Extra Credit Math Question

<@spoonitnow> Okay here's an extra credit problem for you guys
<@spoonitnow> It's folded to you in the SB with QQ. What is the smallest starting stack for an open shove to possibly be -EV?

200 bbs, or whichever stack approximates a calling range of:

Text results appended to pokerstove.txt

441,774,432 games 0.005 secs 88,354,886,400 games/sec

Board:
Dead:

equity win tie pots won pots tied
Hand 0: 49.234% 47.81% 01.42% 211231392 6272802.00 { QQ }
Hand 1: 50.766% 49.35% 01.42% 217997436 6272802.00 { QQ+, AJs+, KQs, AQo+ }

or

Text results appended to pokerstove.txt

339,036,192 games 0.005 secs 67,807,238,400 games/sec

Board:
Dead:

equity win tie pots won pots tied
Hand 0: 43.236% 41.53% 01.71% 140797524 5787306.00 { QQ }
Hand 1: 56.764% 55.06% 01.71% 186664056 5787306.00 { QQ+, AQs+, KQs, AKo }

Originally Posted by a500lbgorilla

200 bbs, or whichever stack approximates a calling range of:

Text results appended to pokerstove.txt

441,774,432 games 0.005 secs 88,354,886,400 games/sec

Board:
Dead:

equity win tie pots won pots tied
Hand 0: 49.234% 47.81% 01.42% 211231392 6272802.00 { QQ }
Hand 1: 50.766% 49.35% 01.42% 217997436 6272802.00 { QQ+, AJs+, KQs, AQo+ }

or

Text results appended to pokerstove.txt

339,036,192 games 0.005 secs 67,807,238,400 games/sec

Board:
Dead:

equity win tie pots won pots tied
Hand 0: 43.236% 41.53% 01.71% 140797524 5787306.00 { QQ }
Hand 1: 56.764% 55.06% 01.71% 186664056 5787306.00 { QQ+, AQs+, KQs, AKo }

Way off.

In what way? You gave no details about the opponent, so the first step is to make the assumption of their calling range. Am I incorrect in assuming that the shorter the stack, the wider the range, and the wider the range the more profitable the move? Thus, as the opponents stack increases, his calling range shrinks, thusly moving our open QQ push to the -EV column?

I was completely expecting you to quote my post and say "nope." or "wrong." so I answered with the hope that you'd discuss the further details with me.

If we assume the opponent is playing perfectly according to the Sklansky-Chubukov rankings, it's 240BB
that's the number where if he only calls AA, KK you lose money

165bb

1326 possible hand combos
12 hands we are big dogs to (AA+KK)

so, with stacks of 165bb:

1314 times we win 1.5bb for total of 1971bb
12 times we lose 165bb for total of 1980bb

This assumes villain folds all but AA and KK obv.

Howzat? I hate maths rounda!

Originally Posted by spoonitnow

<@spoonitnow> Okay here's an extra credit problem for you guys
<@spoonitnow> It's folded to you in the SB with QQ. What is the smallest starting stack for an open shove to be -EV?

I'm going to try this without the pokerstove, without any tools, as I am at work and have nothing handy...

Let's see... I'm in SB, and have QQ. There is probably a mathmatical answer to this question, but without any reads or other table information on the big blind player. This cannot be determined as you have no idea what their calling range would be. But let me try to figure it out...

QQ beats all hands pre-flop except AA or KK.

AK is borderline, but it needs to develop.
6 ways to make AA
6 ways to make KK
1 ways to make QQ

Thus 13 hands beat you. Out of 1325 possible hands (1326 - yours) that is roughly 1/100.

So you will lose your stack roughly 1 time out of 100 (assuming BB always calls). Thus a stack of less than 100 BB would mean the call is -EV?

Am I close?

Originally Posted by iopq

If we assume the opponent is playing perfectly according to the Sklansky-Chubukov rankings, it's 240BB
that's the number where if he only calls AA, KK you lose money

We don't have to make this assumption for the answer to be the same, doucy?

