Had a long discussion in IRC about the merits of 3betting a hand as strong as AQo IP vs. calling. Actual hand that started the discussion doesn't really matter. Basically villain raised to 3.5 BB from MP, Hero 3b's on the button with AQo to 10.5 bb. I went through a long equity calculation to try and justify my play:

Overall villian is a bad reg that runs 22/14, and from this particular position his PFR is 16%. He folds 74% of the time to 3bets, 4bets 4.23% and calls the other 22%. With blockers to AA, QQ, and AK, plus him being OOP, he will likely fold more often, and 4bet less, but for simplicity lets keep those numbers. We are folding to his 4bet, so the EV of the hand if it ends preflop is: .74 * 5bb - .0423* 10.5bb = 3.7 - .44415 = + 3.25585 BB.

Now, the other 22% of his hands are .22*.16 = .0352. Top 3.5% of hands with our blockers is a range of 77+, AQs+, AKo. Now, he would likely 4bet some of those, so maybe we throw in 66 and take out KK. Anyways, against this range, clearly we are not often real happy stacking off postflop. However, we can make most of these hands fold with a cbet. This particular villian has a fold to cbet of 55%, which is probably a low estimate for a 3bet pot. With all the PPs in his preflop calling range, he is likely going to c/f a much larger % of the time.

So, for arguments sake, lets bump up his fold to cbet to 70%. After his call and minus rake the pot is ~ 21.5BB. If we cbet 12BB, .7*22.5 = 15.05 BB but we put in 10.5 preflop so we net 4.55. Assuming we lose that 12BB everytime we get called/raised, we lose .3*(12+10.5) = -6.75BB (not realistic, we should obvi win some percent of the time), the overall EV of our preflop and flop play is:

3.25585+[.22(4.55-6.75)] = +2.7BB. In my database, I currently only have 2 hands that are over +2.5BB or higher per hand on the button, AA and KK.

The play becomes way more profitable if say 10% of the time we get called and win. The EV of this scenario is 3.25585+[.22(4.55+.3*(.9*-22.5+.1*22)] = +3.065 BB. There are also factors like being able to double barrell successfully but that should marginally affect the EV because it doesn't come up too often.

In conclusion, 3b'ing strong hands like AQo is profitable, but we already knew that. The question to answer is whether it is better to flat. Calculating the EV of flatting is very difficult, but I find it hard to believe that it is greater than the EV of 3betting. I now have a headache.