Hi,

What's the calculation required to determine how often we need villain to fold to show a profit?

For example, the hand below. If I estimate I have an average of 8 outs when called (the straight outs), that gives me around 32% equity. With that information, how do I determine how often villain needs to fold to make this semi-bluff profitable?

Is it this:

Pot is 15.6BB after the shove, I can expect 0.32 (15.6) = 5BB back from the pot when called.

I'm risking 5.8BB when I shove, so I need villain to fold 1 - (5/5.8) = Villain needs to fold 14% of the time for this shove to be profitable, if my assumptions are all correct.

Or is this way out? Sorry I'm a math fish.

PokerStars - $2.82+$0.18|40/80 NL (3 max) - Holdem - 2 players
Hand converted by PokerTracker 4

BB: 7.8 BB (VPIP: 64.52, PFR: 17.24, 3Bet Preflop: 16.67, Hands: 31)
Hero (SB): 10.95 BB

Hero posts SB 0.5 BB, BB posts BB 1 BB

Pre Flop: (pot: 1.5 BB) Hero has 5 4

Hero calls 0.5 BB, BB checks

Flop: (2 BB, 2 players) 6 7 7
BB bets 1 BB, Hero raises to 9.95 BB and is all-in, BB calls 5.8 BB and is all-in

Turn: (15.6 BB, 2 players) Q

River: (15.6 BB, 2 players) 7

if you shove and he calls:
cost to shove =5.8
avg. pot equity =32%
total pot when called =15.6

.32*15.6 -5.8 = -0.808

.81 = how much you lose when he calls your shove (relative to folding)
*its the on avg. profit/loss

if you shove and he folds:
you win 3 (relative to folding)
*its the amt. in the pot

folding =0
* you neither win or lose anything when you fold

utilizing the formula
x + (1-x) = 1
* 2 parts of 1 whole when the percentages of each part are unknown

determining the equilibrium folding and calling percentages
x*3 + (1-x)*-.81 =0
3x +.81x -.81 =0
3.81x = .81
x =.81/3.81
x=.2126

you need him to fold more then 21.3% (call less often then 78.7%) of the time for shoving to be more profitable then folding


check:
.2126 *3 + .7874*.81 = 0

ngFTW’s method is fine but he has the wrong shove amount. Villain’s initial stack is 7.8 BB. He posts the blind of 1 BB and then bets 1 BB on the flop leaving him 5.8 BB. So hero has to call the 1 BB bet and then put villain all in with a raise of 5.8 BB for a total shove investment of 6.8 BB. With this amount the required fold equity is 37.6%.

On that particular board, I'm doubting that you really have 32% average equity, if called. It is unlikely he calls if he has complete air. That means he won't be calling without a 6, 7 or pocket pair (or possibly a better straight draw). Since it is a paired board, you might be getting your straight about 32% of the time, but some of that time you are still going to lose.

Hi.

In the PokerStars example you show, there are 2 small bets in the pot prior to the flop (betting round number 1).

And you both have 5.8 big bets in your stacks.

5.8 big bets = 11.6 small bets.

Assuming your opponent calls, when you figure to win 32%, then you must figure to lose 68%.
(Because these per cents must add up to 100%).

If your opponent calls and you win, you're going to win the 2 small bets already in the pot plus the 11.6 small bets of your opponent.
In other words, 2+11.6=13.6 small bets is what you're going to win when you win.

And if your opponent calls, then 11.6 small bets is what you're going to lose when you lose.

.32*13.6-.68*11.6=-3.536. In other words, if your opponent calls with winning and losing hands, alike, then when you shove, you'll average losing 3.536 small bets.

But if your opponent folds when you shove, then you'll win 2 small bets.

So your opponent has to fold to you shove often enough so that your winning 2 small bets happens often enough to off set his averaging 3.536 small bets when you shove and he calls.
3.536+2=5.536

3.536/5.536=63.87%.

Assuming your opponent has 68% equity, and folds the same proportion of winners as losers, I think your opponent has to fold more than 63.87% of the time for your shove to be profitable.

I haven't read the other replies yet.

Buzz

thanks statmanhal for noticing my error and posting the correction.
i agree the correct cost ("total shove investment") is indeed 6.8BB and not 5.8, and consequently that 37.6% is the correct equilibrium folding percentage.




to be very clear, 37.6% answers the question: -what is the percentage the opponent must fold to a shove for shoving to be equal to folding?-
as a consequence of knowing this percentage we know that if he folds to a shove more often then 37.6% then shoving is more profitable then folding and that if he folds to a shove less often then 37.6% then shoving is less profitable then folding.

it doesn't indicate that if he folds to a shove more often then the 37.6% that Hero will profit on the hand, where 'profit on the hand' means to start the next hand with more chips in your stack then were in your stack when you started this hand.

it indicates that if he folds to your shove more often then the 37.6%, you will on avg start the next hand with more chips if you were to shove then if you were to fold.