I forgot were holding 2 cards, so:

1225 possible hands
1213X1.5bb=1819bb (win)
12X152bb=1824bb

so 152bb?

Originally Posted by a500lbgorilla

In what way? You gave no details about the opponent, so the first step is to make the assumption of their calling range. Am I incorrect in assuming that the shorter the stack, the wider the range, and the wider the range the more profitable the move? Thus, as the opponents stack increases, his calling range shrinks, thusly moving our open QQ push to the -EV column?

The part in italics is totally irrelevant - you don't need to know anything about your opponent to answer the question correctly. The part in the bold is correct, but you're thinking about it in a more general way than answering what the question specifically asks. The underlined part is true, but if you ask yourself why that's true (in context of the question asked) you'll be close to answering what your opponent's range is and closer to answering the question.

Also I reworded the question in the OP a little so that it becomes more noticeable what we're after here.

I was completely expecting you to quote my post and say "nope." or "wrong." so I answered with the hope that you'd discuss the further details with me.

Yeah but it really looks like you didn't even try so what do you expect?

Originally Posted by iopq

If we assume the opponent is playing perfectly according to the Sklansky-Chubukov rankings, it's 240BB
that's the number where if he only calls AA, KK you lose money

But QQ isn't a -EV shove to a calling range of only KK+

The calling range can be as wide as I posted earlier and it'd still be -EV

Originally Posted by spoonitnow

you don't need to know anything about your opponent to answer the question correctly.

Painfully, incorrect. Pretend we're playing a drunk FTR game, big blind is fatb.

Originally Posted by a500lbgorilla

Originally Posted by spoonitnow

you don't need to know anything about your opponent to answer the question correctly.

Painfully, incorrect. Pretend we're playing a drunk FTR game, big blind is fatb.

This reply shows that you don't understand what the solution means, which is fine since that means you're here to learn, which is the point. When I post the answer and explanation later, you'll be like "ohhhhhh".

Originally Posted by a500lbgorilla

Originally Posted by iopq

If we assume the opponent is playing perfectly according to the Sklansky-Chubukov rankings, it's 240BB
that's the number where if he only calls AA, KK you lose money

But QQ isn't a -EV shove to a calling range of only KK+

The calling range can be as wide as I posted earlier and it'd still be -EV

Hint: When solving this, you should never be using the range KK+.

How about it's pretty much always -EV unless you have ~10 BB because you can gain more value out of weaker hands with a standard raise and play poker style.

If you always open shove QQ in the SB when it's folded around to you, you'll be making less money than if you play poker with it unless you're a short stacker that people think are loose enough to shove random shit in the SB.

Clar

Why are we trying to find the least possible EV way to approach this? Why aren't we trying to find the most +EV stack to do this to?

This topic shouldn't be in the beginners section if people who have an in-depth poker knowledge aren't able to answer it.

going to guess 88.4

Originally Posted by Clar17y

How about it's pretty much always -EV unless you have ~10 BB because you can gain more value out of weaker hands with a standard raise and play poker style.

While this is false (a play isn't necessarily -EV just because there is a play with a higher EV available), it's also completely irrelevant. The point here is to learn a type of calculation that has implications all over the place, not to solve No-Limit Hold'em.

Originally Posted by wellrounded08

Why are we trying to find the least possible EV way to approach this? Why aren't we trying to find the most +EV stack to do this to?

To the first question, once we find the bounds for the stacks for this to be a -EV play then we know when it is a +EV play. To the second question, this is actually just a more general form of what we are doing with the question I asked, and if you can't find the answer to the question in the OP then you won't be able to solve this either.

Originally Posted by bearing

This topic shouldn't be in the beginners section if people who have an in-depth poker knowledge aren't able to answer it.

going to guess 88.4

Nice attitude towards learning you have there.

Originally Posted by spoonitnow

Originally Posted by a500lbgorilla

Originally Posted by spoonitnow

you don't need to know anything about your opponent to answer the question correctly.

Painfully, incorrect. Pretend we're playing a drunk FTR game, big blind is fatb.

