Monday, June 09, 2008

When are you pot committed to calling a preflop all-in with AK?

UPDATE: I'm pretty sure my original calculations were wrong, so I've revised this post throughout to reflect those changes. Thanks to Greylocks for his help, and I apologize for the mistake.

Calling all-in with AK with 100BB effective stack sizes is not a leak in cash games against an opponent who will push with QQ or better and you've already invested only 21BB.

That's my conclusion after running some numbers based on a recent hand in which I had 4-bet raised to $400 preflop (40BB) with AK and then got pushed on in a 5/10 game. I had to call $600 to win $1,400 in the pot. I was up against KK, failed to spike an Ace and lost a buy-in.

The hand got me wondering whether my call was correct against a villain's hand range of AA, KK, AK and QQ.

When am I committed to calling all-in?

AK is about a 60-40 dog against a range of AA, KK, AK and QQ, according to PokerStove:

equity win tie pots won pots tied
Hand 0: 39.594% 19.26% 20.33% 110831016 116968584.00 { AKs, AKo }
Hand 1: 60.406% 40.08% 20.33% 230565960 116968584.00 { QQ+, AKs, AKo }

The break-even point for calling comes when your opponent shoves for 121BB, and you have to call 79BB more:

0.395(121) - .605(79) = 0


Therefore, against this range, I have to call after only putting in 1/5 of my stack (21 BB) with AK.

What if my opponent's range is AA, KK or AK?

Then my equity drops to 38-62, and I become committed after investing 24BB.

What if my opponent's range is only AA and KK?

Then my equity drops to 20-80, and I'm not committed unless I've already invested 60BB.

Calling an all-in with AK is far easier against someone with a wider preflop pushing range. That changes the odds substantially.

These numbers show me how hard it is to get away from AK preflop against anything but tight players. Against a wider range, I lose the option of folding unless I have a strong read or I'm deeper-stacked.


Greylocks said...

In my mind your way is kind of the odd way to look at it. Figuring pot odds makes more sense to me. According to PokerStove, you're about a 62:38 dog against this range of hands, which comes out to about 1.63:1. If this is correct, you should call if the pot is more than 1.63 times the cost of calling.

This doesn't produce the same numbers you got. If the pot is 135BB and your cost of calling is 65, you're getting better than 2:1, and your EV comes out to .38(135) - .62(65) = +11BB.

So one of us is wrong, or PokerStove is.

Gnome said...

Thanks for helping me figure out this problem. I can feel confident that PokerStove is right and I'm wrong.
I rewrote the post to reflect the new conclusions.

Greylocks said...

The math is tedious and never easy and I have made plenty of similar mistakes myself. That's why I was hesitant to proclaim that I was right and you were wrong.

spritpot said...

It won't make much of a difference, but you'll want to include the blind money, too (and take away the rake, but for the levels you play, I'm sure the rake is less than the blinds).

Also your opponent is not shoving for 121 bbs, he's shoving for 100, forcing you to call off 79 (assuming you started with 100).

So sample hand would be, assuming no rake: CO raises to 5 bbs. For some reason you raise to 21 bbs. CO shoves. Let the payoff to folding equal zero. So RELATIVE TO FOLDING, you will win 100 (CO's stack) + 21 (money you put in) + 1.5 (blinds) = 122.5 bbs. If you call and lose, you lose 79 bbs (the rest of your stack) more than you would have if you had folded.

Also note that there's about a 5% equity difference in most of these calculations between AKo and AKs, so you'd want to take that into account when making your decisions.


Gnome said...

What I meant to write is that an opponent's shove of his entire stack would make the pot odds 121:79.
I ignored the blinds and rake for the sake of simplicity, although I acknowledge they make a small difference.

Weak Player said...

Nice for thought...thanks.

SubZero said...

Very interesting, and so easily applicable too! Will definitely have this in the forefront of my mind next time I get shoved on after I 3-bet with big slick....

Guin said...

How tight would this opponents stats have to be at 6 max before you would fold?

Thinking 16/13?

Gnome said...

It isn't always stat-dependent. You'll find tight players who will go all-in preflop with AK, and loose players who will see a flop.
So the way I've been handling it is to take extensive notes on as many players as I can. Every time I see a preflop all-in, I take a note if my opponent shows down anything but AA or KK. This will help me establish his preflop all-in range for future reference.