Fold equity
 

At the risk of sounding like a donk, could someone please explain fold equity in as much detail as possible.Much love!!

In the most basic sense, during the play of a hand, the cards you hold have a given equity (share of the money) in the pot. That equity has 2 components: Showdown equity (the portion EV of your hand vs your opponent(s) range the percentage of time you to go to showdown), and fold equity (the portion of EV your hand has attributable to the percentage of time you bet and make your opponent fold)

Example. On the turn there is x in the pot and you hold a flush draw. You believe you must make your flush to win and when you make your flush, you will always win.

If that were your only option, your equity in the pot would be 9/46* x.

However, if your opponent checks and you bet y, you suspect your opponent will fold z% of the time. (note 1 > z > 0)

Your fold equity when you bet y is (1-z) * x and here we assume for simplicity that opponent never check-raises.

Of course when your semibluff gets called that creates a new showdown equity value and new EV calc for the river but lets not muddy the waters.

The cleanest real-world example of fold-equity based strategy is ICM in SNGs.

Equal stacks on the bubble, pure equity theory states that 1st in from the SB, it is always +EV to shove any 2 cards and the EV of that shove increases as the blinds increase.

Take a normal 'Stars STT with 4 equal stacks of 3375. Blinds of 100/200/a25. Assume opponent will only call a shove with 15% of hands.When called, assume your random hand will win 30% of the time.

EV

He folds you win = 3650 * .85 = 3102.50
He calls you lost = 0 * .105 = 0.00
He calls, you win = 6800 * 0.45 = 306.00

EV(shove) = 3408.50
EV(fold) = 3250.00

Poker math is often counter-intuitive. In this scenario it is impossible for the BB to reduce your EV to zero by tightening up his range as your fold equity just goes up. Theoretically, if the BB knows you're shoving any 2, the only way he can reduce your EV to 0 (he can never makes it -EV to shove) is to call with any 2.

Late in a SNG, fold equity is god.

I think i might have asked you a little about this before, but i couldnt find the string. I notice that you say that this is the case when 'on the bubble' (thus more people are alive).

I also assume that if you are equal stacks HU - being first to ask you are +EV if you shove with any two? What if you have more chips (say you are winning by a 2:1 ratio) - does that change anything? I am guessing that overall it doesn't because that would just mean that maybe the opponents range increases (he feels more desperate) but as stated that does not bring your +EV down to zero (unless he is willing to call with any two - and i doubt that they will).

Or if you are losing (again by a 2:1 ratio) - does that effect the math? Again, i would assume that we would still be +EV. Even though our opponent might loosen his calling requirements (knowing that we are desperate and trying to pick us off), he almost never will loosen to the point of calling with any two.

Then, if this is all true - if we find ourselves HU with a player we think is better than us we are better off shoving any two cards, at least from the SB.

The example uses equal stacks because then you don't bring stack size issues into play. In the contrived scenario, it is very close to being CEV = $EV. Or better put, everyone has the same risk of ruin.

Obviously if the situation is

UTG: 400
Button:300
Hero: 5000
BB: 7800

You fold all but a very small range (based upon your assessment of BB's calling range). Here CEV <> $EV.

Stack sizes alter push/call ranges. The best players who use ICM are those who use all the information at the table to accurately deduce push/call ranges. That being said, fold equity is still a big part of the calc.

Great post. Just fixing a very small (but important) typo above.

Now my question: What if villain loosens up his calling range not to 100%, as you suggested, but say to 50%? What does that do to the numbers? I'll even increase your win % to 40%, since villain is calling with a much wider range than before.

He folds you win = 3650 * .5 = 1825
He calls you lost = 0 * .2 = 0.00
He calls, you win = 6800 * .2 = 3400

EV(shove) = 5225
EV(fold) = 3250.00

As you said, intuatively, you would think calling with a wider range (but not ATC) would be better, but the math says otherwise. That's awesome.

HU, you have the same effective stacks, but as Kurn said, CEV <> $EV. I think you are trying to compare this conversation to what we were talking about the other day, regarding HU SNGs, and I don't think you can do that.