This reply shows that you don't understand what the solution means, which is fine since that means you're here to learn, which is the point. When I post the answer and explanation later, you'll be like "ohhhhhh".

Originally Posted by a500lbgorilla

Originally Posted by iopq

If we assume the opponent is playing perfectly according to the Sklansky-Chubukov rankings, it's 240BB
that's the number where if he only calls AA, KK you lose money

But QQ isn't a -EV shove to a calling range of only KK+

The calling range can be as wide as I posted earlier and it'd still be -EV

Hint: When solving this, you should never be using the range KK+.

So the answer is completely independent of opponent's decision making?

Originally Posted by a500lbgorilla

So the answer is completely independent of opponent's decision making?

Yes! We just want to find the lowest stack possible when it could be -EV to shove.

For example, it couldn't be -EV to shove with 50bb stacks, but it could be -EV to shove with 100000bb stacks.

Edit: Also this is going somewhere and isn't just a random calculation that nobody really should have to know how to do. But we've got to be able to do this in a basic scenario (the question in OP) before we expand on it.

If we want to find the smallest stack where pushing could be -EV then we should assume that villain has the calling range that hurts QQ most. If stacks are deep this is just the hands that have 50%+ equity against QQ, i.e. QQ+. So, assuming that the stack size S is large enough that AKs would make a mistake by calling (S > 13BB or so), our EV is never less than 1/1225 of

1212 * 1,5BB + 13*(2S*eq - S +0,5BB)

where eq is the equity of QQ against the range QQ+, which pokerstove should be able to tell you. If we set EV = 0 then solving for S

S = 1824,5BB / 13*(1 - 2*eq)

Well yeah I didn't mean to say it would be -EV I just mean when would you actually use this "theory" if you were sitting down at a cash game and always had 100BB on you?

But since you feel that's irrelevant let's just ponder the question some more...

It's independent of villains decision making. I can't see how or why but ok, that means that we can't assign him a range since we aren't allowed to use his thought process.

So we shove x BBs and if villain folds we win 1BB.
If he calls we're risking x BB to win x BB (neglecting rake) and so we win ( QQ vs villain's range ) % of the time and lose the other times.

How can we not include villain's decision making?

The less BB we have, the more inclined villain will be to call . If we start making our stack bigger and bigger the villain will call less and less. Eventually when chip stack gets high enough villain will only call with AA.

So:

1226 combinations, we hold 1.
1219 times we win 1 BB ( or is it 1.5 since our SB counts as dead money?)
6 times (When he has AA) we risk x bb with 18.45% equity.

6x * 0.8155 = 1219 x 1.5
x ~ 373.7BB

That is the definitive number of BBs where it is DEFINITELY -EV to shove since we know that AA is calling if he has common sense.

If we add KK to his calling range then obv that number gets approx halved.

Clar

*edited for clarity*

Originally Posted by bearing

This topic shouldn't be in the beginners section if people who have an in-depth poker knowledge aren't able to answer it.

going to guess 88.4

No doubt. I'm completely lost.

Anything more than 239bbs

Someone please break this down for the noobs. I can handle the math, I just need to understand the decision process.

I'm assuming a calling range of QQ+, AK:
29 combos of this range in 1225 possible hands for villain, and we're shoving x big blinds.

1196 times we're not called and we win 1.5bb = 1794bb.

29 times we're called and we're 40.2% to win a pot of 2x bb (or we could say we're 59.8% to lose this)

So 29*(2x)*59.8% = 1794, so x= ~52bb?
I'm not sure of my use of (2x) in this equation, since we've only lost xbb - obviously here the answer would then be 104bb...

1326 total starting hands, you have exactly 1 of them. So, if the opponent could be dealt some magical hand of jokers which has 100% equity once in 1326 hands then we win 1 bb 1325 times and lose our stack once. Thus, we would win 1 bb for 1325/1326 cases and lose every thing for one. For it to be neutral EV, we'd need a stack of 1325bbs.

