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How do I calculate "I break even if Villain folds this percentage of shoving", including Fold Equity, in heads up play?

I know several factors are involved

*Pot Size before I shove

*How much do I have to call?

*How much am I shoving total?

*Estimated percent equity when called? (0-100)

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This is not hard to calculate once you understand what it means. In poker since you are not expected to be mathematicians, we need to boil the math down to little piece.

First you calculate your average EV in the hand (no folds) as a baseline, then you figure out how much you gain when you opponent folds, then you can calculate what the percentage fold needs to be in order to bring your baseline to zero (break even).

Let's look at an hand and evaluate it:

Let's say you have an equity of 10% in the hand (on the river, your winrate is 1 out of 10). Normally, if the hand is played until the end, then your net win (loss) is:

(p+2b) * 10% - b

If your push is half-pot, pot = $200, your push is $100, then:

(200+2*100) * 10% - 100 = -60

So on average, each hand is -60 for you. Now if you make him fold, that's 60 bucks saved + pot ($200), so you need this percentage of folds, 60/260 = 23%.

Let's sanity check the math:

Fold EQ + Win EQ - Investment

[$300 * 23%] + [$400 * 10% * 77%] - [$100] = -$0.2 (basically broke even)

=========---========

We can confirm that mental process results in correct math, so now it's time to get used to that mental process:

1) Calculate your current total equity in dollars. Let's say your equity in the pot is 25%, then you expect to win 25% * final pot (current pot + 2 bets) minus your investment of one bet.

So let's say you are betting $600, you expect final pot to be $1600, then you are getting $400 ($1600 * 25%), risking $600, your EV per hand is -$200.

2) Calculate your win amount in dollars if he folds. If you get him to fold 100%, you save $200 and you win the current pot (should be $400 if it's heads-up). That's a $600 win if he folds 100% of the time.

3) Calculate the percentage of fold you need. Since you only lose $200, so you need him to fold 1/3 of the time ($200 / $600).

Again we can confirm the math:

Fold EQ + Win EQ - Investment

[$1000 * 1/3] + [$1600 * 25% * 2/3] - [$600] = $0

=========---========

We can even do very complex fold equity calculations with the same process:

Let's say your question is:

"The pot is $256, my bet is $340, 3 players, one of them only have $130 left... How much fold equity do I need? I can win at showdown 10% at best."

1) Calculate your current total equity in dollars. ($256 + $340 * 2 + $130) * 10% - $340 = $ -233.4

2) Calculate your win amount in dollars if he folds: $233.4 + $256 = $489.4

3) Calculate the percentage of fold you need. 233.4 / 489.4 = 0.477

Again we can confirm the math:

Fold EQ + Win EQ - Investment

[($256+$340) * 0.477] + [($256 + $340 * 2 + $130) * 10% * 0.523] - [$340] = $0.0438

=========---=========

I translated all that in purely mathematical terms and come up with this equation, the f come from step 3, which is basically step 1 / step 2. I flipped step 1 to b - e(2b+p) to get a positive number and then divided it by the equation in step 2. This equation isn't derived from pre-existing equations, I took the logic behind the math and wrote it into an equation. (this equation is modified to work for 2 players only):

f = (b - e(2b+p)) / (b - e(2b+p) + p)

f = the fold rate you seek

b = your push bet

p = current pot

e = expected winrate at showdown

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  • 1
    I would like a proof of why this is correct but it seems to work.
    – paparazzo
    Oct 5 '18 at 17:45
  • We could probably break it down in a purely mathematical sense, let me see if I can boil it down. Right now, I only know that logically it works by following the train of thought.
    – Ying Li
    Oct 5 '18 at 18:09
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    I boiled it down to this "f = (b - e(2b+p)) / (b - e(2b+p) + p)"
    – Ying Li
    Oct 6 '18 at 16:37
  • Well, the f come from step 3, which is basically step 1 / step 2. I flipped step 1 to b - e(2b+p) to get a positive number and then divided it by the equation in step 2. The equation isn't boiled down from pre-existing equations, I basically just took the logic behind the math and wrote it into an equation. First you calculate your average EV in the hand (no folds) as a baseline, then you figure out how much you gain when you opponent folds, then you can calculate what the percentage fold needs to be in order to bring your baseline to zero (break even).
    – Ying Li
    Oct 6 '18 at 17:32
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b is bet
p is pot
f is fraction fold
e is equity

EV = -b + f(b + p) + (1-f)e(2b + p) = 0

Ying solved it first but this is how to break it down

-b + f(b + p) + (1-f)e(2b + p)  
-b + fb + fp + e(2b + p - f2b - fp) 
 fb + fp + e(- f2b – fp) + e(2b + p) - b  
 fb + fp + f(- e2b – ep) + e(2b + p) - b  
 f(b + p + - e2b – ep) + e(2b + p) - b  
 f(b + p + - e2b – ep) = b - e(2b + p)   
f = (b - e(2b + p)) / (b + p + - e(2b + p))
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  • The math is correct. My mental process uses the same money and I can do it in the heat of the game at river. I don't think I can use that equation at the same speed.
    – Ying Li
    Oct 5 '18 at 14:05
  • @YingLi But you do use this equation
    – paparazzo
    Oct 5 '18 at 14:24
  • No, but I looked at it and it looks correct. I plugged in some numbers too and it produces correct results. I don't think we can easily isolate the f in your equation, thus it's impossible to use in real life situation, so I agree with the "can't be evaluated to closed form" aspect too.
    – Ying Li
    Oct 5 '18 at 14:26
  • My previous comment is actually: "My mental process uses the same LOGIC" not "money", very strange typo :P
    – Ying Li
    Oct 5 '18 at 14:28
  • What's going nowhere? I am just agreeing with your assessment. I said that your math looks sound, I just can't seem to isolate the f and use it in a poker game. Which you agreed in your answer as well. I think this cannot be tackle with one equation, but rather a series of mental process breaking down the elements.
    – Ying Li
    Oct 5 '18 at 14:31

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