Hi, if I play heads up poker, I open 2.5BB from the SB,

If BB folds 20% of the time, call 60% and 3bet 20%.

What would be the formula to find out, for a hand which would fold to 3 bet, how much equity is needed ? I would not bother about the equity realization question since I would consider it a different concept.

I guess the equation would look something like this :

0.2 (-2) + 0.2 (1.5) + 0.6 (3x - 2(1-x) ) = 0 ??

Hi, it's (0.2)(-2) + (0.2)(1.5) + (0.6)(5x-2) = 0

Thankyou

You have the right idea but hero loses 2.5 with a villain 3 bet and hero fold, or if villain calls and hero loses. With a call and hero win, hero wins 3.5 since his 2.5 bet as SB is a raise of 2. If I did the arithmetic correctly, the needed equity is about 47.2%.

However, I'm sure you know that is exact only if the hand is checked down, which is unlikely, so this calculation only serves as an early indicator of where you stand. With 3 more rounds of betting, you have a greatly expanded game tree and things can change much.

20% x (-2.5) + 60% x 50.11% x 5 - 2.5 + 20% x 1 = -1.3



(vs that range you have 50.11% equity)

If you opened to 3x
20% x (-2.5) + 60% x 50.11% x 5 - 2.5 + 20% x 1 = -1.6

If you min raised
20% x (-2) + 60% x 50.11% x 4 - 2 + 20% x 1 = -1


I ignored pot odds because your stack actually increases by 1bb when he folds and not 1.5bb and I considered that you fold to all of his 3bets.

Now I will do the same except that we will continue vs his 3bets by ONLY CALLING 20% hands which is his 3bet %. Of course, ignoring all further actions and positional advantage.

20% x 80 % x (-2.5) + 60% x 50.11% x 5 - 2.5 + 20% x 1 = -1.12

And this one is according to pot odds:
20% x 80 % x (-2) + 60% x 50.11% x 5 - 2 + 20% x 1.5 = -0.52


Idk if this is correct, but i hope it helped.

Hi, to be honest I really think my equAtion in post number 2 is correct.

ME ?????

The first factor to be established in an EV analysis is the baseline stack. In most cases, people, including me, use the stack size just prior to the last decision. In this case it is hero’s stack just prior to his 2.5 bet. If villain fold hero wins the pot and his stack increases by 1.5.

You can use other baseline stacks but then you must be consistent when comparing EVs. For example, if you use the stack prior to any betting as the baseline, then a hero fold is not 0,but is the amount of the blind if hero is one of the blinds.

POST 2 Eqn.

YOU (0.2)(-2) + (0.2)(1.5) + (0.6)(3X-2(1-x)) = 0

ME (0.2)(-2.5) + (0.2)(1.5) + (0.6)((1.5+2)X-2.5(1-x)) = 0

If villain 3 bets, you have hero folding and losing 2 when he bet 2.5. Why?
If villain calls you have hero winning 3 or losing 2 when he actually wins 3.5 or loses 2.5. Why?

It appears you are use the stack baseline as that amt. at the beginning of the hand and that may be the difference but then you have a 1.5 win for a villain fold.

It's not clear from your question whether you are putting in an extra 2BB or 2.5BB, but either way the general formula is:

0.2*1.5 + 0.2*(-r) + 0.6*[(1-x)*(-r) + x*(1+r)] = 0

where r is the amount extra you are putting in the pot from the SB.

I measured the EV in terms of the effect on your current stack before the raise, eg:

* 20% chance you pick up the blinds (+1.5).
* 20% chance you get 3-bet and lose the extra r BB you put in (-r).
* The 60% of the time you are called and lose (ie: (1-x) of the time), you lose the extra r BB you put in (-r).
* The 60% of the time you are called and win (ie: x of the time), you win double the small blind amount plus the r BB your opponent put in to match you (1+r):

r=2
r = 2.5

For a raise of 2 (ie: raising to 2.5BB) from SB:

0.2*1.5 + 0.2*(-2) + 0.6*[(1-x)*(-2) + x*(1+2)] = 0

x = 13/30 = ~43.3%


For a raise of 2.5BB (ie: raising to 3BB) from SB:

0.2*1.5 + 0.2*(-2.5) + 0.6*[(1-x)*(-2.5) + x*(1+2.5)] = 0

x = 17/36 = ~47.2%


Juk

Hi, I meant SB raising to 2.5 by investing 2BB on top of his SB. (so 2 more).

