This is effectively a heads-up situation, which I just want to point out before we get into this. This is a vital fact when we analyze Hero's 3-bet. We are assuming that all people left to act behind Hero will fold, and this is certainly not always the case (unless Hero is BB).
Villain raises 4BB w/ 146 hands { 22+, ATs+, KQs, AJo+, KQo }
Hero raises to 12 BB w/ x hands { XX? }
If Villain does not fold, Villain will shove all-in. <- This is a major assumption.
Villain folds 96 hands @66% { 99-22,AJs-ATs,KQs,AQo-AJo,KQo } and shoves for 96 BB with 50 hands @ 34% { TT+,AQs+,AKo }.
Hero faces a bet with alpha of 44%, and calls 88BB with 28 hands { JJ+, AKs }.
How many hands does hero fold, so that he's not exploited by the shove?
***
When Villain shoves 96BB into a pot of 17.5BB, Hero wants that to be at most 0EV for Villain.
Villains has { TT+,AQs+,AKo } and bets into Hero.
Villain's EV to shove = (fold%)(17.5BB) + (1 - fold%)(something)
something = (42%)(105.5BB) - (58%)(96BB) = -11.4BB
(Using Equilab for pre-flop equities with those ranges)
substitute back
Villain's EV to shove = (fold%)(17.5BB) + (1 - fold%)(-11.4BB)
Set it less than or equal to 0 and solve for fold%. (We want to remember that right now we're looking at Villain's perspective.)
(fold%)(17.5BB) + (1 - fold%)(-11.4BB) <= 0
distribute the (1 - fold%)
(fold%)(17.5BB) + (-11.4BB) - (fold%)(-11.4BB) <= 0
clean that up
(fold%)(17.5BB) - 11.4BB + (fold%)(11.4BB) <= 0
collect terms with (fold%)
(fold%)(17.5BB + 11.4BB) - 11.4BB <= 0
move the known values to the RHS
(fold%)(17.5BB + 11.4BB) <= 11.4BB
fold% <= 11.4BB/(17.5BB + 11.4BB)
Now just solve it
fold% <= 39%
So now we know that in order to make Villain's 4-bet shove -EV, Hero needs to fold LESS than 39% of the time.
***
Hero will call with 28 hands, from the x hands in { XX? } that Hero 3-bet with.
So x = (y + 28)
where y is the number of hands Hero 3-bets, but folds to a 4-bet shove.
fold% = y/x = y/(y + 28)
and we want that less than 39%
y/(y + 28) <= 39%
y <= (39%)*(y + 28)
y <= (39%)*y + (39%)*28
y <= (39%)*y + 11
y - (39%)*y <= 11
(1 - 39%)y <= 11
y <= 11/(1 - 39%)
y <= 18.2
So Hero wants to fold less than 18.2 hands... let's use 18
Woo hooo. Don't it get you all chubby right around now?
29 + 18 = 47 hands to 3-bet for Hero.
{ 99+, AQs+ } is 44 hands.
(Open to revision... I haven't done this type of calc in over a year.)
Hero raises to 12 BB w/ x hands { XX? }
x <= 47
{ XX? } = { 99+, AQs+ }
(which is 44 hands, which is less than 47 hands, so ensuring Villain's shove is -EV, so hopefully the rake doesn't screw you both.)
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