raising with draws - how much FE needed?
 

I'm sure that many of the more mathematically inclined posters might have
worked this out for themselves, but I thought that some people might
find the following useful. If you don't like math, you probably
don't want to read this. If you are a math geek (like me), you might
check my calculations to see if I am right. I worked this out while
I was analyzing a play that I had made in a live game last night and
I came up with a nice little formula that people might find helpful.
The basic situation is the following.

In NL holdem you have a 8/9 out draw on the flop and your single
opponent bets into you. How much fold equity do you need to make it
correct to reraise all in?

Say that

P = pot on the flop

S = effective stack

B = opponents bet


Now E(fold) = S. That is, the expected value of your stack given
that you fold is S. We want to compare this with E(all in) - the
expected value of your stack if you raise all in. Say that your
opponent will fold x of the time and call 1-x of the time (0<= x <=
1). We want to know how big x must be in order to make raising all
in at least as good as folding.

I will assume that if he calls, you have a 1 in 3 chance of winning
the hand. This is approximately true for many common draws (e.g
flush draw, oesd with a backdoor flush draw)

Then

E(all in) = x(P+B+S)+1/3(1-x)(2S+P)

since if opponent folds, your stack will be P+B+S and if he calls,
your stack will on average be 1/3(2S+P). Therefore raising all in is
better than folding if

x(P+B+S)+1/3(1-x)(2S+P) > S

Rearranging this yields

x>(S-P)/(S+2P+3B)

The nice thing about this is that this formula is just about useable at the table (there is only one division involved - my real time arithmetic skills struggle to cope with most formulae that involve lots of divisions).

As an example, here is the hand that prompted me to work this out.

I was at a final table of the local cardroom tourney and I picked up
Th9h in the big blind. There was 12K in the pot and the flop was
QhJs5c. I had 29K behind and I checked to my opponent who bet 7K
into the pot (he had me covered). So S=29, P = 12 and B=7. Therefore
(S-P)/(S+2P+3B) = 17/74 which is about 0.23.

Thus I needed him to fold at least 1/4 of the time to make the all
in raise better than folding (at least in terms of tournament
chips). Given my knowledge of this particular opponent, I think that
he won't fold 1/4 of his range here so I should have folded. In the
actual event, I reraised all in and busted from the tourney when he
called with QT.

This formula only works if you have close to a 1 in 3 chance of
winning when called, but it would not be hard to modify to other
drawing situations as well. Any comments?

Re: raising with draws - how much FE needed?
 

Yup I just checked it over and the formula is correct .

It's a neat way to calculate the percentage of times one needs to fold for a shove to be positive EV .

Here is the formula if you have 6 outs , or about a 25% chance of winning .

x(P+B+S)+1/4(1-x)(2S+P) > S
4x(P+B+S) + 2S+P -x(2S+P)>4S
x[2S+3P+4B] >2S-P
x>(2S-P)/(2S+3P+4B)

Re: raising with draws - how much FE needed?
 

If you only have a 25% edge, then you need your opponent to fold about 37.7% of the time which is a big jump from 22.97% from the numbers given in the example .

The formula for 5 outs or equivalently about 20% is
x> (3S-P)/(3S+4P+5B)

Again using the numbers provided , you have to assure yourself that your opponent will fold around 44.1% for the play to be profitable .

Re: raising with draws - how much FE needed?
 

BTW Jay, you can you these formula to help evaluate semibluff lines even when your semibluff is not an all in raise. If you have position on your opponent and you are sure that he will check the to you on the turn if he calls, then these formulae still provide a lower bound for the fold equity that you require. You just have to replace S (stack size) with the size of your raise. This is just a lower bound as it doesn't take into account any extra money that you might make when you make your draw.

Re: raising with draws - how much FE needed?
 

O yeah, one other thing, since stack-to-pot ratios are all the rage these days. In the original situation (8/9 draw), say that your opponent makes a pot sized bet on the flop. Then the formula simplifies to

(spr-1)/(spr+5)

where spr is the stack-to-pot ratio. Since we have all read PNL [img]/images/graemlins/smile.gif[/img] and we are all keeping track of the spr, this is very easy to calculate quickly.

Cliff notes: You have a 8/9 out draw on the flop and opponent pots it to you. If you think he will fold more than (spr-1)/(spr+5) of the time, then an all in reraise is better than folding.

Re: raising with draws - how much FE needed?
 