However I notice my equation on post 2 gives the same result than your equation giving 43.3% as a result. So I guess it's a different way to do it. Interesting

Pot odds were bothering me for a while because of that.

According to "my way" you could argue that I have to fold to a pot size bet with a 40% equity hand because I don't have 50% equity which is needed for me to "breakeven".
( if hand was about to end right there )

Of course that I use pot odds in my decisions because folding a 40% equity hand in that scenario is making a favour to my opponent.
But in this case when I am playing heads up and I am making a pre flop plan I think that things should be a bit different.

To make my point easier to understand, lets say that the pre flop was the only street and after pre flop decisions the hand was over. Playability of hands isn't relevant and we are looking just at the raw equity.
If I menaged to make a"perfect" strategy that would be neutral EV for me according to the pot odds, after 10 000 hands I would go broke.
And it is not because of the rake. Rake doesn't exist in this example.
Let's say that we are oppening 100% of our range and our opponent is:
(Blinds 0,5/1)
1) folding 5%
2) calling 85%
3) 3betting 10% to 3x our open (we never 4bet him after)

1)When he folds, we win 1.5bb
2)When he calls, we have 51.32% equity in a 5bb pot which is 2.57bb and that is +EV because we risked only 2bb to get there and 2.57bb > 2bb, so we make +0.57bb there.
3) If he is 3betting 10% to 7.5bb, we need to call 5bb to win the pot of 10bb.
Pot odds: 5 : 15 = 33% equity needed to breakeven.
So when we have something in our 82.5% of top hands, we have 33% equity vs his range of 10% top hands and we can call because we are breaking even by calling.
You may argue that we are calling too much, but those are the correct numbers, I swear. Some of the hands in that range don't have those 33% equity to call but other hands have more than 33% and they make up for weaker ones and on average we have 33%.

So, in the first scenario we are winning 1.5bb, in the second 0.55bb x 80% and the 3rd scenario is neutral.
1.5 x 5% + 0.57 x 85% + 0 x 10% = 0.56bb profit each hand

But the "reality" is different
1) We both start each hand with 100bb and when we raise and he folds our stack becomes 101bb which is 1bb profit
2) We have 51.32% equity in a 5bb bot which is 2.57bb and that is bigger than 2.5bb that we have invested in it from our stack, which makes us a profit of 0.07bb
3) We have 33% equity in a 15bb pot which is 4.95bb. That is less than 7.5bb that we have invested in the pot so we are losing 2.55bb there.

1 x 5% + 0.07 x 85% - 2.55bb x 10% = -0.12bb loss each hand


So, after 100 hand, you will make 56bb "in EV" but you stack will actually decrease from 100bb to 88bb.

Amazing, right?


I'll be pleased to see you thoughts on this, statmanhal.
Since pot odds have been very problematic to me, I came up with some "solutions" to this and I can share them if you are interested in this.
Unless this is an old "issue" that's already been discussed and known in the poker community. I'm a bit new here, so...

It's the same equation if you simplify the bit in square brackets:

(1-x)*(-2) + x*(1+2) = -2 + 2x + 3x = 5x-2

Juk

Sorry but I can’t follow your example because numbers are inconsistent, e.g. your” risked 2BB” in one sentence and in another you have “2.5BB that we have invested” and some other values appear to be typos or come out of nowhere – 57.32%, 82.5%. If the 2.5 includes the 0.5 SB blind posting then you may be inconsistent in the baseline stack you are using.


In any case, let me state that if the EV analysis is done correctly, using a consistent baseline, (something I suspect you are not doing} then it impossible for you to correctly state that you can be +EV but still be a loser. This is so because the very definition of EV is long term average profit/loss, so how can a long term profit end up being a long term loss?