I believe Buzz, (whether he realizes it or not) is trying to give you the percentage your opponent must fold to a shove for Hero to 'profit on the hand'.
which is a literal interpretation of your posted question
but doesn't acknowledge the decision point of the poker hand.

58.4% is the percent that your opponent must fold more often then when you shove for your stack to be greater then 7.8BBs on average to start the next hand.
you can use the same method as used to arrive at the 37.6%, you must just use different values. the cost is 7.8BB, and the amount you win when he folds is 2BB.

Right thanks all.

Yes. That was my intention. However, I misread the opening post. Sorry.

Thank you, ngFTW.
Buzz

I think 13.83% is the answer for required fold equity. (I must be doing something wrong?)

Hero is the one shoving. Is Hero also the one folding? (Seems it must be).

BB bets 1 BB, Hero raises to 9.95 BB and is all-in, BB calls 5.8 BB and is all-in."

When Hero has 32% equity and there are 2BB in the pot before the betting commences, Hero “owns” .32*2BB=.64BB of the pot. That is, a fair split at this point (after the flop but before the betting commences on the second betting round) would give Hero 0.64BB and Villain 1.36BB.

However, if Hero folds to Villain’s 1BB bet, Hero loses his pot equity, which amounts to 0.64BB.

When Hero shoves and Villain folds, Hero wins 3BB.

When Hero shoves and Villain calls, Hero averages losing 1.224BB (which is more than he loses when he concedes his 0.64BB equity).
Because .32*8.5-.68*5.8=1.224

In order for shoving to be Hero’s move, Hero cannot lose more than 0.64BB when he shoves. (Hero can’t afford to shove when expecting to lose 1.224BB).

But Hero wins 3BB if Villain folds to his shove.

So if Villain will be kind enough to fold enough of the time, the shove becomes a good move for Hero. But how much?

If Hero shoves and Villain calls, when Hero wins, it is 8.5 BB that he wins.
If Hero shoves and Villain calls, when Hero loses, it is 5.8 BB that he loses.
If Hero shoves and Villain folds, then Hero wins 3 BB.

Let 1-X be the fraction of the time Villain calls. Then X is the fraction Villain folds.
-.64=(.32*8.5-.68*5.8)(1-X)+3X
-.64=-1.224+1.224X+3X
0.584=4.224X
0.584/4.224=X
0.1383=X

Villain only has to fold more than 13.83% (call it 14%) to make shoving a better play than folding for Hero, assuming Villain calls with equally good and bad hands (and thus Hero, with 32% equity, 32% and loses 68%).

But it seems to me an intelligent Villain doesn't call with equally good and bad hands... he calls more with good hands than with bad hands... and that is the main bugaboo. A secondary bugaboo is Hero doesn't only win when he makes the straight. Thus the 32% seems phony. (I adjust for that and presume you do too).

I thought I understood "fold equity" but since we’re getting different answers I must be missing something.

Doesn't "fold equity" refer to your chances of causing your opponent to fold? In other words, if your opponent will fold to your bet 25% of the time in a $400 pot, then don't you have $100 in fold equity?

The bugaboo seems to be, "How can you possibly know what per cent of the time your opponent will fold to your bet?"

Doesn't the per cent of the time your opponent will fold to your bet depend on
(1) how much the bet is,
(2) your opponent's assessment of the value of his hand relative to yours,
(3) his current impression of your bets (might you be bluffing or unaware),
(4) and various other factors?

The way I play, not that it matters, is when I think I'm behind, I don't shove unless I think there's a very good chance Villain will fold... and I suppose that assessment is based primarily on my general impression of recent past behavior. But there is no number involved.

That is what I was trying to do. However, I misread the opening post in this thread.

What is the "decision point"?

Buzz

OP’s question was

What's the calculation required to determine how often we need villain to fold to show a profit?

The decision point is hero's shove and he is essentially determining what is the minimum fold equity he needs to make the shove profitable.

The EV equation for an all-in bet in this case is the following:

EV hero-shove= Pr(V fold)*Pot +Pr(V call)*EVcall

EV= fe*Pot + (1-fe)*(eq*(Pot + Call + Bet) – Bet)

For this problem, Pot = 3, Call= 5.8, Bet=6.8, eq = 0.32. If you set EV to 0 and solve for the break-even fold equity, you find fe=37.6%,

Thank you. Very clear. I understand.

