we assume that villain is opening 50% of his hands:
22+,A2s+,K2s+,Q2s+,J4s+,T6s+,96s+,86s+,76s,65s,A2o +,K5o+,Q7o+,J7o+,T8o+,98o thats 666 hands
now if he calls a shove with A7s lets say that this is the bottom of his range, so his calling range is:
55+,A7s+,KQs,A8o+,KQo 176 combinations
So he calls with 176 hands out of 666 that makes 26% and he folds 74%
QJo has 35.94% equity against his calling range
So the expected value of my shove :
EV(shove)= 2.5bbx0.74+0.26x(0.3594x29.67bb-0.6406x14.33bb)=
=1.85+0.26x(10.66-9.17)=2.2bb
to find the equity where we are break even we do the equation to zero it gives, that at 16,4% equity
we have the break even point, so 23o has 26,76% equity againt his calling range so its plus EV to shove any2, interesting
hmm
but we dont look for plus EV , we look for maximal value of every hand, so its just the question if you can play your hand
better postflop, or do you have an edge?
If he sees that we push every hand he widens his calling range, so we assume:
22+,A4s+,K9s+,Q9s+,JTs,A5o+,KTo+,QTo+,JTo QJo has 41.6% equity
thats 330 cards which is slightly 50%
so the EV now:
EV(shove)=2.5bbx0.50+0.50(0.416x29.67bb-0.584x14.33)=1.25+0.50(12.3-8.37)=3.215