[ QUOTE ]
O yeah, one other thing, since stack-to-pot ratios are all the rage these days. In the original situation (8/9 draw), say that your opponent makes a pot sized bet on the flop. Then the formula simplifies to

(spr-1)/(spr+5)

where spr is the stack-to-pot ratio. Since we have all read PNL [img]/images/graemlins/smile.gif[/img] and we are all keeping track of the spr, this is very easy to calculate quickly.



[/ QUOTE ]

So, assume 100 in the pot going into the flop, villain has 800 left and you cover. Effective SPR is 8:1. He bets pot, and you have a 2:1 draw if you see both cards.

SPR -1 / SPR +5 = 7/13 so he needs to fold 7/13 of the time to a push to break even. Let's see if it works:

7 times he folds and you win 200 = +1400

6 times we get called. We win 2 and lose 4.
2 times you win 900 = +1800
4 times you lose 800 = -3200

Yep. 3200 - 3200 = 0.

Good work. So, we know the shortcut when we have 33% equity and villain bets pot. How does the SPR -1 / SPR +5 formula adjust when:

You have 20%, 25%, 40%, 50% equity and villain bets pot?

You have 20%, 25%, 33%, 40% and 50% equity and villain bets 2/3 pot.

Thanks

Re: raising with draws - how much FE needed?
 

Binions , just replace P=B in the original formula and divide top and bottom by P .

x> (SPR-1)/(SPR+5)


For 25% , use the formula I provided and again replace P=B

x>(2S-P)/(2S+3P+4B)
x> (2SPR-1)/(2SPR +7)

For 20% , it is

x> (3S-P)/(3S+4P+5B)
x> (3SPR-1)/(3SPR+9)

If villain bets 2/3 of the pot , then you can replace 2/3P=B

for 33% it is :

x> (SPR-1)/ (SPR+4)

Pretty cool , huh ?

Re: raising with draws - how much FE needed?
 

I will make some further simplifications to the formula .

(SPR-1)/(SPR+5) = (SPR +5 -6)/(SPR +5) = 1 - 6/(SPR+5)

That is , the push is neutral EV if you expect to be called 6/(SPR+5) .This is actually quite nice as it's very easy to do this calculation on the spot during a difficult situation .

So in the original example , as long as you get called
6 /7.416666 , then the push makes no difference .If you expect to be called less than that number then the push is positive EV .

Re: raising with draws - how much FE needed?
 

[ QUOTE ]
I will make some further simplifications to the formula .

(SPR-1)/(SPR+5) = (SPR +5 -6)/(SPR +5) = 1 - 6/(SPR+5)


[/ QUOTE ]

I am with you so far. In my SPR=8 example, 1 - 6/(8+5) = 1 - 6/13 = 7/13.

[ QUOTE ]
That is , the push is neutral EV if you expect to be called 6/(SPR+5) .This is actually quite nice as it's very easy to do this calculation on the spot during a difficult situation .

[/ QUOTE ]

Now I see. 6/(8+5) = 6/13. This is breakeven if you think you will be called 6/13.

Re: raising with draws - how much FE needed?
 

[ QUOTE ]

If villain bets 2/3 of the pot , then you can replace 2/3P=B

for 33% it is :

x> (SPR-1)/ (SPR+4)

Pretty cool , huh ?


[/ QUOTE ]

So (SPR +4 - 5)/SPR +4 = 1 or 1 - 5/(SPR +4) so if he calls 5/SPR+4 = breakeven. Did I get that right? (2/3 pot bet and 33% draw equity)

Comparing, in the SPR=8 scenario, when he bets pot if he calls your push less than 6/13 you win. When he bets 2/3 pot, if he calls your push less than 5/12 you win.

The less he bets, the more he needs to fold to your push for you to breakeven (54% v 59%)

Re: raising with draws - how much FE needed?
 

< applause >

nice!

you can get by with a couple memorized numbers. but this is more fun.

someone make an SPR script already and plug it into PAHUD with this fold equity formula.

Re: raising with draws - how much FE needed?
 

Here is the general formula. I used the rule of 4 approximation for calculating equity on the flop.

If
P = pot size on the flop
S = stack size
B = opponents bet size
n = number of outs that you have

let c = ((25-2n)S-P)/((25-2n)S+(25-n)P+25B)

If your opponent will fold to an all in reraise more than c of the time then the all in reraise is better then folding. You can easily rewrite this in terms of spr, as long as you replace bet size by bet-to-pot ratio. Of course this formula is beyond most people's real time arithmetic skills (certainly is for me anyway).

