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To go a little further with the analysis because I believe OP did a really good job of presenting this hand and his thinking.. For the most part the results were removed (remove future cards after the decision next time fwiw). So, well thoughtout threads such as this should be given thought and as much discussion as possible. Oh and I'm bored, so I felt like looking at this situation and this hand represented it well.
Anyways...
(I play around 14/11, which is relatively close to villain. I don't open limp or anything, but I would suspect that villains range from UTG+1 is somewhere around 66+, AJs+, AQo+, KQs. It could be a little tighter, or a little wider, but I think this is a good range to use here).
Blue = likely to bet/fold
Red = Likely to felt by either b/3betting flop or b/calling, check/shoving
Villain's Perceived Range
66 - TT = 30 combos
JJ = 3 combos
QQ = 6 combos
KK = 1 combo
AA = 3 combos
AQ = 11 combos (AQ that will likely b/f)
AsQs = 1 combos (AQ that will continue and likely felt)
AK = 6 combos
KQs = 2 combos
AJs = 2 combos
Total combos = 65 combos
Using villains range we see that villain is likely to felt only 16 combos in his entire range [JJ, KK, AA, AK, AsQs, KQs] (note = KQs is relatively likely to not continue here, but might). This leaves him folding the other 49 combos, which means he folds 49/65 or 75% of the time.
A raise here to $3 means we are risking $3 to win the $2.15. Therefore, a raise here only has to work 3 / (2.15 + 3) or 58% of the time. As we can see he is folding more often that that, and therefore a raise here is profitable based on fold equity alone.
Using the above logic it may seem as if a raise is correct. And while it is +ev, I don't believe it is the most +ev play we can make here. This can be seen by looking at how our actions will alter villains range.
If we call here, villains entire range is forced to see a turn card. We are indeed ahead of the majority of villains range. We are only a dog to JJ, KK, AA (7 combos). And the majority of villains range is drawing relatively thinly, so we aren't too worried about giving a free card. His 66-TT, QQ, KQ, AJ hands have at most 3 outs. And since we are in position we are likely able to extract more value from the some % of this range that will likely either 2 barrel, or make some sort of second best hand.
However, if we raise the flop we fold out the majority of the hands we beat, and surrender the chance of getting another street or two of value from some of those hands. And we are putting ourselves in a spot where we are likely to get allin against a range that we are no longer a favorite to [JJ, KK, AA, AsQs, AK, maybe KQ]. And when we do this we are now drawing relatively thinly when behind, and when ahead villain still has decent equity (3 outs with KQ, and 12 with AsQs).
So imo calling is definately magnitudes better than raising in this spot when we hold AK. Now, if we held a hand like T9 here, then a raise is going to be profitable as I have already shown due to the % the raise needs to work and the % of the time he will actually fold. And this wouldn't be including the % of the time we hit our gutshot and take a bigger pot.
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