You were saying that you like to go push-bot fairly deep, with the blinds still being small, and I said I didn't think that was a good idea. Run some numbers and I think you'll see that your EVs will agree with me.

ie, looking at the obvious extreme example, on the first hand with 1500 stacks and 10/20 blinds, going pushbot here is just stupid. Even if your opponent will call with 10% of his hands (and I'll be nice and give you a 30% chance of winning with your crap against his 10% range):

He folds, you win: 1520 * .9 = 1368
He calls, you lose: 0 * .1 * .7 = 0
He calls, you win: 3000 * .1 * .3 = 90

EV Fold: 1490
EV Push: 1458

It's -EV and you have taken all the skill out of playing. Even if it was EV neutral, the 5% rake would kill you playing like this.

Now, let's make the blinds 15/30 and give you 600 chips and him 2400 chips and say that he'll call you with that 15% of his range. I think this is more where you were saying you go pushbot, at around 20 BBs (even deeper, as I recall). In this case:

He folds, you win: 630 * .85 = 542
He calls, you lose: 0 * .15 * .7 = 0
He calls, you win: 1200 * .15 * .3 = 54

EV Fold: 585
EV Push: 596

Here, your push is actually slightly +EV, but don't forget about the tourney juice and all that good stuff.

Now let's exaggerate and say he'll call you with 25% of his range and you'll win 40% of the time (I think it's safe to say this is not possible):

He folds, you win: 630 * .75 = 473
He calls, you lose: 0 * .25 * .6 = 0
He calls, you win: 1200 * .25 * .4 = 120

EV Fold: 585
EV Push: 593

Again, this is only very slightly +EV, and I think that's probably best case scenario, right?

Me... I'd rather wait for my opponent to screw up. If you started playing like this against me, I'd tighten up my range and calll when I'm confident I'm a decent favorite, and that should make your push -EV (I don't feel like guessing at numbers and doing that math again, but feel free).

Thanks everyone.Its alot to take.I'll be reading your answer again and again and again Kurn to try and have it make total sence to me.I do appriciate all the help.

I was late in a live tourny at Hollywood park , blinds 2000-4000 with 200a and im small blind and it's folded to me .I see A 8 and I shove my 8000ish to the big blind and he looks and calls.He has about 17k.He turns Q 7 off . Is that the right call given the examples above or just stupid? Thanks again.

Assuming 8 players at the table and you having 8000 AFTER your SB (if you started the hand with 10k), there was (200 * 8) + 2000 + 4000 + 8000 = 15,600 in the middle and it was 6000 for him to call. That's 2.6:1, so I wouldn't expect him to fold often.

And if you really started the hand with 8000, then he was getting 3.4:1, meaning he's folding never.

Moral of the story: When you get that short, you have no fold equity, so don't let yourself get that short.

(The push was still correct, of course, but so was his call)

As you said, intuatively, you would think calling with a wider range (but not ATC) would be better, but the math says otherwise. That's awesome.

The magic of fold equity. If you fold at all, that additional equity makes the shove +EV.

Poker truth #1: They can't fold if you don't bet.

OK, now for the non-math, Poker for Dummies side of things... I usually think of fold equity in terms of the size of my stack. The smaller my stack is the less fold equity it possesses. If the stack gets so small in relation to the blinds (like in NG's example from Hollywood Park), then you're fold equity is down to zero and it's correct for someone to call you with nearly ATC. So, obviously, you want to avoid situations where your stack size gets to the point where it has little to no fold equity.

This is why it's important to read/learn/understand the zone and M concepts in HOH. NG, I'd highly recommend that you pick-up HOH and go through those sections as Action Dan does a lot better job explaining those concepts than I could here.

Lets say 6 at the table, then the pot was 7200 before you shoved. If you had 8,000 before blinds and antes, that meant he had to call 3600 into a pot of 12800 so he was getting 3.55:1 on the call. At a bit more than 20% of his stack, that's an easy call for him. In his place if I have a stack of maybe 20,000, I'd call blind.

I mean, getting any more than 4:1 folding any 2 cards is wrong. So this is close.

If he took the time to look before instacalling he's a donk.

I have the book.Time to read it!

Make sure to read both vol. 1 and vol. 2. As a matter of fact, I was thinking I need to pick those up again and re-read them myself!