If you review your posting and correct any typos, state what baseline you are using for EV analysis, and state exactly how much you and villain bet, I’ll try to review your analysis if you still claim you can be a +EV loser.

BTW are you OP using another name?

I'm not OP

I think I didn't make any mistakes, maybe my point is way off.
Anyway, I'll try a different example.

Blinds 0.5/1.
We are opening to 2.5x (SB + additional 2 = 2.5)
BB only calls or folds and when he calls we lose every single time.
This is a bit different but I'm sure you will understand.

So how much do we need him to fold to "breakeven"?
We risk 2 to win 1.5 so... 2 : ( 2 + 1.5 ) = 57.14%

Let's say that he folds 57.14%
These are the final stacks after each scenario:

.....................................US,,,,,,,,,,, VILLAIN
STARTING STACKS:,,,,,,,,,,,100,,,,,,,,,,100
HE FOLDS(57%):,,,,,,,,,,,,,,,101,,,,,,,,,,99
HE CALLS(43%):,,,,,,,,,,,,,,,97.5,,,,,,,,102.5
AFTER 100 HANDS:,,,,,,,,,,,,49.5,,,,,,,,,150.5


If it is "breakeven" and we have to base our strategy around it, why are we down 50.5bb ?

Your numbers are still inconsistent. First you say we are raising 2.5xBB, then that we are risking 2BB. If I assume that you meant a 2x raise, then needing villain to fold 57.14% of the time is correct.

However, you then go back to saying our stack after we raise and lose will be 97.5. Are we raising to 2.5BB or 2BB? Also, when villain folds our new stack will be 101.5, not 101.

Wtf, bro? I dont understand you.
The pot is 1.5 when the blinds are posted.
We are in the SB and when we post our bind we have 99.5bb stack.
If he folds, we win the pot of 1.5bb, so our stack is 99.5bb + 1.5bb = 101bb and not 101.5bb.


I don't undersand why you say that we are actually raising 2x ?
I even said it in the brackets ( 2+ 0.5 = 2.5) because i thought this could be problematic.


When we open raise from CO to $3 in a $0.5/$1 game, it's a 3x raise, right?
I haven't played much HU poker but when we do that same open raise (to $3) from SB attacking the BB it still counts as 3x ? Never seen anyone ever saying it's a 2.5x?


Sorry if this seemed a bit agressive, I didn't want to be harsh, you seem like a good guy.


I want to say what is happening to the money and when we are playing "correctly" according to pot odds, we could still be losing money because after our "breakeven play" our final stack will be lower than our starting stack.

We are raising to total $2.5 from SB in a heads up $0.5/$1 game which is leaving us with $97.5 behind.

When the blinds are posted, the pot is $1.5 dollars and according to the pot odds when we raise to $2.5 dollars ($1.5 more for our opponent to cal), we are risking $2 to win $1.5, right? ****correct me where I'm wrong on this part.

I calculated that we need him to fold ~57% of the time the time to breakeven
The other 43% in this "game" he calls and wins 100% of the time but that doesn't matter because we are "breaking even"...



,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,US,,,,,,,,,, ,,,,VILLAIN,,,,,,,,,POT,,,,,,,TOTAL
STARTING STACKS,,,,,,,,,,,,,100,,,,,,,,,,,,,100,,,,,,,,,,,, ,,,0,,,,,,,,,,200
BLINDS ARE POSTED,,,,,,,,,,,99.5,,,,,,,,,,,,99,,,,,,,,,,,,,,, ,1.5,,,,,,,,200
WE RAISE,,,,,,,,,,,,,,,,,,,,,,,,,97.5,,,,,,,,,,,,99,, ,,,,,,,,,,,,,,3.5,,,,,,,,200
HE FOLDS,,,,,,,,,,,,,,,,,,,,,,,,,101,,,,,,,,,,,,,99,, ,,,,,,,,,,,,,,,0,,,,,,,,,,200
HE CALLS,,,,,,,,,,,,,,,,,,,,,,,,,,97.5,,,,,,,,,,,97.5 ,,,,,,,,,,,,,,,5,,,,,,,,,,200
HE WINS AFTER A CALL,,,,,,,97.5,,,,,,,,,,,,102.5,,,,,,,,,,,,,0,,,, ,,,,,,200


So, if "HE FOLDS" 57% our final stack is $101 which is 1$ more from our starting stack.
If "HE WINS AFTER A CALL" our final stack is $97.5 which is $2.5 less than our starting stack.

