What is the formula to find the "needed equity" for a given pot odds, folding frequency, and desired EV (>0)?

For instance, lets say game is 1/2. and we are in the small blind in a heads up match. We have 12 bb left ($24) and decide it is a push or fold situation. We know opponent will call with top 40% of his hands and fold the rest.

What is the mathematic formula to know how much equity do I need to make this push at least break even?

EV requires 4 numbers

a) how often do you win
b) how much do you win, when you win
c) how often do you lose (this is *usually) 100% - a
d) how much do you lose, when you lose.

EV = a*b - c*d

This is the simplest form of EV - it occasionally gets a little more complicated but this is the principle. In this case, we need to divide it into 2 categories, when he calls and when he does not. Let's call these EV(fold) and EV(call). If he folds you will win EV(fold), if he doesn't, you will win EV(call)

EV(fold) = 1bb
that's the only part of EV(fold) because there is no probablistic component to it, you just collect the pot.

If he calls, you will win with frequency E and lose with (1-E). E is what you're trying to find, your minimum equity. When you win, you will win 12bb, and when you lose, you will lose 12BB.
EV(call) = E*12 - (1-E)*12 = 12E - 12 + 12E = 24E - 12

So, overall EV is
EV = .4*EV(call) + .6*EV(fold)
The .4 and .6 are because 40% of the time he call and 60% he folds.

EV = .4*(24E - 12) + .6*1
EV = 9.6E - 4.8 + .6*1
EV = 9.6E - 4.2

OK. So, to find the break even point, set EV=0, because you break even when your EV is 0.
0 = 9.6E - 4.2
4.2 = 9.6E
E = 4.2/9.6 = .4375 = 43.75%

That is, provided I didn't make a mistake. Let's check.

EV = .6*1 + .4*(.4375*12 - .5625*12) = 0

So yeah, if your equity is 43.75% or greater, your EV>=0

Op did not mention pot amount but there had to be one. In that case,

EV(call) = E*(Pot + 12)- (1-E)*12 = (Pot+12)E - 12 + 12E =(Pot+ 24)E – 12

Therefore the 43.75% is an upper bound assuming Pot>0.

Thank you.

Where did 9.6E-4.8 come from?

Yes, this was a preflop HU situation, there where the blinds in the pot previous to the action. Would like to see your way as well.

From the way it was phrased it sounded like a preflop situation to me. Although actually I messed up and OP can win 1.5bb not 1bb.

0.4 * (24E - 12)

And also I misread that it was 12bb before the hand started, which is wrong. So yeah I made a few mistakes in there.

So to try to correct we actually have
EV(fold) = 1.5

and
EV(call) = E*(12+1.5) - (1-E)*12
because you win 12 plus the pot (1.5bb) when you win
EV(call) = 13.5E - 12 + 12E = 25.5E - 12

So then we're solving
0 = 0.6*1.5 + 0.4*(25.5E - 12)
0 = .9 + 10.2E - 4.8
3.9 = 10.2E
E = 3.9/10.2 = 38.2%

Awesome. This was the missing link in my poker understanding, I believe. This seems like a very critical and overlooked step in range construction. Because if we don't know how to do this, how can we construct ranges for, say, push fold game preflop, without relying in charts? Am I correct?

I arrived to a strange situation.

Just for clarity, lets make this a game of 0.5/1 heads up "push fold" no limit holdem.

SB, first to act. He knows BB will call with any two cards, so he runs the above equation and realizes he needs 47.5% equity againt SB's range to break even. Only a range that suppresses top preflop hands would get less than 50% equity against an any two cards range. If SB pushed with bottom 87%, he would get EV = 0. If he pushed with any two cards (a. K. A. 100% of the time) he would get 50% equity against BB range for an EV of 0.5.

In turn, BB knows SB will push any two cards. Pot odds dictate to BB that he needs 41% of equity as a minimum to call. Again, to reach such low equity against an any two cards range would require to remove top hands. BB's range to get EV = 0 is bottom 60%. If we call with any two we obtain an equity of 50% and an EV of 1.

Is this reasoning correct?

Yeah, your reasoning is correct. If BB will always call your push, that is. However, this doesn't actually mean you'll make money.

The EV you're calculating here is basically compared to folding, which is 0EV. However, when you fold, you really are -0.5bb for the hand. This isn't included in the calculation because it's money you already put into the pot and can not get back without winning. So yeah, if you push every hand and BB calls, it should be obvious that you're not going to increase your stack. It's not a coincidence that your EV is +0.5 when pushing every hand, since this basically just brings you to even after posting your blind.

I get this part. However, what I don't understand is, how is it possible that pushing a range without top hands (this is, removing AA, KK, etc) can be more correct than pushing all hands. This sort of doesn't make sense. Or is it the case that i must push a range where every individual hand meets the minimum equity criteria instead of a range that meets the minimum equity as a whole? (in the latter case, i would have to push 23o even when called 100% of the time, which wouldn't make sense)

It isn't?

Bottom 87% = EV=0
All cards = EV=0.5

0.5 > 0

Can you get an EV>0.5? Sure. Let's say you pushed with equity at 60%.
EV = 13.5*.6 - 12*.4 = 3.3

(And actually I'm trusting you that bottom 87% gives you EV=0, I didn't check myself)

But also, yes, you push hands that are greater than the minimum required equity. It's your hand vs his range, not your range vs his range, because you can see your own cards.