OK, to be clear, if you want your problem to be complete, you need to specify
your hand: AKs
villain's range: for instance you think he plays all pocket pairs the same, i.e. 22AA
villain's calling range if you shove: you said AA, KK, sets, let's say 44,88,JJ on the J84 board zach suggested
that allows us to compute the range he folds. with the above assumptions that's 22,33,55,66,77,99,TT,QQ.
Then we can figure your equity against the range he calls with:
Board: Js 8s 4d
equity win tie pots won pots tied
Hand 0: 32.182% 32.18% 00.00% 4779 0.00 { AsKs }
Hand 1: 67.818% 67.82% 00.00% 10071 0.00 { KK+, JJ, 88, 44 }
We can also figure out the equity against the range he folds:
Board: Js 8s 4d
equity win tie pots won pots tied
Hand 0: 54.413% 54.41% 00.00% 25857 0.00 { AsKs }
Hand 1: 45.587% 45.59% 00.00% 21663 0.00 { QQ, TT99, 7755, 3322 }
We also need to figure the size of both ranges. In this case, there are 48 combos in the range he folds and 15 combos in the range he calls with.
Then we need to know the amount in the pot, let's say A, and the amount left in the effective stack, let's say B.
Your EV when folding is B
Your EV when shoving is A+B when villain folds 48/(48+15) = 76% of the time, and 0.32x(A+2B) when he calls 15/(48+15) = 24% of the time, so in total, your EV when shoving is 0.76x(A+B) + 0.24x0.32x(A+2B) = 0.837 x A + 0.914 x B.
Therefore, shoving is better than folding when 0.837 x A + 0.914 x B > B , i.e. B < 10 x A, i.e. shoving is good practically unless is a huge overbet. Of course we get this result because we started with a large range for villain, which means we get a ton of fold equity. We would need to make this computation for a more realistic range for villain.
Computing the EV of calling is is significantly harder as we need to decide what kind of action we take on a blank turn and what kind of action we take on a turn that gives us the flush.