Something that is interesting to note about these formula is that the amount of fold equity required gets smaller as the size of the all in reraise gets smaller. This seems a bit counterintuitive at first, until you observe that for small all in reraises your opponent will be getting good pot odds to call, so the amount of FE that you actually have gets smaller as your stack gets smaller. To find the right time to use this play you must balance the two things. Your stack must be big to get FE but also must be small enough to ensure that you don't need too much fold equity for the all in to be +ev. The formula in this thread suggest that an spr of between 3 and 7 (spr above will often mean that the villain will have to fold more than 50% of the time) is the optimal range for this all in reraise play to succeed. Of course this depends very much on the precise situation.

Re: raising with draws - how much FE needed?
 

Sorry correction. Should have read

c = ((25-2n)S-nP)/((25-2n)S+(25-n)P+25B)

Re: raising with draws - how much FE needed?
 

[ QUOTE ]
Binions , just replace P=B in the original formula and divide top and bottom by P .

x> (SPR-1)/(SPR+5)


For 25% , use the formula I provided and again replace P=B

x>(2S-P)/(2S+3P+4B)
x> (2SPR-1)/(2SPR +7)

For 20% , it is

x> (3S-P)/(3S+4P+5B)
x> (3SPR-1)/(3SPR+9)

If villain bets 2/3 of the pot , then you can replace 2/3P=B

for 33% it is :

x> (SPR-1)/ (SPR+4)

Pretty cool , huh ?

[/ QUOTE ]

The formula for 40% equity facing a pot bet is (1/2SPR - 1) / (1/2SPR + 4)

At 50%, you need no fold equity to make the push breakeven.

Re: raising with draws - how much FE needed?
 

How can you guys do these calculations in the small time we have to act? Is it merely practice practice practice or we all need to have a good mathematics background?

Re: raising with draws - how much FE needed?
 

Hey Tic , go back to my formula in simplified form .
It's not too difficult to work out 6/(SPR+5) .Anyone can do it and it's going to be the case most of the time so it's worthwhile that you memorize it .

Also remember that this only applies when you have a 33% equity and your opponent bets the pot . Otherwise , you will have to make slight adjustments to the formula .

Re: raising with draws - how much FE needed?
 

[ QUOTE ]
How can you guys do these calculations in the small time we have to act? Is it merely practice practice practice or we all need to have a good mathematics background?

[/ QUOTE ]

There's actually some symmetry to it:

For 20% equity facing a pot sized bet, villain needs to fold more than:

3SPR-1/3SPR+9

For 25%:

2SPR-1/2SPR+7

For 33%:

SPR-1/SPR+5

For 40%

(.5*SPR)-1/(.5*SPR)+4

For 50%:

You need zero fold equity to break even.

When you face a 2/3 pot bet, you simply add a little less to the denominator (ie at 33% its SPR+4 instead of SPR+5).

I have a feeling this will become almost as easy to remember and apply at the table as using the rule of 4.

After all, 3SPR (+9) at 20%, 2SPR (+7) at 20%, 1SPR (+5) at 33% and 0.5SPR (+4) at 40% is a clear pattern. Every time you jump the a categories, you lose 1 SPR and +2 off the demominator, except when you jump from 33 to 40, you lose half SPR and +1 off the denominator. But its the same ratio. If SPR = a +2 loss, then half SPR = +1 loss.

Re: raising with draws - how much FE needed?
 

[ QUOTE ]
I'm sure that many of the more mathematically inclined posters might have
worked this out for themselves, but I thought that some people might
find the following useful. If you don't like math, you probably
don't want to read this. If you are a math geek (like me), you might
check my calculations to see if I am right. I worked this out while
I was analyzing a play that I had made in a live game last night and
I came up with a nice little formula that people might find helpful.
The basic situation is the following.

In NL holdem you have a 8/9 out draw on the flop and your single
opponent bets into you. How much fold equity do you need to make it
correct to reraise all in?

Say that

P = pot on the flop

S = effective stack

B = opponents bet


Now E(fold) = S. That is, the expected value of your stack given
that you fold is S. We want to compare this with E(all in) - the
expected value of your stack if you raise all in. Say that your
opponent will fold x of the time and call 1-x of the time (0<= x <=
1). We want to know how big x must be in order to make raising all
in at least as good as folding.

I will assume that if he calls, you have a 1 in 3 chance of winning
the hand. This is approximately true for many common draws (e.g
flush draw, oesd with a backdoor flush draw)

Then

E(all in) = x(P+B+S)+1/3(1-x)(2S+P)

since if opponent folds, your stack will be P+B+S and if he calls,
your stack will on average be 1/3(2S+P). Therefore raising all in is
better than folding if

x(P+B+S)+1/3(1-x)(2S+P) > S

Rearranging this yields

x>(S-P)/(S+2P+3B)

The nice thing about this is that this formula is just about useable at the table (there is only one division involved - my real time arithmetic skills struggle to cope with most formulae that involve lots of divisions).

