AK allin Pre: Some interesting maths that suprised me

[Pelion]

This is a spot that seemed counterintuitive to me. Its pretty simple back of the envelope stuff so it isnt going to be totally accurate but the result isnt close enough to make that matter.

Someone raises QQ preflop, we 3bet AK and they 4bet QQ to 33bb. At this point there is too much dead money to fold (given that we know they have QQ). We also have no fold equity preflop with only a potsized bet remaining. So we have 2 choices:

1) Shove and flip.
2) Call. Shove if the flop is A or K high and check/fold otherwise.

If we shove and flip then the total hand is breakeven (more or less).

If we call then we will have lost 33bb 2/3 of the time when an A or K doesnt flop. We will win 100bb 1/3 of the time when we shove and he calls. The total EV (from the start of the hand) for this line is

-2/3 * 33 + 1/3 * 100 = -22 + 33 = +11bb.

Obviously better than breakeven even though we are check/folding most flops. In fact in order for shoving preflop to be better they have to fold QQ at least 2/3 of the time on an A or K high flop for a pot sized bet postflop (or fold QQ preflop to a 5bet shove). I suspect most players would feel too committed at this point both preflop and on the flop for us to have anywhere near enough fold equity for shoving preflop to be better than calling.

This might be obvious to some/most of you already but I was under the impression AK "likes to see all of the cards" so it suprised me a little.

Practically this also allows us to lose less to AA and KK the times we miss the flop (alot more often since they hold 2 of our cards). We miss folding weaker pairs preflop but I dont think many people 4bet/fold preflop anyway for 100bbs.


Disclaimer: The maths in this hand has obviously been simplified alot. I have used the total wins and losses of the hand where I should have used the EV from the point of the decision. The reasoning behind this is that, if we can turn a breakeven (or marginal loser) of a hand overall into a winner overall by playing 1 decision differently, then that decision is obviously the more +EV of the options. I have done a more "correct" analysis of the hand too (which actually works out even more in our favour) but would be alot more complicated to post.

[Halv]

If the opponent is sane he'll get away from QQ on A/K high flop 2/3 of the time in a 4bet pot. He only beats a bluff at that point, and he's probably not going to face one 1/3 of the time. Which made me think; can we call 4bets light (ie SC's) and shove 2p+, good draws, plus bluff shove alot of Axx, Kxx flops?

[bode]

If the opponent is sane he'll get away from QQ on A/K high flop 2/3 of the time in a 4bet pot. He only beats a bluff at that point, and he's probably not going to face one 1/3 of the time.

this pretty much sums it up. Even villains at micro stakes can get away from QQ w/ an A or K high flop.

[Pelion]

In response to Halv's idea I dont think it will work

If he always folds an underpair:

On an Axx flop there are 18 combinations of hands that call. (AA/AK), and 24 combinations that fold (QQ,KK) so a shove is just about good if we get to this point, but he doesnt have to call with QQ too often to turn it into a spew. 57% folds so EV is

On a Kxx flop there are 30 combinations that call (AA,KK,AK), and only 12 that fold (QQ) So a shove sucks. 29% folds.

If he is 4betting a wider range preflop there is a bigger chance he will call with an underpair on the flop so I really dont think this part of the plan works.


When we have a draw we will probably have 40% equity over 2 cards. Most of the times we have a draw there wont be a high card so assume pairs call and AK folds 2/3 of the time. 41 combinations call, 11 fold. (20% folds).
So 20% of the time we pick up the 66bb pot. 32% of the time we win 132bbs and 48% of the time we lose the 66bb shove.
total EV is +3.2 + 42.24 - 31.68 = +13.76.

So if we find ourselves with a draw we shove. 2 Pair+ is obviously a super easy shove since we are a favorite to their entire range. EV against a pair is something like +92.8-19.8=+73bb, but he folds AK so the total EV is more like +70bbs.

Overall we flop a OESD OR flushdraw 20% of the time, 2pair+ like 6% of the time and an A high flop around 1/4 of the time (since we dont know if he has an A or not).

SO 20% of the time we win 13.76bb [draw] => +2.8bb
6% of the time we win 70bb or so [2pair+] => +4.2bb
25% of the time we win +6.6bb [A high flop] => +1.65bb

I havent included K high flops since that part came out -EV.

That leaves 49% of the time where we call the preflop 4bet and check/fold the flop for an EV of -21*0.49 => 10.29

The total EV of the move is -1.64bb when we assume he never makes a loose call postflop with an underpair. In reality we are probably quite a bit worse off than this because of loose calls but its interesting that its so close.

[Pelion]

If the opponent is sane he'll get away from QQ on A/K high flop 2/3 of the time in a 4bet pot. He only beats a bluff at that point, and he's probably not going to face one 1/3 of the time.

this pretty much sums it up. Even villains at micro stakes can get away from QQ w/ an A or K high flop.

They can in raised pots, but with only a pot sized bet to go alot of players will feel committed enough to make retard calls when the board has shat on their hand as my AA bluff hand proved. Whether or not this is 1/3 of the time im sure is very opponent dependent. I actually asked trainer_jyms about this when I first thought of it and apparently he makes that call "all the time".

[minSim]

I think players are more scared of Axx flops then Kxx flops (probably right). So maybe QQ is folding a bit more on A high flops. I think I would lay down QQ on A high flops a lot.

Besides that, we always have to work with ranges instead of exact hands. If QQ is in his range, can JJ be? AK? AQ?

Calling when you know villain has QQ+ can be the most +EV, but pushing against AQ+ can be most +EV.

[bode]

If the opponent is sane he'll get away from QQ on A/K high flop 2/3 of the time in a 4bet pot. He only beats a bluff at that point, and he's probably not going to face one 1/3 of the time.

this pretty much sums it up. Even villains at micro stakes can get away from QQ w/ an A or K high flop.

They can in raised pots, but with only a pot sized bet to go alot of players will feel committed enough to make retard calls when the board has shat on their hand as my AA bluff hand proved. Whether or not this is 1/3 of the time im sure is very opponent dependent. I actually asked trainer_jyms about this when I first thought of it and apparently he makes that call "all the time".

i just run into too many villains that dont get the concept of pot comittment. I *rarely* would make the call w/ QQ in a 4-bet w/ an A on the flop if villain is showing alot of strength, even w/ only a pot sized bet behind. Of course this is playing 50nl mostly, and i dont feel that players are tricky enough for this to be -EV. In 200nl+ games, I expect making calls like this to be more standard.