I think what matters here is the minimum fold equity Hero needs to make the shove more profitable than giving up his equity in the pot.

But what is Hero's equity in the pot?

Although the pot is 3BB, I don't think Hero's equity is 0.32*3BB because Hero hasn't yet matched Villain's 1BB bet.

I think what Hero gives up by conceding the pot to Villain's 1BB bet is his equity in the pot to which Hero has already contributed... in other words 0.32*2BB=0.64BB.

In other words, if Hero folds to Villain's 1BB bet, Hero loses 0.64BB, his equity in the pot before Villain's 1BB bet. (If Hero called Villain's 1BB bet, then the pot would be 4BB and Hero's equity in the pot would be 1.28BB)

I think in order to make shoving the proper play for Hero, Hero merely has to lose less by shoving than he would be giving up if he folded to Villain's 1BB bet. In other words, if Hero merely loses -0.64BB by shoving, he's doing as well as if he folded to Villain's 1BB bet.

Ah! Thank you. Very clear.

I don't think we should set EV to 0.

(Because in order to do better by shoving than by folding, Hero merely needs to lose less by shoving than he loses up by folding... and I think what Hero loses by folding is -0.64BB, as explained above).

Buzz

All that is fine, but the question was -- given hero shoves, what fold equity insures a profit. I'm saying the shove is profitable if the EV is greater than zero. I think you're saying the shove is profitable if it is greater than the EV gotten if hero folded, which you calculated to be -0.64bb. But it is pretty well established that a fold has 0 EV for a fold does not affect one's stack size. I'm pretty sure you know this so I don't quite understand your thinking.

I suppose it hinges on your definition of the phrase "to show a profit"

If you mean, is this bet 0ev or better, compared to folding, then you get one answer.

If you mean, will I make a profit on the hand overall, you get another (I don't think this is what people would usually mean though)

Buzz

hero is faced with a decision. his decision is to choose between the options; fold, call or shove. the context of the decision(which i casually called the decision point in a previous post) is--playing NLHE vs. a single opponent, on the flop, in position, with a 6.8 bigblind effective stack, facing a 1 bigblind bet into a 2bigblind pot.

to choose, we compare the benefits of each option, and select the one with the best benefits. that is we try to determine the expectation of of each option, compare expectations and choose the option with the greatest expectation.

the comparisons are:
is shoving better or worse then folding, is calling better or worse then folding, is shoving better or worse then calling.

what you think matters would appear to also be a comparison, but not one of the above comparisons.
how does what you think matters assist in the decision that hero faces?


folding to the opponents bet would result in hero's stack remaining at 6.8bbs, for a change of 0. this is why 'we say' folding = 0.

folding has an expectation of 0.


to determine the expectation of shoving, since when hero shoves villain has 2 possible responses, call or fold, its necessary to provide the frequencies at which he chooses each response. as an alternative, if we provide the equilibrium frequency (the percentage where shoving has an expectation of 0 such that shoving equals folding) we then know that if we can expect our opponent to shove more often than this equilibrium figure, then shoving will have a positive expectation.

shoving has an expectation >=0 if estimated folding frequency is >= the equilibrium folding frequency(FE)

determining the expectation of calling is beyond the scope of this thread.

hero fails to realize it. equity only gets realized at showdown, so he forfiets realizing it. hero doesn't lose it.
hero's stack is not going to change as a consequence of folding. its 6.8bb when he is facing the bet and it will be 6.8bbs when he folds to the bet. hero gains/loses 0 by folding. hero's EV fold (expected value from folding) =0.


opponent's equity is an estimate. its a value one arrives at using one's judgement and experience. its what you do in a game of 'unknown information'. estimating oppponent's equity is necessary to determine the expectation of shoving.
its true the equity value that needs to be used in the equation to determine the equilibrium folding percentage(or minimum FE percentage) is an estimate of the equity hero has vs. the opponents shove-calling range and not the equity hero has vs. the opponents entire betting range. perhaps 32% is a poor estimate, it was a given and VBAces already alerted OP.

I agree the shove is profitable if the EV is greater than zero.

I'm saying if Hero folds instead of shoving, he concedes his equity in the pot, which I think is .64bb.

Thus he doesn't need to make a profit to make shoving a better move than folding. All he needs is to not lose more than he'd lose if he folded.

If the 2bb pot were divided according to EV after the first betting round, and if Hero has an equity of 0.32, then wouldn't Hero's fair share be 0.64bb? Isn't Hero giving up that 0.64bb if he folds?

