Deep Analysis of a Hand (tl;dr)

Going to try something different and do a combination analysis of a hand. Hand reviews like this are something that I intend to start doing regularly so I may as well post them on here so that anyone who reads can pick apart my play. This will be fairly deep and maths intensive so sorry about that, tl;dr

Initial notes: I don't believe my preflop 3-bet was big enough, and allowed villain to profitably set mine against my very tight UTG+1 3-bet range. Villain is an unknown but over ten hands had only played one, so I'm assuming a fairly tight player. So now I'm going to put his range for calling the 3-bet as a pretty loose {99-KK, AQs+, AKo} which is 50 combinations (since I have two aces). Given this range, it should be pretty easy to work out the correct play in this hand, which was my intention in this first analysis.

I think that the vast majority of villains would just call with their entire continuing range on this board; to slow play their strong hands and to play it safe with the marginal AK. So when he does call, I need to refine my view of his range. Villain could certainly flat a single street with TT-QQ but given how strong my range is it's probably safer to fold the flop with the king out there so I'll rule those hands out. That leaves {99, KK, AK} for a total of 12 combinations. The value bet on the turn is actually indifferent. Since we're way ahead (six AK combs) half of the time or way behind (six set combs) half of the time we don't get any value by betting against this specific continuing range. So I think it would have been better to check back. As played, when he raises it's a case of which hands he'd do this with and the pot odds we're getting:
If I assume his raise is all-in (with only $3 behind this is pretty safe) then to call I risk $5.55 to win $(3.95+2.55+8.1)= $14.6 and must be good 28% of the time to justify the call. So he needs to be doing this with at least 3 combinations of AK for calling to be better than folding. I don't think that your average micro player would be turning AK into a bluff here, although some might. This player had given some evidence of tight tendencies so I think that the fold was a good one. An expert player could exploit me in this spot by shoving his entire range, fwiw.

Sorry for this long and heavy blog post, I'll try and tidy the next one up and structure it a little better as that was messy to say the least!