Theory: AK facing a 3-bet

[Ringmaster]

So there have been quite a few AK posts lately. Not really surprising for SSNL as it can be a difficult hand to play. Anyway, I thought I'd share some thoughts on how to play it. Most of this stuff is probably nothing new for the more experienced posters, but hopefully some people will find it useful/interesting.

DISCLAIMER: this is not meant to be a comprehensive guide on playing AK in all situations against all opponents at all levels. I'm going to be considering a very specific situation aimed at small-stakes players, but one which occurs frequently. Details are as follows:

100NL, 100 BB effective stacks, standard TAG opponents.

Hero is OTB w/ A[img]/images/graemlins/club.gif[/img]K[img]/images/graemlins/diamond.gif[/img]

ACTION:
3 folds, <font color="red">Hero raises to 4,</font> <font color="red">SB raises to 13,</font> BB folds, <font color="red">Hero... </font>

At this point Hero has 4 main options:

(a) fold
(b) shove for 100 total
(c) ~pot-sized 4-bet
(d) call 10, see a flop

Let's discuss each in turn.

(a) Fold

Clearly this is not a realistic option. We are being offered 2:1 on a call with position, and even against a range as tight as {QQ+,AK} we have nearly 40% equity. Virtually every player's range will be wider than this anyhow, so folding is LOL.

(b) Shove

How wide does the SB's range have to be for shoving to be immediately +EV?

First we need to know his calling range for a 4-bet shove. This will be highly dependent on player reads, table image and recent history with the villain. For now, let's just assume no history and that both hero and villain have a solid TAG image. Villain has no reason to suspect your are shoving non-premiums, and let's suppose he calls a shove with {QQ+,AK} only.

WARNING, MATH AHEAD:

let p : probability villain folds
E : equity vs villain's call range
Mo : money in pot at decision point
Mh : money remaining in hero's stack at decision point
Mv : money remaining in villain's stack at decision point

EV(shove) = pMo + (1-p)E(Mo+Mv) - (1-p)(1-E)Mh

We have:

E = 38.8% (remembering that we hold Ac and Kd)
Mo = 18
Mh = 96
Mv = 87

We want to know how often villain has to fold in this situation so that EV (shove) = 0. Substituting the known values and solving for p gives us:

EV(shove) = pMo + (1-p)E(Mo+Mv) - (1-p)(1-E)Mh
0 = p(18) + (1-p)(.388)(18+87) - (1-p)(1-.388)(96)
p = 0.500

So if villain folds exactly half the time, pushing will be neutral EV, and if he folds more than this, pushing is +EV. His calling range of {QQ+,AK} has 21 total hand combinations (after accounting for the fact that we hold an A and a K), so if he's 3-betting more than an additional 21 hand combos pushing will be +EV. For example, suppose he 3-bets {TT+, AQ+}. That's an additional 24 hand combos, making a push slighly +EV (as long as he in fact folds TT, JJ, and AQ to our shove). I think it's safe to say that nearly all decent players are 3-betting a range wider than this these days, especially versus a button open.

Now let's look at the EV of pushing against a more realistic range. Suppose villain's calling range is {QQ+,AK} as before, but that he's willing to 3-bet a button open with a much wider range, say 99-JJ, AJ-AQ, and KQ as well as QQ+, AK. In addition, he'll also 3-bet small pairs, suited aces, and suited connectors as a semi-bluff roughly half the time. So his 3-bet range looks like {99+,AJ+,KQ: 100%; 2-88,A2s-A5s,54s-JTs: 50%}. With this range, villain will 3-bet Hero's button open roughly 11% of the time, which seems plausible (although it probably feels like a lot more than this when you're playing!).

What is the EV of shoving versus this range?

The only variable that has changed here is p (probability of villain folding). His 3-bet range consists of 136 total hand combos, of which he folds all but 21. So p = (136-21)/136 = 84.6%.

EV(shove) = pMo + (1-p)E(Mo+Mv) - (1-p)(1-E)Mh
= (.846)(18) + (1-.846)(.388)(18+87) - (1-.846)(1-.388)(96)
= 12.45

So shoving is hugely +EV versus this villain. But wait... a smart villain will adjust his calling range when he figures out what we're doing, right? Let's look at what happens to our EV if villain adjusts perfectly. Suppose he 3-bets, then we show him our hand.

Should we still push?

Let's see what hands villain can profitably call with. There's 114 in the pot and it's 87 more for him to call, giving him 1.31:1 odds. Thus, he should call with all hands that have 43.3% equity or more versus AK. The only hands in his range that fit this criteria are {22+,AK}, 60 combos. So p = (136-60)/136 = 55.9, and our equity versus his adjusted calling range is 42.4%. At this point, you may notice that the EV of pushing must still be positive. Why? Because both our pot equity and villain's folding frequency are higher than in the case where pushing is neutral EV! In fact:

EV(shove) = pMo + (1-p)E(Mo+Mv) - (1-p)(1-E)Mh
= (.559)(18) + (1-.559)(.424)(18+87) - (1-.559)(1-.424)(96)
= 5.31

So, even if villain knows we're pushing AK and he adjusts his calling range perfectly, we still retain just under half the EV we had before. What this means is that, if villain insists on 3-betting such a wide range, shoving with AK is unexploitable! Let me repeat that. Shoving AK in this spot is +EV, and there isn't a damn thing villain can do about it. What's more, this is true regardless of whether or not you are "balancing" your AK pushes with other hands, because it's still +EV even if he can see your hand!

