Fold equity formula correct?

Your formula is basically correct. Regarding the negative values depending on where you're getting them, you may just need to use the absolute values.

Another reason you might get negative numbers is because the EV of actually getting called is positive. In that case, of course we don't need any fold equity to make a profitable bet

Here's how I worked it out as what I came up with looks just a little different, working off a few assumptions:
- This pretty much only applies in case we shove, or;
- We bet where villain will only ever fold or call (never raise) and there will be no further action afterwards, and;
- In case we're not first to act, villain checks to us (though this is relatively easy to work around), and;
- We never split the pot.

The comparison we make then, is:

rFE > |EV if called| / (|EV if called| + Pot)

Where:

rFE = Required fold equity
Pot = Pot size before our bet
|EV if called| = Equity * Total pot - Our bet

So I suppose the entire formula (inequality) could be written as:

rFE > (|E * (Pot + Bet * 2) - Bet|) / ((|E * (Pot + Bet * 2) - Bet|) + Pot)

Where:

rFE = Required fold equity
E = Our equity if called
Pot = Pot before our bet
Bet = Our bet size

So if the pot is 100 and we shove 50 and have 10% equity when called, we get:

EV if called = .1 * 200 - 50 = 20 - 50 = -30

Therefore

rFE > |-30| / (|-30| + 100)
rFE > 30 / 130
rFE > 0.2308
rFE > 23.08%

So we have a positive EV if villain folds more than 23.08% of the time. As proof:

.2308 * 100 + .7692 * -30 = 0

Your formula gives the exact same result so it's correct

Another example:

Pot = 139
Bet = 167
E = .2184

EV if called = .2184 * 473 - 167 = -63.6968

Therefore:

rFE > 63.6968 / (63.6968 + 139)
rFE > 63.6968 / 202.6968
rFE > 0.3142

Proof:

.3142* 139 + .6858 * -63.6968 = 0

Again your formula gives the same results so it definitely works.

Very interesting to look at and see how we both came to slightly different equations which still give the same results.

Hey,

thanks for your reply! I'm confused now - I should not be doing math so late at night...
I just noticed that my math was wrong; I used E*(pot+bet) - (1-E)*bet as the EV in case of a call. It should be, as you have it E*pot - (1-E)*bet = E*(pot+bet) - bet.

How can we then get the same result? This makes no sense ;). I now adjusted my formula and now I get a different result for your values...

EV = p(f)*pot + (1-p(f))*(e*(pot+bet)-bet)
= p(f)*(pot+bet-e*(pot+bet)) + e*(pot+bet) - bet > 0
= p(f) > (bet - e*(pot+bet))/(pot+bet-e*(pot+bet))

For the case with pot=100, bet=50, equity=0.10, I get the slightly different result of 0.2592.

In case of the other example my result is 0.4188, which is a bigger difference ;).

So, either I'm mistaken about this or we both made different mistakes that ended up yielding the same results...
What exactly is your rationale for taking the absolute value of the EV in case of call?
And for the formula:

rFE > |EV if called| / (|EV if called| + Pot)

?

I get sort of degenerate results if for example my equity is close to one (I would expect the result to be close to zero) - large negative values. This is however due to the fact that we are not bounding FE in any way - if we get FE > -5, well,. FE will never take the value -5, in this case it means that any FE will yield a positive EV. Just like FE > 2 means that no fold equity warrants a bet.

The other, related problem is that we are only looking for +EV here, which can be a bit misleading, since what we would really like to do is to find the optimal bet size (we'd have to set the derivative of the EV equation with respect to bet to zero). But since FE is an unknown function of bet, this is not really easily computable.

Interesting to think about, anyway

Not sure how we got the same results, just plugged the numbers into your formula and it gave the same answers

What exactly is your rationale for taking the absolute value of the EV in case of call?
And for the formula:

I'm using absolute values because I'm comparing to a baseline of 0 EV, similar to folding when we don't bet. Of course this is inherently flawed because our EV won't be 0 all the time but it's an assumption we can make to simplify the case a bit. So for simplicity's sake we assume that we either:
- Bet and get villain to just fold or call
- Check and fold if villain bets or always lose if villain checks back (0 EV)

Therefore, for this formula to have any use, the EV of being called has to be less than 0. After all, if our EV if we get called is greater than 0 we won't even need fold equity to make a profitable bet (it only helps make the bet even more profitable). So let's call these A and B:

EV(A) = EV when we get called
EV(B) = EV when we don't bet (= 0)

Basically, we're using the difference between EV(A) and EV(B) to help calculate the fold equity we actually need. If EV(A) is -50 and EV(B) is 0, the difference isn't -50, it's 50. That's why I used the absolute value.

