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Old 12-30-2007, 12:38 PM   #1
MasterLJ
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(Theory) - Amazing Bankroll/Variance Post

I'm sure a few of you remember my horrendous downswing back in September/October. Suffice it to say, I have searched far and wide to really understand variance since then, as I feel you are better suited to cope with the brutal swings of HU Cash if you understand variance.

In 3+ months of searching I finally found an amazing post that pretty much was everything I was looking for.

http://www.liquidpoker.net/news/1213...variance_guide

It's sort of advanced and uses a lot of "basic" statistics. If you haven't learned this stuff before it can be daunting, if you have, the link should be a nice refresher course. Excel provides a lot of these functions (and the author of the thread will point out which functions to use) so I highly recommend playing around with your stats and figuring out where you stand. I have created a spreadsheet that will calculate ROR, probability of being a winning player, winrate confidence intervals, resolution of winrate (how many hands it takes to get within plus or minus X PTBB/100), and bankroll requirements.

Summary and Implications:
It takes a LOT of hands to iron out variance in HU. At 100k hands the average person crushing their stakes are still +/- 4-6 PTBB/100 hands off their true winrate. To get winrate resolution of +/- 1 PTBB/100 it takes well north of 1 million hands given that the average Standard Deviation (SD) is much higher in HU cash. Talking about anything less than 30k hand samples is almost useless from a statistical standpoint. 50k+ is where things really start to converge in a useful manner.

For someone like myself, who has an SD of 82.8 PTBB/100 (I think it's quite high) it will take 2.63 million hands before my winrate converges to within 1 PTBB/100 of my true winrate, with 95% confidence. If you bump it up to 99% confidence the number explodes to 4.548 million hands needed. In english it means that "after 2.63 million hands played I can say with 95% confidence my given winrate (after playing 2.63 million hands) is within plus or minus 1 PTBB/100"

In a practical sense this means that playing a HU cash game with a high SD but a low winrate (say 5 PTBB or less) is absolutely not worth it, especially if you play for a living. This implies that aiming to play in the nosebleeds is almost purely a gamble. Sure, you can have an "edge" but it might take multi-millions of hands for that edge to surface in any meaningful way. Of course if you can find a player playing 300/600 that completely sucks and you have a bankroll that can absorb that sort of risk, then by all means. But Phil Ivey playing Patrik Antonius, or other high stakes HU matches are almost like watching 2 men flipping a coin for $60k a pop since neither can reasonably expect a large edge (I assume) and neither will be able to put in the hands necessary at those stakes to iron out the variance in a meaningful way. This is something I knew intuitively but was not able to quantify. Year after year I saw the absolute top players get beaten back (Townsend is a great example) and I knew that I never really wanted to play higher than 25/50. Now it should be clear why.

Finally, I wanted to share my stats, off of the Excel spreadsheet I crafted, since October (generally what I hold to be my stats since I really figured out HU cash). It's not a lot of hands because I've done a lot of traveling since then, but it might provide some clarity via example:

Winrate: 13.78 PTBB/100
SD: 82.8 PTBB/100
N: 44808 hands
Desired Confidence Interval?: 95%

True Winrate is between: 6.11275 PTBB/100 and 21.44724 PTBB/100
(in english, you can say with 95% confidence that my true winrate is somewhere between those two numbers). This is +/- 7.66724 PTBB/100

If you bumped the confidence interval up to 99% you get the winrate being between 3.703 and 23.8564.

Probability of Being a longterm winner?: 99.97865%
This stat is based off winrate, SD and sample size. If you drop the winrate to 5 PTBB/100 this stat plummets to 89.94%. At 2.5 it's 73.86%.

Winrate Resolution:
Resolution (PTBB/100): 1
Confidence: 95%

Number of Hands: 2,633,643

In English this says, after 2.633 million hands your winrate is withing plus or minus 1 PTBB/100 with 95% certainty. If you drop the SD to 60 PTBB/100 this number drops to 1.382 million hands.


Desired Risk of Ruin: 0.0000001% ( 1 in 10 million... I think ).
Using Low Winrate (6.11 PTBB/100): 116 buyins
Using Upper Winrate (21.44 PTBB/100): 33.12 buyins
Using Current Winrate (13.78 PTBB/100): 51.55 buyins

If you change your desired ROR to 0.01% (1 in 10,000 chance of going broke)
You get:
Using Low Winrate (6.11 PTBB/100): 51.65 buyins
Using Upper Winrate (21.44 PTBB/100): 14.72 buyins
Using Current Winrate (13.78 PTBB/100): 22.91 buyins

If you change your desired ROR to 0.001% (1 in 100,000 chance of going broke)
You get:
Using Low Winrate (6.11 PTBB/100): 64.56 buyins
Using Upper Winrate (21.44 PTBB/100): 18.40 buyins
Using Current Winrate (13.78 PTBB/100): 28.64 buyins


The critical difference between how the author of that post calculates ROR and how most people calculate ROR is the difference between a stagnant BR and one that grows. Most people calculate ROR on a set bankroll that does not grow (e.g. $10k). This is silly, because in practice if you are a winning player, your bankroll grows as you win. You have to be careful because the author obviously makes the assumption you are not taking out any money from your bankroll. The "true" ROR calculation is somewhere between both methods, however I think that the method proposed in the article is more applicable than a stagnant BR.