So, we win $1 57% of the time
and lose $2.5 43% of the time

1 x 57% - 2.5 x 43% = -0.505

That's the average loss of $0.505 per hand which is $50.5 loss per 100 hands



I hope that you understand me now
With this I wanted to show that some people misinterpret "breakeven".

I was just being dumb and didn't realize we were in the SB, sorry and ignore my last post, then.


When calculating EV we have to consider a reference point. The majority of people use the current stacks as a reference point to normalize the EV of folding to 0. We can also reference the stacks at the start of the hand.

Notice that the average loss you calculated is almost exactly equal to our SB. In fact if you hadn't rounded the calculation would be exactly equal to our SB. We are breaking even in the sense that we don't lose any additional money from our decisions after having posted the SB. The SB itself is a sunk cost and irrelevant. You're right that we're not breaking even on the hand overall, but that doesn't matter if we consider the .5BB as already "lost" after we've posted it.

I spent some time on this and rather than go through all the detail, I’ll summarize as follows:

1. OP’s final criterion is to maintain his initial stack size of 100, assuming that is the size prior to his blind posting.

2. But, he develops a required fold equity based on maintaining the stack size prior to his raise of 2.5, and he mistakenly (IMO) uses 2 as his risk amount

3. If you use 2,5 for the risk amount, the required fold equity is 2.5/4.0 = 0.625 and the expected stack size after the bet is 99.5, consistent with the stack size prior to the bet.

4. If you use the criterion of maintaining the initial stack of 100, the required fold equity is given by fe*(101)+(1-fe)*97 = 100; fe = 0.75

5. So OP uses one criterion to develop a required fold equity and then applies that to a different criterion, which seems to reveal an inconsistency. This is a mistake.

I have to admit, I don’t think I have ever looked at the criterion of the blind maintaining his initial stack so Marko’s question is, for me at least, quite useful. Whether it is a good criterion, is another question.

Edit; Just saw brownies comment and I think the above supports that

Hi, doesnt he have to fold 2.5 : 3.5 = 71.43% for us to really "breakeven" meaning we end up with the final stack of $100 ?

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,US,,,,,,,,,, ,,,,VILLAIN,,,,,,,,,POT,,,,,,,TOTAL
STARTING STACKS,,,,,,,,,,,,,100,,,,,,,,,,,,,100,,,,,,,,,,,, ,,,0,,,,,,,,,,200
BLINDS ARE POSTED,,,,,,,,,,,99.5,,,,,,,,,,,,99,,,,,,,,,,,,,,, ,1.5,,,,,,,,200
WE RAISE,,,,,,,,,,,,,,,,,,,,,,,,,97.5,,,,,,,,,,,,99,, ,,,,,,,,,,,,,,3.5,,,,,,,,200
HE FOLDS(71.43%),,,,,,,,,,,,,101,,,,,,,,,,,,,99,, ,,,,,,,,,,,,,,,0,,,,,,,,,,200
HE CALLS(28.%),,,,,,,,,,,,,,,,,97.5,,,,,,,,,,,97.5 ,,,,,,,,,,,,,,,5,,,,,,,,,,200
HE WINS AFTER A CALL,,,,,,,97.5,,,,,,,,,,,,102.5,,,,,,,,,,,,,0,,,, ,,,,,,,200

1 x 0,7143 - 2,5 x 0,2857 = 0,00005 = ~ 0

Meanwhile I realized that this was for 3x open
"4. If you use the criterion of maintaining the initial stack of 100, the required fold equity is given by fe*(101)+(1-fe)*97 = 100; fe = 0.75"


!!!Something else to consider which changed my way of thinking about poker.