13 of 1325 times he calls and you lose 79.299% of your stack (QQ+ calling range) and 1312 times you win 1 bb.

(1312*1)BB/1325 - 13*(S*.79299 - S*.20701)/1325= 0

Stack = 172 big blinds for a neutral EV shove

As you increase his calling range, you decrease the size of the free blinds we win when he folds, but you also increase the amount of your starting stack which you win back in showdown pots. But since these would include hands like AKs which QQ would have a preflop edge on, I'm going to assume that added them into the range will only increase the size of the stack required for a neutral EV push.

This is all a little rough.
When opponent folds, you win 1 bb. When opponent calls, you win S*W% and lose S*L% for a net of S*(W%-L%). L%=100%-W% so net=S*(2W%-100%). Your W% is a function of opponents Call%, so as Call% goes up, W% goes up. And when opponents Call% reaches about 5%, W% reaches 50% and EV is always + at that point no matter how big your stack.
When opponent calls about 1/110, W% is about 18% and net = -.64*S and total won is 109-.64*S.
So when opponents call range is KK+, then break even stack size S = 109/.64 = 170 bbs.
If opponent calls twice as often about 1/55 (i.e. adds AK to call range), W% goes up to about 40%, net = -.2*S, and total won is 54 - .2*S. Then break even S = 54/.2 = 270 bbs.
So 170 bbs is the lowest stack size you can shove and still be -EV. Sound about right?

Hmm... I'm thinking about this further at work and wondering if it is a figure relative to the villians stack... instead of a 'magical figure' that I'm trying to inspire the math in me head to support...

Could it be relative to having the villian covered causing him to not want to call with anything other than AA/KK, hmm... let me ponder that a bit...

Things to consider.

Even against AA and KK we have a small amount of equity in the pot. We'll draw out on these two hands 1 out of 5 times.

We know that our opponent will always call with AA, KK and AKo+. Honestly though, I don't think putting our opponent on a range has anything to do with this problem though.

We win 1.5 BB Every time we shove and our opponent folds.
We stand the chance of losing our entire stack every tome we shove and our opponent calls
We stand the chance of being dominated every time we shove and our opponent has KK, and AA.
We stand the chance of being coinflipped against when our opponent has AK.

I don't have the math right now but this is what I'm thinking as I start working on it.

Originally Posted by spoonitnow

It's folded to you in the SB with QQ. What is the smallest starting stack for an open shove to possibly be -EV?

Holy crap I've got you guys posting and trying and whatnot. This is great. I'm going to go ahead and post how to figure this out and why it works. In a little while I'll post the next question that will build on the ideas thrown around in the answer to this question. But for now:

The way we decide if this play is possibly -EV is to see what EV the play has if Villain knows our cards and plays perfectly accordingly.

With that assumption in mind, now we decide the criteria for Villain to call so that we can decide on his calling range. Let's call S our starting stack sizes, then the pot odds Villain will be getting to call are S+1:S-1, which means each hand in his calling range for the purposes of answering the question must have an equity greater than or equal to (S-1)/(S+1 + S-1) = (S-1)/2S. Clearly QQ+ will be in his range then, so we need to check the next most likely hand, AKs.

AKs has 0.46049 equity, and we can solve for the greatest stack size where Villain can call with AKs by saying 0.46049 = (S-1)/2S and solving for S: 0.46049 = (S-1)/2S, 0.92098S = S-1, -0.07902S = -1, S = 12.7. A small amount of investigation will show that our answer will be greater, so we have our calling range: QQ+.

Now we have to decide how often he holds QQ+. Since we have taken two Queens out of the deck, that leaves 50 cards, or 50*49/2 = 1225 possible combinations. Six of those are AA, six of those are KK, and one of those are QQ, so the chance of Villain calling is 13/1225 (and the chance of Villain folding is 1212/1225).