As an example, here is the hand that prompted me to work this out.

I was at a final table of the local cardroom tourney and I picked up
Th9h in the big blind. There was 12K in the pot and the flop was
QhJs5c. I had 29K behind and I checked to my opponent who bet 7K
into the pot (he had me covered). So S=29, P = 12 and B=7. Therefore
(S-P)/(S+2P+3B) = 17/74 which is about 0.23.

Thus I needed him to fold at least 1/4 of the time to make the all
in raise better than folding (at least in terms of tournament
chips). Given my knowledge of this particular opponent, I think that
he won't fold 1/4 of his range here so I should have folded. In the
actual event, I reraised all in and busted from the tourney when he
called with QT.

This formula only works if you have close to a 1 in 3 chance of
winning when called, but it would not be hard to modify to other
drawing situations as well. Any comments?

[/ QUOTE ]

I don’t know if this is any easier but this is what I came up with a while back to calculate required fold equity.
Called = total pot if called.

W = Stack if (called * Equity) - Initial Stack
Fold % required = (W) / (W - Current Pot Size)

Re: raising with draws - how much FE needed?
 

[ QUOTE ]
I don’t know if this is any easier but this is what I came up with a while back to calculate required fold equity.
Called = total pot if called.

W = Stack if (called * Equity) - Initial Stack
Fold % required = (W) / (W - Current Pot Size)

[/ QUOTE ]

I'm not sure I understand. It seems to me that the numerator is bigger than the denominator here, so you would require that the opponent folds more than 100% of the time????

I might just be misinterpreting your formula. Can you give an example of how you would use this formula to calculate required FE?

Re: raising with draws - how much FE needed?
 

[ QUOTE ]
[ QUOTE ]
I don’t know if this is any easier but this is what I came up with a while back to calculate required fold equity.
Called = total pot if called.

W = Stack if (called * Equity) - Initial Stack
Fold % required = (W) / (W - Current Pot Size)

[/ QUOTE ]

I'm not sure I understand. It seems to me that the numerator is bigger than the denominator here, so you would require that the opponent folds more than 100% of the time????

I might just be misinterpreting your formula. Can you give an example of how you would use this formula to calculate required FE?

[/ QUOTE ]

Sure.

You’re stack size is 150 (opponent has you covered).
Pot size is 50.
You’re equity if called is 20%.

W = ((150 + 50 + 150) * .20) – 150
W = (350 * .2) – 150
W = -80
-80 / (-80 ) – 50 = -80 / -130 = 0.6153846

You need him to fold 62% of the time for this to be chip EV neutral.

Re: raising with draws - how much FE needed?
 

There is some missing information . How much does your opponent bet into the pot ? Does he make a pot size bet or a two thirds size bet ?

If he makes a pot size bet , then the amount of fold equity required is :

(3SPR-1)/(3SPR+9) = 8/18 = 44.444% so clearly 62% is too high .

Re: raising with draws - how much FE needed?
 

[ QUOTE ]
There is some missing information . How much does your opponent bet into the pot ? Does he make a pot size bet or a two thirds size bet ?

If he makes a pot size bet , then the amount of fold equity required is :

(3SPR-1)/(3SPR+9) = 8/18 = 44.444% so clearly 62% is too high .

[/ QUOTE ]

I over simplified my example calculation. W = the total amount you would win if called times your equity minus your initial stack size. So if your opponent bets into you and you have to call that bet first you have to factor that into the total amount you would win if called. Pot is the size of the pot after you match opponent’s bet.

Example:
You’re stack size is 250 (opponent has you covered).
Initial Pot size is 50.
Villain bets 50.
You call 50.
Pot = 150 = (50 + 50 + 50)
You’re equity if called is 20%.

W = ((Pot) +200 + 200) * .2 ) – 250 = -140
-140 / (-140 ) – Pot = -140 / -290 = .48275

Re: raising with draws - how much FE needed?
 

[ QUOTE ]
[ QUOTE ]
There is some missing information . How much does your opponent bet into the pot ? Does he make a pot size bet or a two thirds size bet ?

If he makes a pot size bet , then the amount of fold equity required is :

(3SPR-1)/(3SPR+9) = 8/18 = 44.444% so clearly 62% is too high .

