like to bet smaller with a big hand oop? If vilain want to call with a wide range isnt it the best opportunity to make him pay the price for it? (I think that we are likely ahead pre with AA )

Also, there are other factors that could be considered in our range construction, like how we think we can outplay vilain in a 3 bet pot oop (do we have a hughe skill advantage or not)

Not going to get into specifics about my 3betting ranges obviously, shouldnt be necessary in a discussion about sizing a balanced 3betting range.

Not sure where you take this from but I referred to 9x in an unbalanced situation, not $9.

This is just wrong, all of it. No offense.

Ok, I have no problem to ear that, but can you explain what is wrong and what could be better?