I tried to explain my reasoning above. (It may not be answering OP's question).

Anyhow, thanks for your response.

Buzz

I suppose so.

I wasn't interested in showing a profit.

I was interested in the optimum way to play the hand.

Is folding better than raising?
I think the answer depends on how likely Villain is to fold to a raise. (Hero's fold equity).

When there are 2bb in the pot, what does Hero give up if he folds?

So long as Hero is still active in a hand, I think a percentage of the pot "belongs" to Hero. I think the percentage of the pot that "belongs" to Hero is Hero's "equity." For example, if Hero, at a particular point in the hand, has a 32% chance of winning the hand then Hero has an equity of 32%. If the players decided, at that particular point, to divide the pot, then Hero would be entitled to 32% of the pot.

Thus after the first betting round, when there are 2bb in the pot, if Hero's equity is 32%, then 0.64bb "belongs" to Hero.

Thus if Hero folds to a first round bet, he's giving up the 32% (0.64bb) of the pot that "belongs" to him.

By folding, Hero loses the 0.64bb that "belongs" to him.

If there is a way to play the hand so as to not lose that much, then, whether or not Hero manages to show a profit, I think that's a better way to play the hand.

Buzz

But the only way he can get that 0.64bb is if the hand were checked down. He doesn’t have that option. He has to call villain’s bet, raise or fold. The call or raise involves an additional risk investment on his part (losing chance = 68%), and this is accounted for in the EV analysis that ngFTW initially posted.

I think it fair to say that assigning a negative value to folding based on equity lost contravenes accepted theory.

Exactly. (I agree).

"Decision point" is a good description. I just had never heard "decision point" before and wanted to make sure we were on the same page.

I agree.

The comparison I thought (and think) I was making is "is shoving better or worse then folding?"

If shoving is better than folding, I think Hero's decision should be to shove.

I see what you are doing.

No offense intended, but I think it's flawed.

I think with an equity of 32%, Hero's "share" of the pot is 0.64bb. And if Hero folds, I think he gives up (loses) this 0.64bb that "belongs" to him.

I think the question should be, "Is there a way to play the hand so that Hero does better than losing 0.64bb?" I don't think Hero has to make a profit on the hand to do better. I think Hero merely needs to lose less than 0.64bb to do better. And I think your equation should reflect that. I think you should set the right side of your equation equal to -0.64, rather than 0.

I agree. But in the process of folding, Hero concedes his 32% share of the pot.

OK.

I follow.

I agree. And that is the problem.

Semantics. In my opinion, when Hero forfeits his 0.64bb share of the pot by folding, he loses that 0.64bb share.

Call it whatever you like, when Hero folds he gives up his 32% pot equity... and that's not nothing.

True. But before he folds, his net worth is 6.8bb plus the 32% equity he has in a 2bb pot. In other words, before he folds his net worth is 7.24bb. After he folds, his net worth is 6.8bb. At least that's how it seems to me.

I follow what you have written.

But I think Hero loses his pot equity if he folds. And I think Hero's pot equity is worth something. (When there are 2bb in the pot, I think 32% equity is worth 0.64bb).

Buzz

Thanks for your reply.

I don't know what "accepted" theory is. But you seem to know what you're talking about, so if you say so, fine. In that case I'm proposing an alternate theory. (I can't believe I'm the first one to think of it).

I do see that Hero doesn't have the option of taking 0.64bb out of the pot... well... I suppose if it were just Hero and Villain, after Villain bet the 1bb, Hero could bargain, saying he'd take 0.64bb to fold. Settlements like this are more familiar to me in the context of big money backgammon games where Player A doubles and Player B makes an offer. Might be a new concept for you.

Are we in agreement that Hero's fair share of a 2bb pot when Hero has 32% equity would be 0.64bb? Thus in this situation in a heads-up game when Villain bet 1bb, Hero could offer to concede 1.36bb from the 2bb pot in exchange for folding. I think that would be a fair settlement. Do you agree?

And if you do agree, then perhaps you can see that Hero is giving up his fair share of the pot if he folds. In other words, he's not giving up nothing.

