I'm wondering how profitable cold 4bet as a bluff is.


let's say HJ open 3bb, CO 3bet to 9bb and Hero (BTN) 4bet to 22bb.
if HJ and CO go all-in with AK,QQ+ and fold everything else.
Since the range is 2.6% does this means that if CO is 3betting 5% he will fold 50% of the time ?

thanks

We can write the EV of this out explicitly:

EV = F*13.5 + (1-F)*(-22) = 35.5F - 22

where F is the probability that both fold. If Fh and Fc are the folding probabilities of the HJ and CO, then F = Fh*Fc. Ignoring card removal effects, if Rh and Rc are the ranges of the HJ and CO (in percentage), then

Fh = (Rh-2.6)/Rh = 1 - 2.6/Rh
Fc = 1 - 2.6/Rc

All together:

EV = 35.5*(1 - 2.6/Rh)*(1 - 2.6/Rc) - 22


Suppose the HJ is opening 20% and the CO is 3betting 8%. Then, plugging those numbers in gives:

EV = 35.5*.87*.675 - 22 = -1.15

So in that situation you would be losing 1.15 bb on average each time you do this. If the CO is 3-betting 10% the EV is +.85 so it would be (slightly) profitable.

thanks a lot, very interesting