HU, blinds $0.5/$1
We open to $2.5, BB 3bets us to $7.5 and we call.
He has two red AA, and we have JT suited.

The flop comes Q92 rainbow and we have OESD and no backdoor flush draw.
Now we don't care about our ranges, let's just look at these 2 exact hands.
We have 34.24% equity and he has 65.76% equity.
If we had a different hand and we plugged it in vs his range and got this number of 34.24%, it would be the same (let's put playability, positions and ranges aside).

1)The pot is $15 and he cbets for half pot ($7.5) and he shows us his hand.
We can either call or raise this but let's just concentrate on the calling option.
According to pot odds, we are breaking even by calling a half pot bet if we have minumum 25% equity. 7.5 : ( 15 + 7.5 + 7.5 ) = 0.25 = 25%

So, in this case we have more than enough equity to call ( 34.24% > 25% ) so we aren't breaking even, we are actually making money by calling. Or maybe not ?

To make this simple, let's say that after we call that flop bet, the hand ends right there.

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,US,,,,,,,,,, ,,,VILLAIN,,,,,,,,,POT,,,,,,,TOTAL
STARTING STACKS,,,,,,,,,,,,,100,,,,,,,,,,,,,100,,,,,,,,,,,, ,,,,,0,,,,,,,,,,200
BLINDS ARE POSTED,,,,,,,,,,,99.5,,,,,,,,,,,,99,,,,,,,,,,,,,,, ,,,1.5,,,,,,,,200
WE RAISE,,,,,,,,,,,,,,,,,,,,,,,,,97.5,,,,,,,,,,,,99,, ,,,,,,,,,,,,,,3.5,,,,,,,,200
HE 3BETS,,,,,,,,,,,,,,,,,,,,,,,,,97.5,,,,,,,,,,,,92.5 ,,,,,,,,,,,,,,10,,,,,,,,,,200
WE CALL,,,,,,,,,,,,,,,,,,,,,,,,,,92.5,,,,,,,,,,,92.5 ,,,,,,,,,,,,,,15,,,,,,,,,,200
HE CBETS,,,,,,,,,,,,,,,,,,,,,,,,92.5,,,,,,,,,,,,,85,, ,,,,,,,,,,,,,,22.5,,,,,,,,200
WE CALL,,,,,,,,,,,,,,,,,,,,,,,,,,85,,,,,,,,,,,,,,,85, ,,,,,,,,,,,,,,,,30,,,,,,,,,,200
WE WIN(34.34%),,,,,,,,,,,,,,115,,,,,,,,,,,,,,85,,,,,, ,,,,,,,,,,,,0,,,,,,,,,,200
WE LOSE(65.76%),,,,,,,,,,,,,85,,,,,,,,,,,,,,,115,,,,, ,,,,,,,,,,,,0,,,,,,,,,,200
AFTER 100 HANDS,,,,,,,,,,,,,-373,,,,,,,,,,,,573,,,,,,,,,,,,,,,0,,,,,,,,,,,,200

So, we either win or lose $15.
34.24% x 15 - 65.76% x 15 = - 4.73 which is a loss of $4.73 per every hand

After 100 of these hands, after 100 times that we are put into this spot, we are going to be down 4.73 x 100 = $473 which is shockingly massive.

So, if we are losing a **** tone of money there, why would we even call that flop bet??
If you are confused by this, just continue reading and I will try to explain the reality of this spot as best as I can.
It's not that we have position in the hand or implied odds that are going to make us money because we are making a "profitable play".
We legit have the worse hand in a heads up pot where both players invested the same amount of money and therefore, the only player who is making money here is the player in the BB who has the best hand at the moment.
So, if you are maybe wondering, if BB checks here or uses a different bet size, he is going to make money anyway and we are going to be losing money whatever we do.

2)Let's make a different scenario.
We have the same hands, same flop and same action unless here BB cbets for $30 which is an 2x pot overbet.

