This hand was posted in the NL forum & I argued in favor of calling with AK vs. a range of AA/KK in this particular situation.

V1 opens for $12, V2 3! to $25 & Hero 4! to $75. He started with $225. V1 folds & V2 goes all-in. Hero has $150 behind.

I didn't get any response either way, so I wanted to confirm my thinking/math with you guys/gals.

What happens when Hero folds pre-flop? He loses $75

What happens when Hero calls the $150 vs. a range of KK/AA?
[We'll assume the $7 in rake came from the blinds & $4 of the $12 V1 o/r for. We'll also assume Hero tipped the dealer a $1, when he won or split the pot, leaving $7 of V1's money in play.]

He loses his $150 80.4442% of the time = -$120.67
[his $75 raise became part of the pot when his V went all-in, so it's not part of his investment towards competing for the pot when he calls V's all-in.

He wins $307 17.3922% of the time = $53.39
He ties 1.0818% of the time & wins 1/2 of his $75 + the $7 from V1 = $0.44
Net loss: $66.84

$66.84 is less of a loss than $75, so calling vs. a range of AA/KK is correct.
You lose $6684 long-term, vs. $7500 when you fold.

Have V going all-in 50% of the time that he has 1 of the 2 AKs & Hero's equity increases to 22.622%, however, he ties 13.88% of the time. Still:

He loses his $150.00 63.498% of the time = -$95.25
He wins $307 8.742% of the time = $26.84
He splits the $82 13.88% of the time = $5.69
Net Loss: $62.72

Have V going all-in 100% of the time with AKs & Hero's equity is 25.732%, but winning outright only 2.253% of the time.

He loses $150 50.789% of the time = $76.18
He wins $307 2.253% of the time = $6.92
He splits the $82 23.479% of the time = $9.63
Net Loss: $59.63

Red shows an inconsistency. You either count the $75 in both cases or in neither case. If you don't count it, then folding loses $0, which is better than calling and losing $67. If you count it, folding loses 75 and calling loses 75 plus another 67 for a total of 143. No matter how you slice it, calling is $67 worse than folding.

But you shouldn't need an EV calc to know this is a fold. From basic pot odds, you should know you can't call a pot-sized bet with only 20% equity.

The correct and obvious answer is never.

Except if the pot is right and their stack is small enough.

Sure, but stack size only matters here if they are the one betting, not calling. And any opponent who offers them 4:1 pot odds to call would be a very bad player.

It would be very rare for sure. But I think OP's scenario is an example, if V2's stack were smaller. Suppose V2's starting stack was 110. If it were just Hero vs V2, Hero would obviously reraise to 110 instead of 75. But if V1 has Hero covered, maybe Hero plans to fold to a V1 shove, in which case by raising to 75 he'd only lose 75 instead of 110. Once V1 folds and V2 shoves, Hero is committed to call the 35 into a pot of 162.

In the other thread, you're still confused, so I'll explain that post further.

If you're looking at it from the present, folding would lose Hero nothing. If you look at it from the start of the hand, Hero loses $75. Neither perspective is wrong, but consistency (lack of contradictions) is the key, because when comparing EV's you need to compare apples to apples.

The pot is ~160 and Hero's equity is ~20%.

The standard approach is to begin from the present, so once Hero raised 75, that 75 went into the pot and was no longer his. Therefore right now, folding is 0ev.

With that approach, E(call) = .2(310) - .8(150) = -58

But your approach works too if done correctly. You say folding = -75ev, that's fine, but then the equation for E(call) must change: E(call) = .2(235) - .8(225) = -133
(Notice that 133-75 = 58.)

You stand to win 235 instead of 310 because you said 75 is already yours.

You stand to lose 225 instead of 150 because you said 75 is yours to lose. If it's yours to lose when you fold, it's also yours to lose when you call.

You're getting 2:1 pot odds. You need 33.3% equity to call. That's all there is to it.