Re: Pf confusion
These types of questions are really tough to answer in a vacuum because the correct play depends so much on the villain's range. But let's look at what might happen if we push.
Suppose that we have a villain who only 3-bets with the goods. Give him a range of {AA, KK, QQ, AK}. That's 3 +3 +6 +9 = 21 combinations of hands he could hold (given that we hold AK). Let's assume that he folds all but AA and KK to a push. You therefore have about 18.5% equity when called.
If he folds:
we win $90 x (15/21) = $64.3
If he calls and we win:
we win $460 x (6/21)(.185) = $24.3
If he calls and we lose:
we lose $434 x (6/21)(.815) = $106.1
So the EV of pushing is $64.3 + $24.3 - $106.1 = <font color="red">-$17.5</font>
Clearly, pushing is -EV against this range. It's also pretty easy to see that calling will be -EV as well, so folding is our best option by far.
Suppose now that villain 3-bets more liberally, with a range of {AA, KK, QQ, JJ, TT, AK, AQ}. Let's assume in this case that he calls a push with QQ+ (it's easy to see that if he instead folds queens as in the last case, our EV of pushing will be quite a bit higher). So there are 45 combinations, 12 of which he calls with. Our equity versus his calling range is 30.9%.
If he folds:
we win $90 x (33/45) = $66.0
If he calls and we win:
we win $460 x (12/45)(.309) = $37.9
If he calls and we lose:
we lose $434 x (12/45)(.691) = -$80.0
So the EV of pushing is $66.0 + $37.9 - $80.0 = <font color="green">+23.9</font>
So pushing in this case is +EV, and by quite a bit (remember, we assumed that he calls with QQ; if he folds it our EV will be even higher). So folding is out, and the only question is whether calling to see a flop would be even more +EV than pushing.
Unfortunately, the analysis for calling is quite a bit more complicated. There are several things we need to take into consideration, like our position, our pot odds, how often villain C-bets, if he makes big post-flop mistakes, etc. In this hand, I'd be inclined to call for 3 reasons:
1) we have position
2) we're getting 2:1 on a call, which is enough to try to spike an A or K on the flop.
3) metagame considerations (I'd play AA and KK the same way).
If we were to assume that we win the current pot every time we pair up on the flop, than calling would be EV neutral. However, some of the time we pair up we'll lose to a set, or tie with another AK, etc. On the other hand, some of the time we pair up we'll win an extra bet from an underpair, or AQ (perhaps more than one bet). We may also be able to out play JJ, TT or another AK or AQ even if we don't pair up. It's hard to put a $ value on these considerations, but I think that the positives outweigh the negatives (ie. calling is +EV). Whether it is more or less +EV than pushing is debateable, and probably depends quite a bit on the villain in question. Even if it's slightly less, however, it might still be worth it if it allows you to play other hands like AA and KK more optimally.
So basically, this post was a long winded way of saying that there is no "standard" way to play AK preflop.
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