In conclusion, I hope some of you can use and process this information as much as I have. If I get enough requests I might make my spreadsheet available somehow so that you can play around with it and see where you stand.

Last edited by BrandysB; 03-12-2009 at 06:32 PM. Reason: link changed due to OP's request
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Old 12-30-2007, 12:49 PM   #2
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Re: (Theory) - Amazing Bankroll/Variance Post

Great post, nice example.

I would love to play around with the spreadsheet.
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Old 12-30-2007, 01:08 PM   #3
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Re: (Theory) - Amazing Bankroll/Variance Post

Quote:
Originally Posted by janh View Post
Great post, nice example.

I would love to play around with the spreadsheet.
I second that
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Old 12-30-2007, 01:23 PM   #4
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Re: (Theory) - Amazing Bankroll/Variance Post

Link to the file: http://www.savefile.com/files/1292691
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Old 12-30-2007, 02:29 PM   #5
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Re: (Theory) - Amazing Bankroll/Variance Post

It would be great for anyone who uses the Spreadsheet to post your results here as sort of a guide to others. I did some quick browsing through PT and it seems HU SD ranges from about mid 40's to high 80's with 60 being a number I've seen quite a bit, although bigger winners generally have higher SDs.

I think I'm going to add the author's version of "What is the chances of having a DS in X amount of hands?"
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Old 12-30-2007, 02:53 PM   #6
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Re: (Theory) - Amazing Bankroll/Variance Post

Yeah, I wrote a simulation a few months ago and the results are similar. 50k hands can vary a lot. Even a million hands vary somewhat. When going up to 10 millions of hands the variance seems to stabilize.
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Old 12-30-2007, 02:55 PM   #7
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Re: (Theory) - Amazing Bankroll/Variance Post

Quote:
Originally Posted by Kharlog View Post
Yeah, I wrote a simulation a few months ago and the results are similar. 50k hands can vary a lot. Even a million hands vary somewhat. When going up to 10 millions of hands the variance seems to stabilize.
Were you the guy who did this all in R? Forgive me if it was someone else.

I'm actually having a bit of trouble with the author's Probability of Losing X buyins after Y hands, the results are not what I expected.
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Old 12-30-2007, 04:29 PM   #8
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Re: (Theory) - Amazing Bankroll/Variance Post

This is exactly why Im not a huge fan of HU cash. Pokey made a very similar post with simulations in the SSNL sticky I believe.
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Old 12-30-2007, 04:30 PM   #9
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Re: (Theory) - Amazing Bankroll/Variance Post

Great post, thanks,

Just curious where you play and what stakes.

The question I have has to do with HU, variance and game selection. Right now, I'm playing NL 200 HU. The variance is higher than six max; however, with game selection, I'm able to almost always (at least i strive to) be sitting with a bad player. So at at six max table, best case scenario 2 of my 5 opponents (40%) might be bad, and that's not always possible, whereas at HU, 100% of my opponets will hopefully be bad. Shouldn't this aspect of what's possible at lower HU reduce standard deviation?
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Old 12-30-2007, 04:52 PM   #10
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Re: (Theory) - Amazing Bankroll/Variance Post

Quote:
Originally Posted by Lucky View Post
Great post, thanks,

Just curious where you play and what stakes.

The question I have has to do with HU, variance and game selection. Right now, I'm playing NL 200 HU. The variance is higher than six max; however, with game selection, I'm able to almost always (at least i strive to) be sitting with a bad player. So at at six max table, best case scenario 2 of my 5 opponents (40%) might be bad, and that's not always possible, whereas at HU, 100% of my opponets will hopefully be bad. Shouldn't this aspect of what's possible at lower HU reduce standard deviation?
It increases your winrate, it doesn't necessarily reduce your SD
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Old 12-30-2007, 04:59 PM   #11
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Re: (Theory) - Amazing Bankroll/Variance Post

Yea what coh said, what a lot of you should take from this is that if you have experienced gut wrenching variance yet, you've been running hot so get ready for it!
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Old 12-30-2007, 05:08 PM   #12
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Re: (Theory) - Amazing Bankroll/Variance Post

How do I find SD? I knew at some point, but I've forgotten. Isn't there somewhere on PT to find it?
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Old 12-30-2007, 05:24 PM   #13
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Re: (Theory) - Amazing Bankroll/Variance Post

Quote:
Originally Posted by abcjnich View Post
How do I find SD? I knew at some point, but I've forgotten. Isn't there somewhere on PT to find it?
Session Notes -> More Detail
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Old 12-30-2007, 05:25 PM   #14
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Re: (Theory) - Amazing Bankroll/Variance Post

Quote:
Originally Posted by abcjnich View Post
How do I find SD? I knew at some point, but I've forgotten. Isn't there somewhere on PT to find it?
In PT go into Session Notes tab, look to the upper right (not all the way up though) and click on the "More Details" button.
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Old 12-30-2007, 05:30 PM   #15
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Re: (Theory) - Amazing Bankroll/Variance Post

Quote:
Originally Posted by Lucky View Post
Great post, thanks,

Just curious where you play and what stakes.