The rest is a basic equity calculation:

Let S be our starting stack before posting the blinds. For large stacks, Villain's best calling range is QQ+. Our value comes from three scenarios:

Villain folds 1212/1225 * 1.5 = 1.4840816
Villain calls, we win 13/1225 * 0.20701 * (S+1) = 0.0021968S + 0.0021968
Villain calls, we lose 13/1225 * 0.79299 * -(S-0.5) = -0.0084154S + 0.0042077

Then the break even point for when the shove has 0 value is:

0 = 1.4840816 + 0.0021968S + 0.0021968 + -0.0084154S + 0.0042077
0 = 1.4904861 + -0.0062186S
S = 239.68

So an open shove with QQ can be -EV with any starting stacks greater than 239.68. But that's too trivial, so let's try a harder one building on what you've learned:

Question 2: It's folded to you in the SB with some hand randomly chosen from {AQ+, TT+}. Your stacks are 15bb before the blinds are posted, and you raise to 5x the big blind with the hand that was randomly chosen from the given range. Villain can then fold, or go all-in for 15bb. Then you can call or fold. What range should Hero call with to maximize his value against a Villain who knows his opening raising range (but Villain doesn't know that we know he knows that)?

op knows that we're opening AQ+ TT+?

Ok I kinda understand what you mean now.

In question 2 we gotta take the same approach. You're saying we get a random hand from {AQ+, TT+} and villain knows that this is our range. So presumably if he's playing correctly he will go all-in with anything that has >50% equity against that range.

Then we only need to call 10bb to win 20bb so we need 33% against his range to call.

If villain is playing perfectly, then he's shoving everything that is >=50% equity against {TT+, AQs+, AQo+}.
Not quite 50% because he has 1bb invested and only has 14bb left. So instead of 50% it's more like 46.7%.

What has >46.7% equity against our opening range?
JJ, QQ, KK, AA, AKs, AKo.
(TT only has 35.8%)

What hands in our range have 33% equity or higher against {JJ+, AKs, AKo}?
TT, JJ, QQ, KK, AA, AKo, AKs

So we call with {TT+, AKo, AKs}. amirite?

Clar

Originally Posted by spoonitnow

Originally Posted by iopq

If we assume the opponent is playing perfectly according to the Sklansky-Chubukov rankings, it's 240BB
that's the number where if he only calls AA, KK you lose money

We don't have to make this assumption for the answer to be the same, doucy?

because you said "smallest"

Originally Posted by a500lbgorilla

op knows that we're opening AQ+ TT+?

Yeah but I'm going to break it down one more time for clarification.

1. We're opening a random hand from {AQ+, TT+}.
2. Villain knows #1.
3. Villain doesn't know we know #2.

Originally Posted by spoonitnow

Originally Posted by a500lbgorilla

Originally Posted by spoonitnow

you don't need to know anything about your opponent to answer the question correctly.

Painfully, incorrect. Pretend we're playing a drunk FTR game, big blind is fatb.

This reply shows that you don't understand what the solution means, which is fine since that means you're here to learn, which is the point. When I post the answer and explanation later, you'll be like "ohhhhhh".

Originally Posted by a500lbgorilla

Originally Posted by iopq

If we assume the opponent is playing perfectly according to the Sklansky-Chubukov rankings, it's 240BB
that's the number where if he only calls AA, KK you lose money

But QQ isn't a -EV shove to a calling range of only KK+

The calling range can be as wide as I posted earlier and it'd still be -EV

Hint: When solving this, you should never be using the range KK+.

You're right, it's QQ+

If he assumes we're never folding since we're getting 2:1, he should shove with a range that doesn't lay us appropriate pot odds. QQ+ (which has 68.18% equity against our opening range).

And we should call with a range of hands that then do get the appropriate pot odds or TT+,AQs+ (which has 34.76% equity against QQ+)

Originally Posted by spoonitnow

AKs has 0.46049 equity, and we can solve for the greatest stack size where Villain can call with AKs by saying 0.46049 = (S-1)/2S and solving for S: 0.46049 = (S-1)/2S, 0.92098S = S-1, -0.07902S = -1, S = 12.7.

