Two Plus Two Newer Archives - EV of calling a preflop 3bet with small/medium pairs

A recent review session threw up the question - can you profitably call a standard preflop 3bet with 100BB stacks, when you hold a small/medium pair?

This may well have been covered elsewhere - or be obvious to all - but here are my calculations/conclusions. Please let me know if my math(s) is off.

NL100. Hero has 66 in MP, Villain is on the Button. 100BB stacks. Villain is a TAG who CBs 2/3 pot 100% of the time. He never steals or squeezes preflop.

Hero raises to $4 with 66
Villain on the Button re-raises to $14 (he has a big pair or AK)
blinds fold.
Hero calls $10

Is this a +EV move if Hero is playing for set value alone?

Best case scenario – Villain stacks off 100% of his range every time

To do an EV calc, we need to make a few assumptions:-

Villain bets $20 on every flop when checked to.
Hero check/folds without a set; check/calls or check/raises with a set or better.
Hero gets stacked 100% if he hits a set and loses.
When Hero’s set holds up, he wins $18 preflop + $86 post flop = $104
When Hero loses set over set, he loses $96

A set wins 82% vs. AA-JJ
A set wins 95% vs AK
A set wins 87% vs AA-JJ, AK range
(Lets assume the blinds are lost to rake.)

EV when Hero hits a set and Villain (with AA-JJ or AK) goes broke 100% = (0.87 * 104) + (0.13 * -96) = $78

Hero hits a set 11.8% of the time, so

Best case EV of call for set value alone = (0.118 * 78)+(0.882 * -10) = $0.4

Conclusion
With 100BB stacks, calling for set value alone is only +EV if Villain always stacks off with his entire range.

Another way of looking at it.

The chance of losing set over set is very important here. Hero is 7.5:1 (11.8%) to hit a set, but Hero loses with a set 13% of the time vs. an AA-JJ, AK range. You are 7.5:1 to hit a set, but (0.118*0.87 = 0.103) 8.7:1 to hit a set and win. Hero will get stacked ~1.5% of the time. It breaks down like this:

88.2% of the time Hero loses his $10 call
10.3% of the time Hero wins a big pot
1.5% of the time Hero loses his stack

In our typical HU example Hero has 96BB left before his call and the call is 10BB, Hero needs to win the following amount for the call to be 0EV

(0.882 * -10)+(0.103 * X)+(0.015 * -96)=0
X = $100

So Hero needs to win 10 times his call ON AVERAGE to make it +EV for set value alone. With 100BB stacks, he needs to win a further $82 – but Villain only has $86 left.

Hero needs to stack Villain nearly 100% of the time with his entire range, if playing solely for set value.

So what is the EV of a call here for set value alone if we make a more reasonble assumptions of how much Villain will lose with his AA-JJ, AK range?

Let’s say

AA-JJ lose whole stack every time there are no overcards by the turn (optimistic, maybe)
AA loses stack every time
KK only loses 50% of remaining stack on average (after the turn) when an Ace flops/turns
QQ loses 50% of remaining stacks when an A or K flops/turns (QQ is less worried about a K, than KK is of an Ace).
JJ loses 40% of remaining stacks on average when an overcard flops/turns (hates an Ace; K or Q not so bad)
AK loses 30% of remaining stack on average past the turn, when it gets to the turn UI
Villain CBs every flop

When we flop a set, Villain has $66 left on the turn, Hero can win a maximum of (18+20+66)=$104

Average win when Villain has:-

AA (0.82 * 104) + (0.18 * -96) = $68

KK 0.76 * [(0.82 * 104) + (0.18 * -96)] + 0.24[(0.82 * 71) + (0.18 * -96)] = $62

QQ 0.56 * [(0.82 * 104) + (0.18 * -96)] + 0.44 [(0.82 * 71) + (0.18 * -96)] = $56

JJ (0.40 * [(0.82 * 104) + (0.18 * -96)] + 0.60 [(0.82 * 64) + (0.18 * -96)] = $48

AK (0.34 * 104) + (0.66 * 61) = $76

Note:
With AA Hero (only) wins ~82% of the time he flops a set (this fact is important)
With KK there is a 24% chance of an Ace by the turn i.e A6xx board
With QQ there is a 44% chance of an Ace or King by the turn
With JJ there is a 60% chance of an Ace, King or Queen by the turn
With AK there is a 34% chance of AK hitting TP or better by the turn

This gives a weighted average win of $66 when we hit a set vs a range of AA-JJ, AK.

Hero flops a set 11.8% of the time; so

EV of call for set value alone = (0.118 * 66)+(0.882 * -10) = -$1.0

I did some algebra which indicates that Hero can only profitably call a 3bet to 12.7BB, under the above assumptions. Of course, as soon as we give Villain a wider range than AA-JJ, AK, the situation for Hero is worse, as Villain is going to pay off a smaller proportion of his stack on average with worse hands. Also, we cannot expect to take 100% of Villain’s stack every time he has JJ and no overs come. Some players will be able to get way from this vs. a TAG who calls a preflop 3bet and a big CB.

Conclusion
If you have a small/mid pair and are facing a standard sized 3bet by an aggressive Villain, you cannot profitably call for set value alone with 100BB stacks – whether he has a tight or loose 3bet range – even if he CBs 100% of the time. If you do call, you will need to find some +EV spots postflop without a set.