Situation: You are playing a hypothetical aggressive opponent who is raising his button 2/3rd of the time. Blinds are 25/50, effective stacks 1250 (25 BB).

Villain makes his raise. You have:

1) 2 2
2) A 2
3) J T
4) 7 5
5) 3 2 (for ****s and giggles)

Which of these hands are good to shove over his raise? Depends on what he's calling with:

Scenario 1: Villain is a solid (but a bit too tight) player, this is your first three bet shove, and he's going to respect your first raise. He's calling with 66+, ATo+, A9s, KQ.

This is 10.4% of hands. He's folding 1 - 10.4/66.7 = 84.4% of the time. You get 150 chips for free when this happens, so add 126.61 to your cEV for this move.

What if he calls, though? Here's the equity for each of those hands agaisnt this tight range:

1) 2 2 37.94% win, 61.31% lose
2) A 2 29.59% win, 66.68% lose
3) J T 36.52% win, 62.67% lose
4) 7 5 32.29% win, 67.07% lose
5) 3 2 25.46% win, 73.82% lose

So the 15.6% of the time you are called, you're obviously a dog. Here's the loss in cEV for each scenario (equals 15.6%*1250 chips lost * (%lose -%win)):

1) 2 2 -45.57
2) A 2 -72.33
3) J T -50.99
4) 7 5 -67.82
5) 3 2 -94.30 (!)

What have we learned from this? Well, if your opponent is opening wide and calling this tight, 3-bet shoving ANY TWO is +cEV. Also, against a tight calling range, the suited connector hands play better than the weak ace, but the pair is of course the best hand to shove against a tighter calling range. This is also why raising to 3x at 25 BB effective is generally a leak -- if you raise to 100 or 125, the P(fold) chip expectation falls to 84.4 and 105.5 respectively. As you can see, in the former case shoving 32o is now bad (lol). Of course, minraises will get called a bit more often, but this post will not deal with post-flop play for now. It's already going to be long enough.

Of course, you can't just keep shoving over this raise, because a smart villain is going to adjust, and a dumb villain is going to get pissed off that you're shoving so much and call lighter anyways.

Scenario 2: Either you've shoved a couple times over the villain in scenario 1, or you're playing someone who doesn't respect you as much. Villain is now calling 44+, A8o+, A5s+, KJo+, KTs.

We repeat the calculations. Villain is now calling with top 15.8% of his hands. You still get a fold 1 - 15.8/66.7 = 76.3% of the time, for a P(fold) chip EV of 114.47. Wooooo. Let's see how your shoving hands are holding up now if called.

1) 2 2 win 40.38%, lose 58.74%, cEV = -54.39
2) A 2 win 31.40%, lose 62.13%, cEV = -91.04
3) J T win 39.05%, lose 60.06%, cEV = -62.24
4) 7 5 win 34.51%, lose 64.75%, cEV = -89.59
5) 3 2 win 27.62%, lose 72.44%, cEV = -132.78

Everything but that 32o is still +EV here. The most surprising find is that A2o fares worse than the low suited connecter here, and that JTs isn't too far off the pair in equity. This is because A2o is crushed by all your opponent's range but the Kx hands, while JTs is still doing OK against the weak aces and low pairs. Moving on...

Scenario 3: Villain is tilting or just likes to gamble. He's calling your shove with any pair, any ace, KTo+, K9s+, QJ, QTs, JTs.

You know the drill: This is top 26.1%, so

P(fold) = 1 - (26.1/66.7) = 60.9%. cEV for folds is 91.3. Your shoving ranges will fare as follows:

1) 2 2 win 43.8%, lose 54.97%, cEV = - 54.59
2) A 2 win 32.1%, lose 53.3%, cEV = -103.615
3) J T win 31.12%, lose 53.09%, cEV = -107.38
4) 7 5 win 37.08%, lose 62.05%, cEV = -122.04
5) 3 2 win 27.95%, lose 71.06%, cEV = - 210.70

Yikes, now only the pocket pair is profitable against this range. Let's add a stronger but not amazing Ax hand to this mix:

6) A 8 win 41.98%, lose 47.75%, cEV = -28.2

A ha! Against a wide calling range, a middling A-rag hand fares pretty well. Better even than that low pair. Let's do two more examples before I get to the point:

Scenario 4: Villain is a drunk monkey. He's calling that shove with any pair, any ace, K7o+, any suited king, Q8o+, Q6s+, J9o+, J7s+, T7s+, 97s+, 87s, 76s.

This is a whopping 39.7% of hands! You only get a fold 1 - (39.7/66.7) = 40.8% of the time, and only have 60.72 in +cEV in folding. Let's see how your hands fare. I think I've convinced you that connector hands are bad against a wide call range, so I'll throw out 75s and 32o, and show 22, A2, A8, and JTs, along with a slightly better pair (55):

1) 2 2 win 45.71%, lose 53.04%, cEV = -54.242
2) A 2 win 41.24%, lose 49.54%, cEV = -61.42
3) J T win 43.56%, lose 53.39%, cEV = -72.74
6) A 8 win 48.88%, lose 44.41%, cEV = +33.078 (!)
7) 5 5 win 51.81%, lose 47.26%, cEV = +33.67 (!)

You can now 3-bet shove A8 and 55 type hands for VALUE here, never mind the small chance you have of a fold! The difference between A8 and A2 and 55 and 22 is HUGE if villain is calling you mega-wide! One more example:

Scenario 5: Villain is tight from the button. He's only raising 25%, and calling the same range as the villain in scenario 1.

The equity calcs are the same, but your fold EV drops a lot:

P(fold) = 1 - (10.4/25) = 58.4%, cEV of a fold is 87.6.

Against this villain it is wrong to 3-bet shove A2, but not JTs. Hmmmmmm. Your edge against this opponent comes from him playing mega-passive on the button, of course.

What have we learned from this example?

1) Against an opponent you suspect is not calling your 3-bet very often, your edge in 3-bet shoving comes from FOLD EQUITY. On the off-chance you are called, it is best to have a pair or a middling suited connecting hand than a bad ace.
2) Against an opponent you suspect will call your 3-bet shove somewhat wide but not a lot, your edge is still in your fold equity, but suited connector hands drop a bit in value, and marginal aces increase in values.
3) Against an opponent you suspect will call with a lot of his raising range, your EV in shoving comes from the fact that a middling Ax hand or a low-ish pair is a FAVORITE against whatever trash he's calling with. Suited connectors should not be shoved against these opponents.
4) 3-bet shoving any pocket pair over a 3x raise with 25 BB effective is almost never a mistake.