Okay i'm clearly missing something from the process here. MY lunchbreak has basically gone whilst i figure this shit out, but the above has me beaten. How on earth have u gotten to the figure 12.7 using those numbers??? It's been toooo many years since I did any real math (I have been racking my meagre brain tho, honest!)

I will use rounded numbers kettle, but the main gist is:

0.46=(s-1)/2s
0.46 * 2s = s-1
0.92s = s-1
(s - 0.92s) - 1 = 0
0.08s = 1
s = 1/0.08 which is roughly 12

[quote="spoonitnow"]

Originally Posted by spoonitnow

With that assumption in mind, now we decide the criteria for Villain to call so that we can decide on his calling range. Let's call S our starting stack sizes, then the pot odds Villain will be getting to call are S+1:S-1, which means each hand in his calling range for the purposes of answering the question must have an equity greater than or equal to (S-1)/(S+1 + S-1) = (S-1)/2S. Clearly QQ+ will be in his range then, so we need to check the next most likely hand, AKs.

Where does the above equation come from?

I am also a little fuzzy on the pot odds figure. If the villian can see our cards and play perfectly, he will only call with even odds or better, which would be S:S to S+1:S, wouldn't it?

Maybe I'm math illiterate, but it has been over 15 years since I did any serious math.

Spoon did all this math when he could have just posted the Sklansky-Chubukov rankings.

Question 2: It's folded to you in the SB with some hand randomly chosen from {AQ+, TT+}. Your stacks are 15bb before the blinds are posted, and you raise to 5x the big blind with the hand that was randomly chosen from the given range. Villain can then fold, or go all-in for 15bb. Then you can call or fold. What range should Hero call with to maximize his value against a Villain who knows his opening raising range (but Villain doesn't know that we know he knows that)?

Let me get on that

Question number two. Well he's going to shrink his range to one better than we are playing so we should shrink our range too.

I know we can profitably call AKo+ AA, KK to a 10bb reshove. What throws me is that we're getting 2-1 now, and against his re-pop range, does QQ win often enough in the times it's dominated and the times it's a coinflip to be only a 33% dog. I think it is.

Although AQ does not.

sooooo
QQ+
AKo+

We put in 5BB, villain shoves 15BB
villain should shove QQ+,AK (if he folds AKo it's a different problem... but he should think that he's getting called here with AQ even though he's not because we know that he knows our range)

Total pot is 30BB if we call for 10BB
our call is 33% of the total pot
we need more than 33% equity

apparently TT+,AK is our calling range since TT has 36% equity against his shoving range

Question 2: It's folded to you in the SB with some hand randomly chosen from {AQ+, TT+}. Your stacks are 15bb before the blinds are posted, and you raise to 5x the big blind with the hand that was randomly chosen from the given range. Villain can then fold, or go all-in for 15bb. Then you can call or fold. What range should Hero call with to maximize his value against a Villain who knows his opening raising range (but Villain doesn't know that we know he knows that)?

I'm going to do this one in a few parts to give people who are having trouble with it a chance to work on it with a little bit of new information at a time.

First, let's consider our pot odds. With 15bb stacks before the blinds are posted, we put in 5bb and Villain puts in 15bb, so we are calling 10bb to win 20bb. This means that any hand we call with should have at least 33.333% equity against Villain's range.

Second, let's consider some sample calling ranges based on possible shoving ranges from our Villain:

Code:

Villain Shoves With      Hero +EV Calls With
--------------------------------------------
AA, KK                   AA
AA, KK, QQ               AA, KK, AKs
AA, KK, QQ, AKs          AA, KK, AK
AA, KK, QQ, AK           AA, KK, QQ, JJ, TT, AK
AA, KK, QQ, JJ, AKs      AA, KK, QQ, AK
AA, KK, QQ, JJ, AKs      AA, KK, QQ, JJ, AK

Third, one starting point for investigation after gathering this preliminary information would be to figure out what range Villain can play this way to maximize his own profit. That's where I'll be picking up from sometime tomorrow.

QQ+, AK you can call with TT because of how much AK he has in his range

ez thread