[Theory] Fold equity and Semi-bluffing

Introduction for Beginners

Fold equity is a basic aspect of poker that many of us rarely think about but very commonly utilize. FE is the probability that your opponent will fold to your bet and you win the pot right away. Any time you raise on the button with a marginal hand like A7o, you are doing so for many reasons (positional advantage, balancing the strong hands you raise, etc.), but the most immediately profitable reason is that your opponents fold often enough to make it +EV. Any time you make a continuation bet with a missed hand, you do so with the assumption that you have, with an average opponent, about 2/3 FE because that is the probability that the flop will miss your opponent. Without FE, these plays no longer make sense and you are forced to play showdown poker.

So whether you realize it or not, FE is ingrained into your most basic preflop and postflop plays. However, it doesn't stop there. FE is most essential in calculating your expected value in semi-bluff situations. A semi-bluff is where you make a bet or raise with some sort of draw. If called, you can still when the pot by completing your draw, but the only reason it is profitable is because your opponents will fold a certain percentage of the time. When Phil Ivey 5-bets all in with Q8 on a J77 board, it's because he has calculated both his pot and fold equity against his opponent's range and he has decided that it is a profitable play. All good players know this concept intuitively, whether it is by having made the calculations and/or by playing a crapload of hands.


A Brief Primer on Expected Value

Unfortunately, to make use of the concept of FE, you have to know some math. So just to review, your total expected value is the probability you will win the pot multiplied by how much is in the pot when you win it, minus how much you lose multiplied by the probability that you lose. That looks pretty confusing in words, so let me give a crude example:

You have KK and your opponent just raised all in preflop for $50 and you have him covered. Everyone folds to you. Your opponent flips over JJ before you make your decision. With a full 5 cards to come, your chance of winning is about 80%. What is your expected value for each possible action?

Well, folding does not change your EV because you have no chance to lose or win anything. When you fold, your EV is always 0.

EV(folding) = 0

EV(calling) = 0.8*50 + 0.2(-50) = 0.6(50) = $30

Your expected value for calling the all-in is $30, which means that you expect to win that much on average every time you make the call. EV(calling) > EV (folding) and the play you should make is clear. Notice that you only care about your net profit or loss from the time when you make your decision. Pretty easy, right?

Calculating how much fold equity you need to make semi-bluffing all-in +EV


Now, let’s say that you are playing 100NL 6-max with 100BB effective stacks and you have the button. LAG on the CO raises to $4 and you repop to $14 with 6d7d. The blinds fold and the CO calls. The pot is $29.50.

The flop comes: Jd8s2d

The CO leads into you for $20. That makes the pot $49.50 and you have $86 behind. You have 9 flush outs (for right now we'll discount that he might have a higher flush draw or be bluffing , and the fact that you have backdoor draws), which is good for about 35% pot equity. What is the EV of moving all in right now?

EV = FE(49.50) + (1-FE)[(0.35*137.50) + (0.65*-86)]

So how much fold equity do you need to have to make this play breakeven?

0 = FE(49.50) + (1-FE)[(0.35*137.50) + (0.65*-86)]
0 = 49.50(FE) + (1-FE)(-7.8) = 49.5(FE) -7.8 + 7.8(FE)
57.3(FE) = 7.8
FE = 0.136

You need to have 13.6% fold equity to break even with an all-in semi-bluff in this situation. If your opponent will fold greater than 13.6% of the time, you have a positive expectation with this move. Playing around with the numbers a little bit, you can see that you can semi-bluff all in with LESS fold equity the stronger your draw is (more pot equity) and/or the smaller your raise is in relation to the pot (which is related to fold equity usually). If, for example, you had AdQd instead and some of your overcard outs might be clean, then you have a clear shove with so much already invested.

Quantifying Fold Equity

Okay, so you need >13.6% fold equity. How can you know if your opponent is willing to fold that percentage of the time or not? This is where the math kind of breaks down and it's player dependent. Here are some factors that you might want to consider when estimating your actual fold equity:

1. What is your opponent's WtS%? If this number is high, he is likely the be a calling station and your fold equity might be adversely affected. Likewise, if it's small, you can consider it favorable.

2. What is your opponent's aggression factor? If this is very high, when he bets it is less likely that he has a strong hand to play for stacks, and your push might be just what he needs to lay it down.

3. What is your image? Obviously, this only applies when your opponents are somewhat observant. If you've been playing 30/25 at the table and raising with lots of hands (even if you had the goods), you are going to get looked up more often. At lower levels especially, lots of players think that a maniac preflop equals a maniac postflop. That's a bad generalization of course, but you have to take that into consideration. Even if you are a nit postflop, your continual raising preflop will make many opponents think you play that way postflop as well.

