It depends on how widely the BTN 3-bets the CO, and how often he 5-bet jams.
Against moderately aggressive regs, I call it off with AQ.

Another option is to just call. Your hand dominated a bunch of the hands he 3-bets, and the SPR is stll high. Most regs will 3-bet hands like QKo, AJo, AJs, A5s etc. every time, and you'll get to keep those in.

Decide what matters more in certain spot: a marginal all in spot, or having to play OOP without initiave.

Im too lazy to do you math for this spot, but 4 bet range consists of range percentage that is pure value and you will call off 5 bet allin and percentage that you wil fold, more air type and semi bluffs type hands like Q8ss, 77 and alike what is of your own preference 

Hand AQ is too strong hand to put in your semibluff range that you will fold to a 5 bet unless you are playing against very very loose player, in which case you will put AQ in 4 bet call 5 bet allin value range

That is usually not the case, so as Chael mentioned above, best option is to flat and keep wide BTN 3 bet range most often dominated , and ofcourse depending of players assumed tightness or wideness ( known from players stats or previous history ) you play postflop accordingly, and if unknown can assume worst, that player is  tight- til provided with more info to adjust your play better

A more precise way to phrase this question is 'Under what conditions is 4bet/calling or 4b/folding AQ more +EV than flatting with AQ?'.  I think it's always going to be +EV to 4b AQ, but almost never dominating to 4b/x with AQ.

Some people also have a flatting 4b range so it gets even more complicated to calculate the ev of a 4bet. Obviously the exact ev of calling a 3bet is impossible to calculate.

to Ben, what do u mean almost never dominating to 4b/x with AQ?  I didnt understand this.

to mkae 4betting best it has to be better than flatting

I set it up like this, following your assumptions (w/ blinds 5/10 and 100bb stacks):

With these ranges, the EV of 4bet/calling AQo comes out similar to calling (32.46 vs 34.26), so these conditions would be the cross over point.  For simplicity I've just assumed checkdown mode postflop.  Not perfect, but hey...

A smarter opponent however, would shove AA-22,AKs-AQs,AKo-AQo (~8.3%) and absolutely crush AQo.  This seems like a very easy and straightforward adjustment.

I'll admit I don't play a ton of NLHE, but this example seems a bit artificial to me.  For BN to 3-bet that 11% range he'd have to be playing a pretty weird style, and assume CO opens super wide.  Which is probably a correct assumption if hero stacks off with AQo.  But if he has that knowledge he should probably 3-bet a different range.  

Either way, I don't see how 4-bet/calling AQo can ever be even close to the best play against a rational opponent.

Can anyone improve on this?

Edit: What wasn't included in the calculations above was how much equity is realized for CO when calling.  If we assume he realizes 80% then the EV of calling becomes 15.41.  The effect of this change is that villain gets to shove a slightly tighter range, essentially taking out half the combos of KQo.  

I am quite a beginner doing those calculs but I will try and feel free to correct me. Here I share how I will doing it for a given 3betting range.

So we opening 3bb on the cut-off  with AQ, otb 3bet 9bb with his 3betting range.We choising between flatting his 3bet or 4bet 22bb and call off his 5bet (assuming he will never call the 4bet)

For this exemple I give to vilain this 3 betting range  TT+,AK for "value"  (46 combos) and (k9,q9,q8,j9,j8,t8,t7,97,96):xx for "bluff" (36 combos).  For a total of 82 combos (6.18% 3b range).

AQ have 47% vs this 3bet range and 31,6% vs the value part

1) 4bet 

ev = (36/82)*13.5bb + (46/82)* (0.316*101.5bb+0.684*(-78bb))

ev = 0.44*13.5 + 0.56 *(32.074-53.352)= -5.98bb

 2) call 3bet

pot is 13.5bb and we have to call 6bb. Giving a 19.5bb pot if we call

AQ have 47% vs vilain's range. So if we realise 100% of our equity (r%).  ev = (19.5bb* 0.47*1)-6bb = 9.165-6 = 3.165bb

I dont really know how to estimate r% but we can guess witch factor are in our favor or not. In this spot we are oop, without initiative .... so probably r% < 100 %. 