[/ QUOTE ]

I over simplified my example calculation. W = the total amount you would win if called times your equity minus your initial stack size. So if your opponent bets into you and you have to call that bet first you have to factor that into the total amount you would win if called. Pot is the size of the pot after you match opponent’s bet.

Example:
You’re stack size is 250 (opponent has you covered).
Initial Pot size is 50.
Villain bets 50.
You call 50.
Pot = 150 = (50 + 50 + 50)
You’re equity if called is 20%.

W = ((Pot) +200 + 200) * .2 ) – 250 = -140
-140 / (-140 ) – Pot = -140 / -290 = .48275

[/ QUOTE ]

Therein lies part of the rub.

The formula derived by OP & JayShark figures the fold equity you need when you come over the top not when you call.

So, in your example, with 20% equity, a 50 final pot preflop, a pot sized bet into you, and 250 effective stacks going into the flop:

1. the SPR is 5 (250 to 50)
2. the 20% formula for a pot sized bet is 3SPR-1/3SPR+9

So, villain needs to fold at least 14/24 or 58.3% when you come over the top to break even for the times when he calls and you are drawing at 20%.

Let's see if it works.

14/24 he folds and you win 100 (the 50 in the pot and his 50 bet)
2/14 he calls and you win 300 (50 in the pot + his 250)
8/14 he calls and you lose 250 (the 250 in your stack on the flop)

1400 + 600 = 2000
8*250 = 2000

Yep, it works.

Re: raising with draws - how much FE needed?
 

[ QUOTE ]
For 40%

(.5*SPR)-1/(.5*SPR)+4

For 50%:

You need zero fold equity to break even.

[/ QUOTE ]

Does this mean with 14-15 outs on the flop, it is generally correct to go All-In? Of course, all 14-15 outs are not always solid, but can someone post a pointer to a discussion thread on the pros/cons of pushing AI with 14-15 outs? Thanks!

Re: raising with draws - how much FE needed?
 

If you have 15 clean outs on the flop and your opponent has no redraws, then you have approximately 54% chance of having the best hand at showdown - i.e. you have the best hand (in terms of showdown equity). Any time that you have a hand that is favourite to win a showdown, then pushing all in is superior to folding. Of course, this does not necessarily mean that pushing all in is correct. There might be other ways to play the hand that have a higher expected value.

Re: raising with draws - how much FE needed?
 

[ QUOTE ]
If you have 15 clean outs on the flop and your opponent has no redraws, then you have approximately 56% chance of having the best hand at showdown

[/ QUOTE ]

FYP.

Run AhAs v 5c6c on a 7c8c2d board on pokerstove (56.3%). Or, do the math [1- (30/45*29/44)] = 56.1%

Re: raising with draws - how much FE needed?
 

[ QUOTE ]
[ QUOTE ]
If you have 15 clean outs on the flop and your opponent has no redraws, then you have approximately 54% chance of having the best hand at showdown

[/ QUOTE ]

FYP.

Run AhAs v 5c6c on a 7c8c2d board on pokerstove (56.3%). Or, do the math [1- (30/45*29/44)] = 56.1%

[/ QUOTE ]

FYP [img]/images/graemlins/smile.gif[/img] You are assuming that you know your opponents cards. So if you don't know the opponents cards, but somehow know that you have 15 outs, then your probability of winning is

1-(32/47)*(31/46) = 0.541165.... or approx 54%.

Re: raising with draws - how much FE needed?
 

[ QUOTE ]
[ QUOTE ]
[ QUOTE ]
If you have 15 clean outs on the flop and your opponent has no redraws, then you have approximately 54% chance of having the best hand at showdown

[/ QUOTE ]

FYP.

Run AhAs v 5c6c on a 7c8c2d board on pokerstove (56.3%). Or, do the math [1- (30/45*29/44)] = 56.1%

[/ QUOTE ]

FYP [img]/images/graemlins/smile.gif[/img] You are assuming that you know your opponents cards. So if you don't know the opponents cards, but somehow know that you have 15 outs, then your probability of winning is

1-(32/47)*(31/46) = 0.541165.... or approx 54%.

[/ QUOTE ]

Yes, but when you assume "clean outs v foe with no redraws," you should use 45 and 44 unseen cards.

Also, see Barry Greenstein's Ace on the River at p. 299 for why 45-44 unseen cards should be used instead of 47-46 even when your foe's cards are hidden.

Re: raising with draws - how much FE needed?
 

Sorry to bump this these few days later, but has anyone used this over the past few days and could provide further hands, or just experiences.

Are you mostly using this in a tournament setting?