4. What is your opponent's hand range? If your opponent has taken a very strong line that makes you think he might have a set or something similar, you can put your fold equity at almost zero. However, if many other hands in his range are foldable to your push, then that is favorable.

Conclusion
I know I probably made some mistakes so please let me know if you actually managed to read through all that. For some players I think a lot of this stuff is elementary, but I'd like to hear some thoughts/disagreement/expansion, especially on the last part, factors you take into account when quantifying fold equity.

ez call imo

sort of float too if an A or K flops and he checks cause it sounds like JJ

it's hard to believe you'd only need 13%

in fact, I think so often not only do you not have 65% to win because he could have Adx

but when he has set he has redraws AND WHA"TS WORSE, he he has AdKd or just a higher flush draw you are drawing kinda slim for a 200$ pot

edit: oh yea and the Adx or Kdx also has a redraw if the flush hits / ie it doesn't just take out a single out

it's really kind of annoying that you just make a blanket statement without showing where my math is wrong. And also, if you didn't notice, I made this stipulation in the calculation:

"(for right now we'll discount that he might have a higher flush draw or be bluffing , and the fact that you have backdoor draws)"

For the sake of simplicity I was just trying to show the basic math of a hypothetical situation. I realize that he could very well have AdKd, etc.

EV = FE(49.50) + (1-FE)[(0.35*137.50) + (.065*-86)

i don't get it, is this what you wanted me to do? / I never said your math is wrong but the assumptions that lead to the math may very well be... and I can only give a blanket statement to state that.

also i didn't see this in your main post / but i still think it can't just be ignored, mathematically you could pokerstove the hand range and add these in, because i think if AdKd is just 3% likely it'll double the FE you might need... not sure though but the effects i feel will be drastic. I just don't want people to get the wrong idea and start shoving every FD after reading this thread

"(for right now we'll discount that he might have a higher flush draw or be bluffing , and the fact that you have backdoor draws)"

he's showing people merely the process of how to do the maths. if i get bored later i might work out the general relationship between winning odds, pot odds and fold equity with some examples along the curve, as that 13% surprised me and i find it interesting. i hope someone else does it first though.... (hint)

btw thanks tachy

It's easy to shove 86 into 49.5. How about standard raised (but not reraised) pots and deeper stacks?
BTW Nice post.

kako mozes pricati ova sranje bre????

nije lako shove, jer ako on ima bolje ruke njego tebe, on ces inta-call

ako vi ste vrlo LAG i ova je heads up i imas to image, onda to je nesto drugacije

whoops, sry about that gibberish post

but unless you're playing ultra LAG shorthanded and have history, it's not an ez push because villian's preflop range to call a RR OOP might be AA KK, AK QQ only

Good article, the 2+2 forums use this kind of theory to know the eV.

On May 25 2007 18:37 Donald wrote: whoops, sry about that gibberish post but unless you're playing ultra LAG shorthanded and have history, it's not an ez push because villian's preflop range to call a RR OOP might be AA KK, AK QQ only

wtf donald

good post, but i'd like to see a more complex equity calculation with different hand ranges.

put in AA, KK, QQ, JJ, 88, 22, 9Td, AJo, AJs, KQd, AKd, AQd in his range, considering it the most brutal possible handrange, and see what FE we need against that equity.

I've just done the math on similar scenario with only difference being this is not 3bet pot so stacks are deeper:

NL100 6-max $100 stacks

LAG raises to $4, you call with 6d7d, blinds fold
(pot:$9.5)
flop: Jd8s2d
LAG leads $8 ... (pot:$17.5) you have $96 behind
How bad or good is shove here? And how often he should fold to make this EV+?

EV = FE(17.5) + (1-FE)[(0.35*105.50) + (0.65*-96)]

So how much fold equity do you need to have to make this play breakeven?

0 = FE(17.5) + (1-FE)[(0.35*107.50) + (0.65*-96)]
0 = 17.5(FE) + (1-FE)(---25.475) = 17.5(FE) --25.475 + -25.475(FE)
42.975(FE) = 25.475
FE = 0.592 = 60%

With deeper stacks required FE grows exponentially. However the real question is how will this affect our opponent. How likely is he to call (example 1) $66 in $135.5 pot or how likely is he to call (example 2) $88 in $113.5 pot.

great post i think this should be added to the articles section