POKER MATH: Calculating Fold Equity

sunirico

sunirico

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We are on a holy mission in the name of the poker gawds to discover the lost and secret art of quantifying fold equity. Seeking math wizards and numerological mages who have reached enlightenment in these arts.

I am using the formula

EV = P(f) + (1-f)[EV(P+R+R) - S]

this has the same result as

EV = P(f) + (1-f)[EV(P+R) + (1-EV)(-S)]

and we are trying to find the value for f (fold equity) where EV = 0 or break even.

The following once in a lifetime scenario plays down at a poker table in some obscure and secret underground location.

BB posts 400,
SB posts 200,
pot 600,
Action folds down to BU
BU raises 800,
pot 1400,
SB raises 800 all-in,
pot 2800,
BB folds,
BU calls 800 (maybe or maybe not),
pot 3600

At some point the SB collected the following variables and proceeded to do a most simple mathematical calculation.

Pot if fold (P) = 1400 (3.5BB)
Stack if lost (S) = 1400 (3.5BB)
Raise (R) = 800 (2BB)
EV = 56.09 (66 vs K8o)

= P(f) + (1-f)[EV(P+R+R) - S]
= 1400f + (1-f)[0.5609(1400+800+800) - 1400]
= 1400f + (1-f)[0.5609(3600) - 1400]
= 1400f + (1-f)[2019.24-1400]
= 1400f + (1-f)619.24
= 1400f + 619.24 - 619.24f
f = -619.24 / (1400 - 619.24)
f = -619.24 / 780.76
f = -79%

Eureka we have found the answer, fold equity is -79%. BOOM!

Keeping his best poker face to not reveal any tells of his complete and utter confusion the SB quickly runs the calculation again, from a less than ideal spot this time.

EV = 25%

= P(f) + (1-f)[EV(P+R+R) -S]
= 1400f + (1-f)[0.25(1400+800+800) -1400]
= 1400f + (1-f)[0.25(3600) -1400]
= 1400f + (1-f)[900-1400]
= 1400f + (1-f)-500
= 1400f - 500 + 500f
f = 500 / (1400 + 500)
f = 500 / 1900
f = 26%

Snap! Dude on BU you should fold 1:4 times for us to break even, who's your daddy. But the BU called and he watches his chips get shipped away...

Now, the question of the universe and everything can anyone PLEASE tell me what on earth -79% is supposed to mean?
 
U

UncleConRon

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My Opinion

You really need to do all that. Its just all-in pre-flop and you know these hole cards should hit. Because they lost the last two times you had them. Vice-versa the hole cards just hit so fold them. Here's another angle. You know weak hole cards hit when the dealer button was on some spot. Wait for good hole cards and the dealer button in same spot. Give or take a spot for standard deviation. Remember, it hits 75 percent of time in a standard deviation with a coin flip. with 13 cards its even less of an error. Flushes and straights are a little exception to the rule. With those cards you need to see the flop. You want an open end straight or four cards to flush. Count it a third of the time. Your should hit your straight or flush a third of the time. If you have suited connectors and just hit a straight or flush best to fold them. Not even look at the flop.
 
sunirico

sunirico

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You really need to do all that.
No I don't think you need to do it but I do have it on good authoroty that the kewl kids can work this out on the fly. Crazy new world we're trying to make a living in!
Its just all-in pre-flop and you know these hole cards should hit. Because they lost the last two times you had them. Vice-versa the hole cards just hit so fold them. Here's another angle. You know weak hole cards hit when the dealer button was on some spot. Wait for good hole cards and the dealer button in same spot. Give or take a spot for standard deviation. Remember, it hits 75 percent of time in a standard deviation with a coin flip. with 13 cards its even less of an error. Flushes and straights are a little exception to the rule. With those cards you need to see the flop. You want an open end straight or four cards to flush. Count it a third of the time. Your should hit your straight or flush a third of the time. If you have suited connectors and just hit a straight or flush best to fold them. Not even look at the flop.
Wow thank you for this gold mine of information. I'm going to have to read through it again line by line there are just so many gems.

What I really want to know, perhaps you did answer that and I missed it, my apologies, what does the answer -76% fold equity mean?

Is it 4:3 or does it stay 3:4 and we just discard the minus, or do we read it as 1:4 instead? Is it even still relevant to folding or does it now favour calling?

Is there anyone that knows how to interpret this? If my EV is >50% the answer tends to be negative.
 