We have 34.24% equity in the pot.
Versus this overbet, according to pot odds we need to call $30 to win the previous pot of $15 + his flop bet of $30. 30 : ( 30 + 15 + 30) = 30 : 75 = 0.4 = 40%
We need 40% equity here to breakeven.

If we want to calculate our average loss per hand here:
34.24% x 37.5 - 65.76% x 37.5 = - 11.82 , we lose $11.82 each hand
(after we call the pot size is $75 and both players invested half of that which is 37.5, we either win or lose $37.5 at the end of the hand )


3)And now, let's see what happens if we decide to end this torture on the flop and fold right away ( even to to BB's bet size that is "profitable" for us to call, which is bet size that requires us to have at least 34.34% equity to call according to pot odds, e.g. half pot bet).
When we fold, his bet size doesn't matter because we lose the same amount of money regardless, anyway...

If we fold to his cbet
,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,US,,,,,,,,,,, VILLAIN,,,,,,,,,,,,POT,,,,,,,,TOTAL
STARTING STACK,,,,,,,,,,,,,,100,,,,,,,,,,,,,,100,,,,,,,,,,, ,,,,0,,,,,,,,,,,,200
HE CBETS,,,,,,,,,,,,,,,,,,,,,,,,92.5,,,,,,,,,,,,,85,, ,,,,,,,,,,,,,,22.5,,,,,,,,200
WE FOLD,,,,,,,,,,,,,,,,,,,,,,,,,92.5,,,,,,,,,,,,,107. 5,,,,,,,,,,,,,0,,,,,,,,,,,200

we will be down $7.5 every hand

4)Last example finally, to make this comparison of average loss in every scenario look good.
He cbets the flop pot of $15 with $16.31

For us to "breakeven", 16.31 : ( 16.31 + 15 + 16.31 ) = 16.31 : 47.62 = 0.3425 = ~0.3424 = 34.24% equity is needed.

,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,US,,,,,,,,,,, VILLAIN,,,,,,,,,,,,POT,,,,,,,,TOTAL
STARTING STACK,,,,,,,,,,,,,,100,,,,,,,,,,,,,,100,,,,,,,,,,, ,,,,0,,,,,,,,,,,,200
HE CBETS,,,,,,,,,,,,,,,,,,,,,,,,92.5,,,,,,,,,,,,,76.1 9,,,,,,,,,,31.31,,,,,,,,,200
WE CALL,,,,,,,,,,,,,,,,,,,,,,,,,76.19,,,,,,,,,,,,76.1 9,,,,,,,,,,,47.62,,,,,,,,200

We either lose or win $23.81 and we have the same old 34.34% equity.

23.81 x 0.3424 - 23.81 x 0.6576 = - 7.5 , which is a loss of $7.5 every hand on average.

SUMMARY
1)25% equity needed to call and we had 34.24% --> average loss of $4.73
2)40% equity needed to call and we had 34.24% --> average loss of $11.28
3)We decided to fold,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,--> average loss of $7.5
4)34.24% equity is needed to call, we had 32.24$--> average loss of $7.5

CONCLUSION:
-In the scenario 4), it is obvious that both calling and folding would bring us the same result.
-In the scenario 1) pot odds are forcing us to call because we are above the "breakeven limit". They are forcing us to call because not because we are making money by calling. We are just losing less by calling, if we folded we would lose even more and for that reason, we have to call.

WHAT'S THE SOLUTION, WE ARE ALWAYS LOSING ?
-If we are an underdog, we need to orientate by the pot odds and try to cut our losses, try to lose as less as we can. By doing that, our opponent is going to win the minimum from us.
-In the long run, both us and our opponent will have sometimes AA and sometimes JT in this spot. What we need to profit is to win more money the times we have AA and lose less money with JT (in comparison to our opponent)


Ty if you made it to here, this was a bit too long and I was explaining it in more details so even a dummy like me could understand it.
I have some strategy theories beyond this but I didn't work them out completely. It all came up because I was trying to find my own way to play profitably as opposed to mostly listening to others how to play.

Good luck with your theories.