For r%= 80%, ev = (19.5bb*0.47*0.8)-6bb = 1.332bb

I have a couple comments to people's replies and questions:

@Chael Sonnen: I'm not sure what games you play in but I can't imagine you are +EV vs a "moderately aggressive reg" by 4 bet calling AQ in this spot? at midstakes in the games I play I would almost want to be against a maniac to do this. (I am somewhat tight preflop getting it all in so maybe I'm wrong though).

@Aleksandra Bozic: I don't like ruling one hand out of your bluffing range just because it's a strong hand. If the situation is right for a 4 bet bluff to be profitable and more +EV than calling, I don't think it should matter what hand it is. You should just do what is the most +EV. To give a very realistic example, let's say the button will 3-bet the co 10% of the time and fold to 4-bet 70% of the time. let's say his 10% is 99+, KQo+, KQs+, AJo, AJs+ (7.5%) and a few bluffs (57s - T8s, 67s - T9s for 2.5%). He continues facing a 4 bet with only JJ+ and AK. Let's also assume this is a reg that is going to make life suck for you postflop if you just call. A 4 bet / fold is clearly +EV. Is a call +EV? maybe vs this villain you determine it's not. Say you don't spot a huge imbalance of his where he puts too much or too little pressure on. So if you are unsure of whether a call is +EV, then why not 4 bet / fold it? this is the perfect hand to do this with since it blocks the most hands he will continue with. Plus for the occasional time he calls, you at least have two good cards you can hit. 

@ jonna102: you say that a smart opponent would adjust and use this as his range to crush AQ: "AA-22,AKs-AQs,AKo-AQo". This may be true but this range sucks pretty bad vs people who are not 4 bet bluffing here. and if he is getting in 22-88 a bunch vs someone who likely is gonna be getting in TT+, AK+ he is not going to be doing well overall (even if he does do well vs AQ). Also the original 3 bet range you assigned is probably pretty uncommon. I think a lot of people are pretty happy to call with small pocket pairs on the button preflop. Also I think 11% 3-bet OTB to a co open is not really weird and it's totally standard / not unusual. I see between 7 and 14-15% all the time. And I sort of get how your program came up with the calc for the EV of calling AQ but that's way to simplistic and can vary wildly based on the type of player you are facing.

@OP: if you are playing especially 200nl and lower, don't forget to include rake into your calcs when you do call off and win. That's going to change your EV a bit. Also I am totally in favor of doing whatever is +$0.001 EV but if you're the type to not do well following a swing like this (either up or down), if it's close you shouldn't do it (I know that's a whole separate conversation though).

@OP: fwiw here are a few thoughts as to what I do in practice. I'm sure there's a better way (and I look forward to reading more as this is a common spot). as i said before I'm somewhat nitty preflop when it comes to getting all in. I basically never 4 bet-call this hand vs a reg. I can prob think of a few times I did this year but they might have been game-flow related or some really weird stat I saw. If I see the guy is folding to 4 bets in this spot > 60% of the time, I'm not going to rule out 4 bet bluffing. If it's less, I'm probably not going to 4 bet. I could probably push this to 55% given the blockers?? (as you pointed out 57/58% is the breakeven point depending on sizing). but like I said I'm a chicken and I like to have a few points "cushion" to my bluffs. 

So anyway if I don't rule out 4 betting, then I decide should I 4 bet or call (or fold). the more I know about the guy's range and the more I know about what he likes to do postflop, the more info I have on a decision. (does he have a high bet/fold? does he have a bunch of broadways in his range? does he pot control a lot and make it easy to get to showdown? does he fold a lot to donks? does he just bet bet bet 85%+ of the time where you'll feel very comfortable calling down if you hit a piece? etc etc). If I don't have great reads or don't feel comfortable I just 4 bet / fold. (it's possible I'd fold if I just felt for whatever reason based on gameflow he wasn't bluffing here much. maybe I look at vs hero stats and notice he is 3 betting me 7% here and his normal is 11%? or maybe I just 4 bet him a few times and sense he is 3 betting me mostly with value hands? I dunno. these are just be guesses though...I'd say I usually just look at the numbers...just wanted to mention this. Also if I've been losing a lot, some people may think I'm tilted and my bluffs might work less often. I might lean towards just folding if I think I can't call at this point. this prob applies more to live poker or maybe on a site where the player pool is small and you know people are paying attention to that - or maybe you just lost two stacks to the guy on the btn or something where he can't miss that info).