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beriantiger

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Overthinking

Seems like the classic case of overthinking.
 
vinnie

vinnie

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We are on a holy mission in the name of the poker gawds to discover the lost and secret art of quantifying fold equity. Seeking math wizards and numerological mages who have reached enlightenment in these arts.

I am using the formula

EV = P(f) + (1-f)[EV(P+R+R) - S]

this has the same result as

EV = P(f) + (1-f)[EV(P+R) + (1-EV)(-S)]

and we are trying to find the value for f (fold equity) where EV = 0 or break even.

The following once in a lifetime scenario plays down at a poker table in some obscure and secret underground location.

BB posts 400,
SB posts 200,
pot 600,
Action folds down to BU
BU raises 800,
pot 1400,
SB raises 800 all-in,
pot 2800,
BB folds,
BU calls 800 (maybe or maybe not),
pot 3600

At some point the SB collected the following variables and proceeded to do a most simple mathematical calculation.

Pot if fold (P) = 1400 (3.5BB)
Stack if lost (S) = 1400 (3.5BB)
Raise (R) = 800 (2BB)
EV = 56.09 (66 vs K8o)

= P(f) + (1-f)[EV(P+R+R) - S]
= 1400f + (1-f)[0.5609(1400+800+800) - 1400]
= 1400f + (1-f)[0.5609(3600) - 1400]
= 1400f + (1-f)[2019.24-1400]
= 1400f + (1-f)619.24
= 1400f + 619.24 - 619.24f
f = -619.24 / (1400 - 619.24)
f = -619.24 / 780.76
f = -79%
= P(f) + (1-f)[EV(P+R+R) - S]
= 1400f + (1-f)[0.5609(1400+800+800) - 1400]
= 1400f + (1-f)[0.5609(3000) - 1400]
= 1400f + (1-f)[1682.7-1400]
= 1400f + (1-f)282.7
= 1400f + 282.7 - 282.7f
= 1117.3f + 282.7
f = -25.3%


Eureka we have found the answer, fold equity is -79%. BOOM!

Keeping his best poker face to not reveal any tells of his complete and utter confusion the SB quickly runs the calculation again, from a less than ideal spot this time.

EV = 25%

= P(f) + (1-f)[EV(P+R+R) -S]
= 1400f + (1-f)[0.25(1400+800+800) -1400]
= 1400f + (1-f)[0.25(3000) -1400]
= 1400f + (1-f)[750-1400]
= 1400f + (1-f)-650
= 1400f - 650 + 650f
f = -650 / (-2050)
f = 31.7%
{Fixed the numbers here also}

Snap! Dude on BU you should fold 1:4 times for us to break even, who's your daddy. But the BU called and he watches his chips get shipped away...

Now, the question of the universe and everything can anyone PLEASE tell me what on earth -79% is supposed to mean?
OK, you had some math errors, but the end result is the same. You end up with a negative percentage for f with your original numbers and showdown-EV. Why is that, you ask? Well, you have to remember the original question you are asking.

You are solving: tEV = P(f) + (1-f)[EV(P+R+R) - S] when tEV = 0.

{The left side EV is total EV. I changed it to tEV make it obvious that the EV inside the equation is different. The EV in EV(P+R+R) is showdown EV, that percentage your hand has of the pot when you are forced to show it down.}

Why? Because you want to find out how often your opponent must fold for you to make money by shoving. If you opponent folds exactly that percentage, you don't make any money (but you don't lose any either). If he folds more often, you benefit from that. If he calls more often, you lose.

Now, in that light, do you see why it's negative?

You picked a hand with a positive expectation when called. You benefit when he calls. If you had your way, you would force him to never fold. When he folds, you lose equity that you had from the showdown side of things. You want him calling 100% of the time, more even (if such a thing were possible outside mathematical equations).

Compare that to the second hand, where your cards are not the favorite. In this case, if he calls 100% of the time, you will lose money on the shove. Now, you rely on him folding often enough that the money you earn from his folds compensates for the money you lose when he calls. He now needs to fold more than about 32% of the time, or else you lose money by shoving.

I hope this makes sense and helps. When your hand is a favorite (greater than 50%), you prefer the call. You don't benefit from fold equity because it reduces your showdown equity. It hurts your total EV when the other guy folds. When your hand is a dog (less than 50%), you need a certain amount of fold equity to make up for the lack of showdown equity.
 
