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1/2$ 6max NLHE game. I am UTG and I raised by 6$ with AKo. Everybody folded except BB who raised by 11$. Pot is now 20$. Now, if I go All-in my opponet will fold everthing except AA and KK in which case UTG has 18.48% against 81.53% equity. How deep or better say short stack should be to have +EV in this case? How to calculate that?

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How do you know your opponent will fold anything except aces and kings ? what about queens ? – Radu Murzea May 2 '12 at 13:04
Are you saying that you were UTG and raised with AKo, or that there is another player in UTG who also has AKo? – Jeffrey Blake May 2 '12 at 13:48
I am UTG and I just to want to know how calcualtion works, not did I make correct estimation on my opponent range – Shark4Ever May 2 '12 at 13:55
@SoboLAN: It's pretty common for someone to fold everything but AA/KK to a 4-bet preflop. It's opponent dependent, but this is a relatively easy read to develop if you've spent some time playing with him. Even more so if you have stats in online play. – Jeffrey Blake May 2 '12 at 14:25
up vote 5 down vote accepted

If you know your post-flop equity, your cards are irrelevant to the calculation here.

That said, we have one crucial bit of information that is lacking: what range of hands is your opponent reraising with preflop? If he reraises with 100% of his cards and then folds everything but AA/KK, then it's pretty easy to make this profitable. If he only ever reraises with AA/KK in the first place, then it will never be profitable.

Once you have a judgement for the hands he is reraising-but-folding, you can plug it in to the equation for calculating your EV on a heads-up shove, as follows, assuming you both have $200 stacks:

EV = (F% * $20) + (1 - F%) * ((.1848 * ($20 + ($200 - $11 [e.g. the amount your opponent has already put in the pot]))) - ((.8153) * ($200 - $6 [e.g. the amount you have already put in the pot])))

That's the equation to start from. But your question was about what stack size is needed to make this profitable. So to solve the equation, we need to make an assumption about how often he is reraise+folding. I'm going to say 75% for the example below, but that could be way too high for a lot of players. That's saying that there are 4 times as many hands that he reraises with preflop as there are combinations of AA/KK. For this to be accurate, you need to have a good idea of the actual frequency of his reraises.

Using 75% as an example, and simplifying the equation a bit...

EV = (.75 * $20) + (.25 * ((.1848 * ($9 + S$)) - (.8153 * (-$6 + S$)))

Which can be simplified further to:

EV = $15 + (.25 * ((.1848 * $9) - (.8153 * $6) - (.6305 * S$)))

And then further to:

EV = $15 + (.25 * -$3.2286) + (.25 * (-.6305 * S$))

And finally to:

EV = $14.19285 - (.157625 * S$)

We can then set EV to $0 to find the breakeven point:

$0 = $14.19285 - (.157625 * S$) => .157625 * S$ = $14.19285 => S$ = $90.04

So, if your opponent is reraising 4 times as many hands as he will call your shove with, then a shove is profitable for any amount under $90.04. Replace the .75 and .25 in the second equation to give a starting point for solving for a different frequency of reraise vs call.

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