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As far as shoving goes, that's actually a math question, although God only knows if I remember enough algebra to do this right.
First, we determine a range for villain. OP has the most insight into that. Then, we break this range down by hand combinations (three for AA, three for KK, nine for AK, six for QQ, etc.) Finally, we determine what he'll call a shove with. Since this is your first time four-betting him, a reasonable range would be QQ+, or maybe KK+. We then calculate our shove equity.
For instance, let's assign him a completely made-up three-bet range of {99+, ATs+, AJo+, KQs}. There are 58 total hand combinations in that range. And let's say that he calls a shove with QQ+, which constitutes 12 of those hands. Therefore, 12/58 = 20.6% of the time, villain will call the shove, and 46/58 = 79.3% of the time, villain will fold to the shove.
When villain folds, hero picks up the pot of $9.5. When villain calls, hero risks $57.85 to win a pot of $109.6 X% of the time, where X is equal to hero's equity against villain's range. In this case, Pokerstove gives AKo an equity of 30.9% against villain's calling range of QQ+, which means hero's dollar equity is about $33.87, a loss of $23.98.
Therefore, the EV of a shove is determined by the following:
0.793(9.5) - 0.206(23.98) = 2.59,
or a profit of $2.59.
Intriguingly, a shove can still be +EV against a tighter range. Let's say that villain is raising a TAG range of {TT+,AK} and that villain will call with KK+. Now there are 33 hand combinations, and of those 6 will call your push.
27/33 = 81.8% of the time he folds
6/33 = 18.2% of the time he calls
Your equity against a range of KK+ is, of course, much worse, only 18.474%. This means that your dollar equity is only $20.24 when you get called, a loss of $37.61. So:
0.818(9.5) - 0.182(20.24) = 0.92, or a profit of $0.92.
(Which, realistically, you lose to rake. Still, nifty, and probably less exploitable than either calling or folding.)
If villain is ever reraising with AQo preflop, a shove is completely justified.
Anyway. I hope this was helpful. I found it useful as a mental exercise. If any of my math looks wrong, somebody please let me know.
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