# Rake Aware - is 3betting really saving rake vs. regs?

So I see and hear a lot of comments that 3betting is the default in a high rake environment.
BUT given that most players can download pretty optimal ranges relatively easily, and these ranges advocate for defending 50% of the range, I am a bit confused on how it is actually saving us \$\$.

Three points / assumptions to make:
1) I only want to discuss the rake-specific implications, and not benefits for range of 3betting mediocre hands in position.
2) blinds are folding in this example but assuming in general they're not squeezing too often and it doesn't hurt us too much (either not super wide given utg raise or we've developed some sort of anti-squeeze defense to exploit over enthusiasm).
3) when we 3b, I'm assuming our opponent would have a call or fold strategy. Obviously not balanced but I am assuming the "rake benefits" of 4bets (i.e. we don't pay any rake when we fold) are offset by the extra rake when we defend our 3b about 50% of the time etc.

A bit of math:
Option A: utg raises to 2.2xbb, hero in BU calls 2.2bb, sb folds; bb folds. Pot is 5.9bb, rake is 5%, or 0.295bb.
Option B (i): utg raises to 2.2xbb, hero in BU raises to 7.5bb, sb folds; bb folds, utg calls. Pot is 16.5bb, rake is 5%, or 0.825bb
Option B (ii):utg raises to 2.2xbb, hero in BU raises to 7.5bb, sb folds; bb folds, utg folds. Rake is 0%, or 0.0bb.

According to this math, is utg flatted 3b half the time, we'd be saving about 30% of the rake bill preflop --> 0.295/(0.825*.5)=71.5%.
Utg would need to fold to 3b 65% of the time for the rake effects to break even, and this doesn't get into the effects of a single bet on flop etc. This fold to 3b seems too high in today's game.

So other than the times sb/bb can squeeze, what am I missing here about the anti-rake strategy?