Last edited:
vinnie

vinnie

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I made a mistake above and it's too late to edit it.

It's not true that we want him to call, even with a negative f. 100% of the 1400 that is already out there is better than 53% of the total pot at the end. But, it is true that we'll show a profit with the first hand no matter what. We don't need the person to fold to show a profit. If they call 100% of the time, we still profit. If they fold 100% of the time, we show more of a profit. It's like holding Aces pre-flop. Sure, it's great to play for stacks. But, if you could get you opponent to put in 95% of his stack and then fold for the remaining 5%, you'll make more money than showing it down.

Basically, if f is negative, we show a profit even when called 100% of the time. If f is positive, we need them to fold that much or more to show a profit with the hand.
 
sunirico

sunirico

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OK, you had some math errors, but the end result is the same.
Oops copy and paste errors, I redid these calculations over and over so many times that I started to just copy the previous equation and edit it. At some point I remember trying total pot, it must've gotten stuck there. Well spotted!!!
You end up with a negative percentage for f with your original numbers and showdown-EV. Why is that, you ask? Well, you have to remember the original question you are asking.

You are solving: tEV = P(f) + (1-f)[EV(P+R+R) - S] when tEV = 0.

{The left side EV is total EV. I changed it to tEV make it obvious that the EV inside the equation is different. The EV in EV(P+R+R) is showdown EV, that percentage your hand has of the pot when you are forced to show it down.}

Why? Because you want to find out how often your opponent must fold for you to make money by shoving. If you opponent folds exactly that percentage, you don't make any money (but you don't lose any either). If he folds more often, you benefit from that. If he calls more often, you lose.
We are attempting to find the break even point,

with EV <50% we loose money when called but win money when they fold and f is a positive number that indicates how much folding will compensate for the deficit.

is it then also correct to say that:

with EV >50% we win money when called and when they fold so f is a negative number that indicates how much not folding will compensate for the excess.

Not sure if that even makes sense....
Now, in that light, do you see why it's negative?

You picked a hand with a positive expectation when called. You benefit when he calls. If you had your way, you would force him to never fold. When he folds, you lose equity that you had from the showdown side of things. You want him calling 100% of the time, more even (if such a thing were possible outside mathematical equations).

Compare that to the second hand, where your cards are not the favorite. In this case, if he calls 100% of the time, you will lose money on the shove. Now, you rely on him folding often enough that the money you earn from his folds compensates for the money you lose when he calls. He now needs to fold more than about 32% of the time, or else you lose money by shoving.

I hope this makes sense and helps. When your hand is a favorite (greater than 50%), you prefer the call. You don't benefit from fold equity because it reduces your showdown equity. It hurts your total EV when the other guy folds. When your hand is a dog (less than 50%), you need a certain amount of fold equity to make up for the lack of showdown equity.
Agreed,
<50% we loose when called so we need them to fold enough to make up for what we loose when called,
>50% we already win when called so the result is negative and there is no folding required to make up for anything.

I made a mistake above and it's too late to edit it.

It's not true that we want him to call, even with a negative f. 100% of the 1400 that is already out there is better than 53% of the total pot at the end. But, it is true that we'll show a profit with the first hand no matter what. We don't need the person to fold to show a profit. If they call 100% of the time, we still profit. If they fold 100% of the time, we show more of a profit. It's like holding Aces pre-flop. Sure, it's great to play for stacks. But, if you could get you opponent to put in 95% of his stack and then fold for the remaining 5%, you'll make more money than showing it down.

Sure, if everyone folds we win the pot 100% of the time every time, no variance involved.

I think a light may just have gone on.... you said if they call we still show profit but when they fold we show more profit. From this perspective "not folding" to reduce excess does start to make sense.

Basically, if f is negative, we show a profit even when called 100% of the time. If f is positive, we need them to fold that much or more to show a profit with the hand.
This is the part I was struggling with, not the positive result which made sense, if f is 25% then we need them to fold 1:4 to break even, it was the negative results that didn't make sense. I could see that because of the strong equity we don't lose anymore when called so obviously the resultant f value could not still suggest folding more. If we are already winning why would we need fold equity!

Would I be making a leap if I assume what you are saying is that we only equate positive results to fold equity and should consider any negative results to be 0%, no folding required?

Thank you so much vinnie for taking the time to discuss this with such clarity and detail. I am still perplexed as to what the negative value actually means but am much more confident with the results thanks to you.