In this case our baseline happens to be 0 EV so we can simply take the absolute value of the EV of getting called, since we always compare it to 0. But there are also cases where we might have a baseline of say -20 EV (so even a -10 EV play would be better than our baseline play). If our EV then was -50, we'd use the difference between -20 and -50, being 30.

However in this relatively simple scenario that we're looking at now, that will never happen. Our baseline is always 0 EV.

The other, related problem is that we are only looking for +EV here, which can be a bit misleading, since what we would really like to do is to find the optimal bet size (we'd have to set the derivative of the EV equation with respect to bet to zero). But since FE is an unknown function of bet, this is not really easily computable.

Yea you're absolutely right, it's very limited. Let's not forget poker is a game of incomplete information though. If we knew all folding frequencies, specific ranges and combo counts and whatnot, or if everyone played perfect GTO we could calculate things like the perfect bet size.

Unfortunately (from a mathematical point of view anyway) none of this is the case so the best we can do is make approximations like this.

Very interesting stuff indeed though

Originally posted by TJtheTJ
I'm using absolute values because I'm comparing to a baseline of 0 EV, similar to folding when we don't bet. Of course this is inherently flawed because our EV won't be 0 all the time but it's an assumption we can make to simplify the case a bit. So for simplicity's sake we assume that we either:
- Bet and get villain to just fold or call
- Check and fold if villain bets or always lose if villain checks back (0 EV)

Therefore, for this formula to have any use, the EV of being called has to be less than 0. After all, if our EV if we get called is greater than 0 we won't even need fold equity to make a profitable bet (it only helps make the bet even more profitable). So let's call these A and B:

EV(A) = EV when we get called
EV(B) = EV when we don't bet (= 0)

Basically, we're using the difference between EV(A) and EV(B) to help calculate the fold equity we actually need. If EV(A) is -50 and EV(B) is 0, the difference isn't -50, it's 50. That's why I used the absolute value.

I still don't quite understand how you get to your formula from here. In my book the total EV is

EV = p(f)*pot + EV_call

like you said.
But here you use

EV_if_called = Equity * Total pot - Our bet

when it should be

EV_call = (1-p(f))(Equity * Total pot - Our bet)

since 1-p(f) is the probability of the call occurring.

Thus, in your terms:
EV = p(f)*pot + (1-p(f))*EV_if_called
= p(f)*pot + EV_if_called - p(f)*EV_if_called
= p(f)*(pot - EV_if_called) + EV_if_called > 0
=> p(f) > -EV_if_called/(pot-EV_if_called) = EV_if_called/(EV_if_called-pot)

but you seem to have a plus in front of the pot.

If EV_if_called is positive, p(f) > 1, because we want our opponent NOT to fold. Sensibly p(f) will only be smaller than one if EV_if_called is negative, since only then we want fold equity. Negative values also make sense - in that case you simply always want to make the bet.

I don't have an intuition on what taking the absolute values of EV_if_called means here, but with the formula above a large pot still could turn the whole thing negative.

But I might be misunderstanding you whole approach here ;).

As for the other thing: You are absolutely right, yet it bugs me when I watch people play and they just say "here I'm getting/not getting enough fold equity to call...", but they never elaborate on how much fold equity they actually need. I think there is some improvement to be made here...not for doing there calculations live but to get a better intuition on the factors and proportions when playing. For instance, we could approximate FE as a function of the bet size somehow and then set the dEV/dbet = 0. This would give us the optimal bet size AND the approximated fold equity, but we of course would need to find an appropriate FE(bet). Since it would also have to be texture and opponent-dependent that might prove diffcult, but not impossible (you may adjust for this with a texture/agression-coefficient ).
As I said, I'm less interested in an exact solution than shedding light on the dynamics of fold equity.