If I do rule out 4 betting as a bluff (and the guy isn't a lunatic to where I wanna 4-bet / call), I look for similar postflop stats to decide if I should call or fold. If the guy is 3 betting 11% as you said, it's going to be pretty hard to find a fold here. If he is 3 betting like 6 or 7% and seems really solid postflop, I might just "f*kk it fold". 

sorry my above answer is not really game theory great or whatever but it's just what I do. I'd love to see what others think / do. btw keep in mind, this is based on having all of these stats. sometimes you don't have enough of a sample size to really get accurate numbers in all of these spots...in which case you are just gonna have to guess some things based on what you know about him / other similar regs.

to answer the question of whether 4 bet / calling, 4 bet / folding, calling the 3 bet, or folding is best there are just a ton of variables to consider. It is going to depend on what range the btn 3 bets with, how often he 5 bets (or folds or calls), what hands he 5 bets, how well he plays postflop in this situation, how well you play postflop in this situation. So you can def run simulations with different numbers here but it's going to take a lot more than one. Even if you can just conclude that say you both play equally well postflop in this spot, you're still going to need to run a bunch of different calcs to find the "sweet" spot of where you should 4 / bet / call. Let me know if you figure it out though :) hah.

btw this doesn't even factor in the repercussions of any of these actions to future action in this spot (which I know wasn't your question...but if you start to 4 bet / call AQ vs a guy you play with a lot, your action is going to go a certain way in the future. that may be something you are looking for or maybe not?).

hah yeah I def agree with that.

I suppose if someone wanted to calculate the EV of a call here, you can just make a spreadsheet with your postflop equity vs several different ranges. (like 10% polarized, 10% unpolarized, then break it down further if the guy is 3 betting hands like K9s vs 67s...or if a guy is using offsuit aces or PP to "bluff" with...then just do one for different 3 bet percentages (like say 5-15%). 

then you would have to decide how much of a disadvantage your being oop is. I suppose you can start with a 10-15% disadvantage? (that's just sort of a guess). And scale it up vs better players than you and down vs worse?

It would take a few hours to do this but it's not really that difficult. I guess some of those programs could do that a lot quicker. The biggest question I wouldn't know the answer to would be what to set the "oop disadvantage" to. 

@ candide - i prefer my 4 bet range to consists of value that wil call 5 bet allin and some air that will bet fold to a 5 bet as a bluff

Hand AQ imo is too strong, to 4 bet fold, i prefer to play that hand post flop,  wouldnt like to waste its value that way, and as you said - 'He continues facing a 4 bet with only JJ+ and AK. Let's also assume this is a reg that is going to make life suck for you postflop if you just call. A 4 bet / fold is clearly +EV' - but what if he flats your 4 bet instead 5 bet allin - wil make ur life difficult indeed when you face flop play after you 4 betted AQ and was flatted beside losing its potential when u folded to 5 bet

Ofcourse there is merit of moving AQ to 4 bet , call 5 bet getitin range if situation is right, and we re against looser player with wider range,  but in standard play id prefer just to flat, tho 4 bet folding it against tight opponent cant be wrong, im just not sure if its more + ev then flat calling 3 bet instead and playing it post flop 

3-4-5 bet range  isn't static, but dynamic dependable of opponents we are facing and their tendencies, not sure tho how to do exact calculation for each situation, i play it by feel, ..and in more cases then other i put AQ in flat 3 bet instead 4 bet it, against v tight or very loose opponent id 4 bet fold it or 4bet get it  in 

Ok, I'll give it another go with more complete ranges.  I play mostly PLO so I'll take the liberty of using someone else's ranges.  I've borrowed them from Janda's book.  I can't say if they're reasonable ranges or not, but they seem like a decent starting point.  You guys can probably adjust them better than I can.