Buzz

yes, when you fold you forfeit the value of your hand, you also forfeit the opportunity to see the completion of the board, to integrate new board cards with your hole cards to make your best 5 card poker hand, and you also forfeit the opportunity to show down your hand with your opponent to win or lose the pot.

this seems a fair description of folding. I doubt anyone (but perhaps you,Buzz) would argue/debate/question it.

i'm not debating and i'm pretty sure stamanhal isn't debating the hand you fold has a value.

what we are saying is that the value isn't relevant to determining the equilibrium folding frequency(minimum fold equity).

that what is relevant to making the comparison between folding and shoving, a comparison of expectations, is the expectation of folding and the expectation of shoving.

the expectation of folding is zero
and the expectation of shoving is dependent on your estimation of the opponents folding to shove frequency.


simply:

ev(fold) = 0 - is relevant to thread

"hero's fair share of a 2bb pot" = .64bb - not relevant to thread


accept this, or don't.

No. Why would villain accept this when he would profit 2bb for a fold and more than 1.36bb in EV if hero calls or raises.

Take a simple case where both hero and villain have stacks of 1bb. Villain either accepts the Buzz Bargain or bets the 1bb.

Buzz Bargain Accepted: Hero final stack = 1.64. Villain final stack= 2.36

Villain bets; Hero folds: Hero final stack = 1. Villain final stack = 3

Villain bets; Hero calls: Hero final stack (EV) =1+ 0.32*3 -0.68*1 = 1.28. Villain final stack (EV)= 4 - 1.28 = 2.72

From villain’s standpoint, the Buzz Bargain is a bad deal. If he refuses it, hero must fold or call and villain profits more than the Buzz Bargain provides for both of hero's options.

Thank you.

And if it has value, doesn't that imply it's not worth nothing?

And if it's not worth nothing, then what is it worth?

(I think it's worth 0.64bb when the pot is 2bb and your eqity is 32%).

OK, but I'm trying to get at the actual worth of your hand when you have an equity of 32% and there are 2bb in the pot. I think forfeiting the pleasures (and agonies, which you didn't mention) of playing your hand confuses the issue.

OK. I'm not arguing/debating/questioning it. I think what is relevant to the point I'm trying to make is simply "when you fold you forfeit the value of your hand." Moreover, I think we can assess a monetary value based on the equity of your hand and the size of the pot.

Thank you.

We seem agreed that the hand has value. If the hand has value, what is its value?

(I think at the point on the second betting round where Hero is confronted with a 1bb bet from Villain, with 2bb in the pot before that bet and with a 32% equity, Hero's hand is worth 0.64 bb. If Hero called Villain's 1bb, then the pot would be 4bb and then his hand would be worth .32*4=1.28bb).

You lost me. Sorry. I've read that ten times and I still don't understand what you're saying.

First, I don't know what "equilibrium folding frequency" is. I never before heard or saw that expression. (If I had to guess, I'd guess you mean the minimum fold equity Hero needs to expect from Villain in order to make shoving a better play than folding).

However, I think the value of Hero's hand IS relevant in determining the minimum fold equity Hero needs to expect from Villain in order to make shoving a better play than folding.

Agreed.

Agreed.

I'm not going to accept something I think is incorrect. No offense intended but I see relevancy, whether you're able to see it or not.

Simply: I think if Hero folds, he gives up 0.64bb, his "fair share" of the 2bb pot if there were a "fair settlement" after this flop. Therefore in order to make an alternative (to folding) plan of action favorable, Hero merely needs a way to play the hand so that he doesn't lose 0.64bb.

Buzz

Good point. Villain might not accept that settlement.

The reason Villain might accept a settlement is if the hand played out, Villain, even though a 68% favorite, would be expected to lose 32% of the time. Villain might rather take 1.36bb for a sure profit than take the risk of losing.

In this case, Hero is forced to call. (Because if Hero folds he ends up with 1 chip, but if he calls, he averages ending up with 1.28 chip).

But if we just give both hero and villain stacks of more than 1.125bb, then Hero has to fold to an all-in shove by Villain.

Yes. I see.

In this case, the fair settlement would be .28 chip for Hero and 1.72 chips for Villain.

Do you follow?

No matter. I can see where you're likely to go to go next. If we make the chip stacks their actual sizes, and if Villain knows his equity is 68%, then of course Hero would have to fold to the 1bb bet.

It's not the same as a fair settlement in backgammon because Villain can bet again if Hero calls.

I can see that Hero needs fold equity to shove. Villain cannot know he is a 68% favorite or of course he would not agree to the settlement.

I guess my supposition that Hero's hand has value is erroneous. In order for Hero's hand to have value, he would have to be all-in... or have very little remaining, in this case less than 0.125bb.

My notion that Hero's hand, with 32% equity, has value is incorrect, since Hero is not all-in (or very close to all-in).

Thanks statmanhal and thanks ngFTW.

I was wrong. My apologies.

Buzz

No need to apologize. It was a good discussion and hopefully a learning experience for those who followed. I know it was for me.