Of course, villain could adjust his initial 3-betting range. However, in order to make shoving AK -EV, villain would have to constrain his range significantly, which obviously benefits us in other ways. Let's take a worst case scenario where villain only 3-bets hands that have positive equity versus us: {22+,AK} so he can call our push with his whole 3-bet range. What happens?

In this case our equity is 43.5% and obviously p = 0. Thus:

EV(shove) = E(Mo+Mv) - (1-E)Mh
= (.435)(18+87) - (1-.435)(96)
= -8.57

So now shoving is -EV, but only in isolation. Villain is now 3-betting us at a much lower frequency than he was before (only 6.6% of the time in fact), an excellent benefit to Hero which likely outweighs the -EV of shoving by itself, in the form of easier blind steals. In addition, we might get a little sneaky and start mixing in a shove with AA some of the time. Let's say we continue to shove AK, but mix in a shove with AA only a third of the time we hold it in this spot. What now?

Our average equity rises to 48.4%, yielding an EV of:

EV(shove) = E(Mo+Mv) - (1-E)Mh
= (.484)(18+87) - (1-.484)(96)
= 1.28

Now we're +EV again! The moral of the story is that adjusting his initial 3-bet range to one that makes pushing AK -EV will be of dubious value to villain, and more than likely a bad idea, all factors considered. So if we accept that he won't be willing to adjust his initial 3-bet range to make this play unprofitable, he's screwed, because we just saw that adjusting his calling range isn't enough.

Therefore, not only do we know that pushing AK in this situation is +EV, but there's nothing villain can do to stop us (unless he wants to lose even more money elsewhere). So pushing is clearly better than folding, but is it the best option though?

(c) Pot-sized 4-bet

What about making a smaller, pot-sized 4-bet? While this mostly has the same effect as pushing, there are a few small differences which may be noteworthy. If we reraise to 36 instead of pushing, there will be 50 in the pot and it will cost villain 23 to call. So we're giving him more options: he can fold or push as before, but now he also has the option to see a flop OOP with slightly better than 2:1 odds.

Presumably, villain should be folding and pushing with the same hands as in the previous case. One might argue that a 4-bet could have more FE than a push, however, since it looks like we're trying to milk him with AA or KK. This is especially true if we actually play AA and KK this way and villain knows it. If this is true, it helps the EV of 4-betting AK since he'll be more likely to dump pairs (which we are behind).

OTOH, villain may think he has FE with a push since we're not all-in yet, and consequently he might bluff with some portion of his range that he would othewise have folded had we simply pushed in the first place. Of course, we're not going to be folding getting better than 2:1 on a call with AK, since even if he's pushing only {KK+,AK} we have odds to call with 37.1% pot equity. If he decides to bluff-push some frequency with non-pair hands like suited aces or connectors (which have the best equity when called by a big pair), that's fine with us, as AK is favoured over those hands. We'd prefer he fold his small and medium pairs rather than bluff with them, however. Overall, AK is likely to be ahead of any range he's bluff-pushing with though, so this potential effect is a positive one for us.

What if he flat calls the 4-bet? Both Hero and villain will only have about a pot-sized bet left on the flop, but a potential problem arises since villain will get to see the flop before deciding whether or not to commit the rest of his stack. If he knew we held AK, a smart villain could stop-n-go with {22-QQ} by calling the 4-bet and pushing any non-A or K flop which doesn't also make him a set. Assuming he's never pushing non-pair hands on the flop (or at least, not often enough for us to call profitably), we would have to fold most of the time as we're only getting about 2:1 and therefore don't have odds to spike an A or K on the turnrn/river. What this means is that, unlike pushing, 4-betting only AK is exploitable. However, this problem is easily solved by 4-betting non-AK hands in order to disguise our hand. For example, if our 4-bet range is {KK+,AK}, villain is not going to be (profitably) pushing small pairs into us on the flop since we're holding AA/KK nearly half the time. Opening our 4-bet range slightly thereby denies villain a profitable stop-n-go strategy. As long as we're 4-betting AA and KK with a high enough frequency, a pot-sized 4-bet strategy with AK should be at least as profitable as 4-bet shoving with it, perhaps more so.

(d) Calling

The fourth and final option, calling the 3-bet (with the intention of shoving over a c-bet on most flops), is the most interesting IMO, and has the potential to be the most profitable. Unfortunately its also the most complicated, and since this post is already way too tl;dr I'm going to make a separate post about it some other time. [img]/images/graemlins/cool.gif[/img]

Cliff notes:

- folding AK is LOL
- 4-bet push with AK = teh moniez
- pot sized 4-bet also g00t, but watch out for tricksy villains
- call 3-bet / shove flop = ???

Thanks for reading, any comments/input/corrections are welcome.

[smellmuth]

depends on position stack sizes image etc

[Nielsio]

3-bet% is important

[rakes.a.beach]

nice post. more cliff notes please.

[ikestoys]

this is actually a really good post, i'd admit i was expecting something worse from a newb

[zaphod]

Nice post!

[AggJedi]

Very nice, thanks for going through the trouble to do all of this.

[Wooden Ta Sheng]

Thought provoking.

[Xanta]

Wow, really good. I was really questioning the +EV of shoving AK preflop and wanted to post about it the other day but this answers most of my questions

Quick thought though, in the example you're in position (being 3b from the blinds)

How do the pros and cons change when you're OOP for
1) Pot sized 4betting
2) Calling

Cheers, nice post.

[dragonystic]

nice post, thank you for this