Assumptions:

CO opens ~24%
(AA-22,AKs-A2s,KQs-K6s,QJs-Q7s,JTs-J8s,T9s-T8s,98s-97s,87s-86s,76s-75s,65s-64s,54s,AKo-ATo,KQo-KJo,QJo)

BN plays ~21% vs CO open, 3-betting ~7% and flatting the rest.  One could argue that BN plays a mixed strategy with some hands, flatting AA,AKo for example, but let's say for simplicity he 3-bets this for now:
(AA-JJ,AKo,ATo,KJo,QJo,AKs,A7s-A2s)

CO continues with ~8% vs the 3-bet
(AA-99,AKo-AQo,KQo,AKs-AJs,KQs-KJs,QJs,T9s,98s,87s,76s,65s)

Lets also say that a default player would 4-bet (KK-QQ,AKs,T9s,98s,87s,76s,65s) and flat the rest, and now we want to know the effect of moving AQo from the flatting range to the 4b/call range.

We're also assuming that BN is a good player and clairvoyant vs our range (as suggested by sauce).  After we 4-bet he makes an unexploitable shove against our range, and we make a perfect call.  And finally, let's just assume CO realizes 80% of his equity, and that the hand is checked down after flatting.

Now, with AQo in the flatting range, the EV is ~$44.

Well, there are a few options, but let's say BN shoves with ~6% (AA-JJ,AKs,A7s-A3s,AKo,KJo,QJo).  That makes the EV of 4-bet/calling AQo be $49, and thus better than calling (which now has an EV of $23).  

It's certainly possible to find a more exact margin case, but I don't know how interesting that will be.  It still seems pretty unrealistic to expect 4-bet/calling AQo to be better than just flatting.

J-

Those are very similar to the numbers I get when I analyze spots like this, and the numbers other high stakes players get.  Nice work!

This thread is a good example of a spot where we have nearly the strongest hand in a reasonable flat calling range, and nearly the weakest hand we could put in a 4b/call range.  If that's true, then 4b/fold/call3b, and 4b/call should have similar EVs.  But it looks clear that flatting is better by a fair amount.  So, our original assumption that it was close between 4b/fold, 4b/call and flat the 3b should be thrown away, and we should put AQ in our flat 3b range.  Doing this process in many spots makes our strategy stronger over time.

Why i can not arrive to same results when i put all those assumptions in Crev ? 

I did put the same ranges you stated, but this are my results.

I claim for a decent video about how to use CREV.

I adjusted all the things you told me. Now i have more accurate results. Thank you !

Have you any of you guys checked how many times a hand like AQo see a SD when called a 3b OOP ? 

if we flat the 3bet OOP with AQo , we gonna play this 3bet pot out of position and without the initiative, that lead me to believe that we will not realize our equity most of the time (unless we flop an A or Q in which case the hand will be easy to play)

So do you guys still think its best to flat call the 3bet instead of 4betting and if yes do we have to make some moves postflop on some kind of scary flop for villain's range or playing fit or fold ?  for my part, I don't think playing fit or fold on the flop will be EV+ and should be even worse than 4betting.

Why do we need to play fit or fold on flop? i dont think  its good way to think about the hand

agreed...you prob don't have to fold hardly any flops. dry with a K or super wet (especially midling card) flops, I'd be folding. J-high flops would be my borderline.

hopefully you have some sort of read to go off of postflop. and yes, playing OOP without hitting the flop is never easy.

like I said before (some disagree) I don't see anything wrong with using AQ to 4 bet bluff though if you look at a guy and think you'll just be totally lost postflop...especially if you have a stat that he's folding to 4 bets a reasonable amount. (the amount he will fold to 4 bets will go up if you use AQ since you block several of his nut combos...also if you have a low 4 bet % yourself, he'll just have to fold).