You rox!!!
 
vinnie

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I totally messed things up with my first post by focusing on 50%. We could have a positive expectation even if we're not a favorite to win. If the pot is large enough. This happens when we have a hand that is 40% to win, but we're getting 2-1 on our bet. So, we can ignore the majority of that that mess. It was 2am and I was tired.

It would be more correct to day that if "[ EV(P + R + R) - S ]" is positive, we don't need the other player to fold to show a profit getting the money in.

An example, say we have 40% chance to win a pot of $4000, with a raise of $1000 and a stack of $2000.

[40% * (4000+1000+1000) - 2000]
[ 0.4 (6000) - 2000]
[ 2400 - 2000]
[ 400 ]

So, even though we aren't a favorite to win the hand, we get enough money from the pot that we don't need a fold to show a profit.

Would I be making a leap if I assume what you are saying is that we only equate positive results to fold equity and should consider any negative results to be 0%, no folding required?

Yes. This would be correct.

For the first example, assume they never fold.

tEV = 1400 * 0 + (1 - 0) ($282.70)
tEV = $282.70

In this case, we have a positive expectation even if we never get a fold. We don't show a negative expectation until they fold less than 0% of the time, which is impossible in the real world.

In the second example, if they never fold:

tEV = 1400 * 0 + (1 - 0) ( -$650)
tEV = -$650

When they never fold, we lose money. In this case, we need them to fold a certain amount to show a profit.
 
sunirico

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We don't show a negative expectation until they fold less than 0% of the time, which is impossible in the real world.

It may sound comical when you put it that way but actually this is all I needed to realize.

Your work here is done! Thanks again!!!

Luv the sig, makes me smile every time...
 
sunirico

sunirico

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Just for completion's sake and to conclude:

We calculated the fold equity required for a hand with 25% EV when raising all-in by 800 into a 1400 pot risking a 1400 stack to break even (tEV = 0) at 31.7%.

Pot (P) = 1400
Stack (S) = 1400
Raise (R) = 800
EV = 25%

tEV = P(f) + (1-f)[EV(P+R+R) - S]
tEV = 1400f + (1-f)[0.25(1400+800+800) - 1400]
tEV = 1400f + (1-f)[0.25(3000) - 1400]
tEV = 1400f + (1-f)[750 - 1400]
tEV = 1400f + (1-f)-650
tEV = 1400f - 650 + 650f
650 = 1400f + 650f - 0
650 = 2050f
f = -650 / -2050
f = 31.7%

if they never fold (fold equity = 0%):

tEV = 1400 * 0 + (1 - 0) ( -$650)
tEV = -$650

When they never fold, we lose money. In this case, we need them to fold a certain amount to show a profit.

If they fold at least 32% of the time we make $6.

tEV = 1400 * .32 + (1 - .32) ( -650)
tEV = 448 - 650 + 208
tEV = $6
 
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if

if they fold

they dont fold - 1 they have a hand
2 they think you are bluffing
3 they hope to catch and they do
 
vinnie

vinnie

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ARISE DEAD THREAD! COME BACK TO US!!!


Man, I remember this thread. A good lesson on why I should not go and try performing math and making logical statements after 2am. I really screwed the pooch on this one. It's a good reminder for those who automatically think my math is always right. I do make mistakes.

Not exactly sure why those 3 points matter. The times they don't fold are all accounted for with the equity formulas for the times they call. Actually, the second two points mean we probably have more equity than we originally assumed, because we base our pot-equity on the odds of improving to the best hand with the assumption that we don't currently have it. When we already have the best hand, it's even better for us.
 
sunirico

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ARISE DEAD THREAD! COME BACK TO US!!!

IT"S ALIVE!!!!

Man, I remember this thread. A good lesson on why I should not go and try performing math and making logical statements after 2am. I really screwed the pooch on this one. It's a good reminder for those who automatically think my math is always right. I do make mistakes.

Even after 2 am you've been more help than most! Tx again...
 
thetick33

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would love to figure this out will study a bit more
 
Vilgeoforc

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Fold equity is the probability. And the probability can't be greater than 1 and less than zero. Can be a negative expectation in BB or $, but not in percentage.
 
darthdimsky

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Thank you so much! :D

Was scouring the net for the math behind fold EV when all along it was right under my nose. Will take a while to digest and, hopefully, apply to practice.
 
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