Only about a quarter of the field is gone so we aren't taking any ICM considerations into account. This is a live tournament with a fairly soft field, so I believe keeping a stack that leaves us room to maneuver has a bit of value.

At first glance I think the price he's giving us makes this an easy call against anyone 3betting more than JJ+,AK. As the villain adds more bluffs, at some point with a hand like AQ shoving will become better than calling.

The first thing we can do is figure out when shoving becomes better than folding for cEV:

We're risking 33,000 to win 8,600, and against a suspected calling range of JJ+,AK we have 28% equity.

.28 * 72100 (total pot when called) = 20,188. 33,000-20,188 = 12,812 which is the amount we lose when called.

12812 / (12812+8600) = 0.6. we need him to fold to a shove 60% of the time to break even in chips.

Taking into account blockers, his calling range consists of AK (12) JJ-AA (18) = 30 combos. So he needs to have at least 30 / .4 - 30 = 45 bluff combos.

Here's a reasonable bluff range with 45 combos that takes blockers into account - A2s-A8s (21), KQo (12), KJo (12). At that point the cEV of shoving = cEV of folding = 0.

Next, we need to think about the EV of calling and compare...

Against villain's hypothetical range of AA-JJ,AK,A8s-A2s,KQo,KJo, our hand (AQo) has 52% equity.

He made his raise very small so we only have to call 2,500 to play a pot of 11,100, meaning we need to take 23% of that pot to make calling better than folding.

23/52 = .44 = the amount of equity we need to retain.

this is the "r" concept from Lefort's videos if that helps. r changes with position and each player's range composition, but it's pretty hard not to get to 44%. I won't go into detail here but my estimate for the r we can manage is 65% with AQo.

.65.52 = .34 of the pot. .3411100 = 3,774. subtract our investment: 3,774-2,500 = 1,274, the amount we profit with a call.

In the initial description we said the villain may have been mindlessly aggressive, so let's widen his 3bet/folding range to include AQ (9),KTo(12),QJo(9). now he has 45+9+12+9 = 76 bluff combos

he's 3b/folding 72% of the time now. cEV of shoving:

.728600 - .2812812 = 6192 - 3587 = +2,605

cEV of calling with r = .65 with our equity now at 55% instead of 52%:

.65.55 = .36. .3611100 = 3996. 3996 - 2500 = +1,496.

I do think our r gets higher the more bluffs he adds to his range. what r do we need to achieve to make calling better than shoving? now we work backwards...

x - 2500 = 2605 -> x = 5105
y11100 = 5105 -> y = .46
.46 = .55r -> r = .83

optimistic, but not outrageous especially if we think our opponent is a weak player. also remember that since we're in a soft field our tournament life and maneuvering room are worth something. I think even against this new wide range I prefer a call. if he starts adding more hands like K7s and A9o to his 3betting range, a shove becomes too good to pass up.

anyway, I hope this post/book I just wrote was helpful to some and I hope some others will chime in on the spot.

Standard for me is to call. I recommend this PG video on the topic.

I'll re-read and think about this some more on my flight today, but I don't think there is really ever a spot where 4b shoving AQos 44BB deep is going to be the best play. We are plenty deep enough to have post flop play, even being OOP.

It's horrible when we get called, because who really wants to be flipping for 44BB? If he is adding hands like A9o to his 3b range, then a call should be quite a bit more profitable, as we crush those hands. We don't want to fold them out by shoving.

Not sure folding is ever really an option either. Just makes no sense to raise fold AQ here.

Too deep, agree with flatting. Raising puts us in a situation where we fold out worse and get called by better. 40BB is heaps in live WPT's. No need to panic or spazz.

Really not sure what you did with that first calc but something is way off or i`m confused from the wall of text and not reading it correctly.

To calculate the EV of a shove you need to take into account the following :
- size of the pot before a shove
- times villain calls or folds
- times what you win when he calls or folds

EV = F($Pot) + C(%W$W) – C(%L$L)
“F” stands for “times villain folds” and “C” stands for “times villain calls”.
$Pot = the size of the pot BEFORE you shove
$W = what you would win the times you get called and win
$L = what you would lose the times you get called and villain wins

There`s 72 combos in his 3b range from your assessment, and he calls with JJ+, AK which is 30 combos or 41.6%.

EV = .58 (7800) + .42 (.28 * 71300) - .42 (.72 * 35000)

However, i think the range you gave him is too tight and he also might just be playing 3b/f depending on the SB+BB tendencies and their stacks.

For the sake of argument, let`s say they never squeeze and he does have a flatting range of T9s-ATs, 55-TT, QJs-KQs, AJs, AQs, AJo for example, he still can be 3betting all of the A9s-A2s,K9s-K8s,K5s-K3s,Q9s-Q8s,J9s-J8s,T8s+,97s+,86s+,75s+,65s,ATo-A2o,KTo,QTo+,JTo maybe wider if you are perceived as a nit, which means your FE is high to a point you can almost jam ATC.

That being said, with these stacks and if i thought i had a solid edge over the field i would always flat, but shoving can certainly never be bad vs this guy.

when he folds we take down 8600 (our 2000 raise, his 4500 reraise, 1200 in the blinds, and 900 in antes). total pot when called is 72100 (35000+35000+1200+900), but for $W you need to subtract your investment because we want our net win on the hand, not the gross # in the pot: 72100-33000 = 39100.

and finally, in the third term of your EV calc you should use 33000 instead of 35000 for $L because our 2000 raise is already part of the pot.

so I have...

EV = .58(8600) + .42(.2839100) - .42(.72*33000) = 4988 + 4598 - 9979 = -392.84

if you change .58 to .6 I believe it comes out closer to 0. I think I got it right, no?

using your equation we only need him to fold ~15% which definitely isn't right.

EV = .58(8600) + .42(.2839100) - .42(.72*33000) = 4988 + 4598 - 9979 =
-392.84

When he calls and you win, you win 35000+35000+2100.

total pot when called is 72100 (35000+35000+1200+900), but for $W you
need to subtract your investment because we want our net win on the
hand, not the gross # in the pot.

No we dont subtract our investment, its the same as when you calculate the EV of a river call vs range.I.E, if there`s $20 in the pot and villain bets $20 on the river, we need 33% or 2:1, 20/(20+20+20).

you are wrong. when we shove and win we end up +39100, not +72100. here's a CREV tree of the hand. under the UTG+1 raise allin it gives EV of -276.22 (basically 0 when we're talking about a 70k pot).

I don't know if that image is big enough...here's a bigger one: http://i1176.photobucket.com/albums/x327/nuggetz87/runitonceAQhand_zpsba90079c.jpg

Yeah we win 1x effective stack + blinds and antes, i just now saw i wrote 2x stack win and 1x stack loss.
Not doing any math when tired from now on, sorry for confusing you :).

Great description OP. in spite of the equation discussion about EV of jam, I think we all agree call is best. Now what's next? :)