The question I have has to do with HU, variance and game selection. Right now, I'm playing NL 200 HU. The variance is higher than six max; however, with game selection, I'm able to almost always (at least i strive to) be sitting with a bad player. So at at six max table, best case scenario 2 of my 5 opponents (40%) might be bad, and that's not always possible, whereas at HU, 100% of my opponets will hopefully be bad. Shouldn't this aspect of what's possible at lower HU reduce standard deviation?
What COH said. Standard Deviation isn't going to change but you're going to swing less severely because you are winning more.

I have been a 1/2 regular on FTP for a long time, making the full time transition to 2/4 and it has been going very well. I definitely will still be playing 1/2 when 2/4 sucks, and I might throw in some 5/10 and 3/6 as well since my bankroll is pretty large for the stakes I play.
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Old 12-30-2007, 05:35 PM   #16
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Re: (Theory) - Amazing Bankroll/Variance Post

According to MoP, if you don't add to your bankroll, you are nearly 95% likely to go broke, even if you were rolled to start with.

In English, if you start with a 10k roll and you earn 4k in one month, your risk of ruin increases exponentially for every dollar that you withdraw. If you withdraw 4.1k a month, then you are guaranteed to go bust. 99.5% actually. You should always add SOMETHING to your roll to ensure that you will not go bust. To be fair, I think that they used calculations with a higher ROR.
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Old 12-30-2007, 07:47 PM   #17
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Re: (Theory) - Amazing Bankroll/Variance Post

I use this spreadsheet for my HUSNG's. Its very interesting, once u set the information up it auto-imports tournament summaries from PT, and then gives u all sorts of information, with winning confidence, profit with a certain cofidence level, SD/tourney and BR needed to play a certain stake according to ur results with a 0.5 % ROR.

http://www.mediafire.com/?ftvotjzwst0
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Old 12-31-2007, 05:50 AM   #18
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Re: (Theory) - Amazing Bankroll/Variance Post

Interesting post. However, after 2.4 million hands I would think you are not the same player you were when you started, meaning a true win rate could never be calculated. The math would work with a poker playing computer that never changed.
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Old 12-31-2007, 06:00 AM   #19
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Re: (Theory) - Amazing Bankroll/Variance Post

Great post, I definitely added the bankroll page to my favorites. I'm surprised gems like this only pop up in the HUforum and not all over the rest.

Quote:
Originally Posted by daveT View Post
According to MoP, if you don't add to your bankroll, you are nearly 95% likely to go broke, even if you were rolled to start with.

In English, if you start with a 10k roll and you earn 4k in one month, your risk of ruin increases exponentially for every dollar that you withdraw. If you withdraw 4.1k a month, then you are guaranteed to go bust. 99.5% actually. You should always add SOMETHING to your roll to ensure that you will not go bust. To be fair, I think that they used calculations with a higher ROR.
Sorry, I keep trying to respond to this and I'm not sure I understand what you're saying. Are you saying that with the model if you win 4k and withdraw 4.1k every month you are 99.5% likely to go bust? That may make sense because the model most likely calculates your winrate at -100 per month, the .5 may just be regression inaccuracy when dealing with infinity.

The other part is that you indeed, are guaranteed to go bust. The only reason that you wont is because you'll die first. If we extend time out to infinity in addition to your lifespan, regardless of your bankroll or profit you will experience a 1 in 1 billion trillion jillion chance eventually where you go busto chango.

Last edited by dxu05; 12-31-2007 at 06:11 AM.
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Old 12-31-2007, 06:33 AM   #20
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Re: (Theory) - Amazing Bankroll/Variance Post

Is this post referring only to HU NL or to other games such as HU LHE as well? I can't see any of this really applying to HU LHE where variance should be quite low for a player with a high winrate. And isn't caculating a "true winrate" for HU NL kind of pointless since you are always playing vs different opponents who you have a different edge against? There is no way to measure the variance of your opponent quality, you could play 50k hands vs players you have a large edge on and then 50k hands vs players who have an edge on you. This will surely result in what looks like monster 'variance' when infact your opponent is the biggest variable.

I hope it doesn't sound like im trying to bag on the OP because thats not my intention at all, I just found the post very thought provoking and the material confusing to me.
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Old 12-31-2007, 06:42 AM   #21
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Re: (Theory) - Amazing Bankroll/Variance Post

Chester,

You have an average winrate that you can use to calculate. This average winrate will be reflective of your table selection, thus reflecting "variance" of your opponents. Effectively, to simplify it even more, just take out table selection and assume all your past opponents were random. As long as your future opponents are random and not more difficult than usual your bankroll required to dodge ruin can still be correctly estimated.
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Old 12-31-2007, 09:07 AM   #22
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Re: (Theory) - Amazing Bankroll/Variance Post

Quote:
Originally Posted by chesterfieldking View Post
Is this post referring only to HU NL or to other games such as HU LHE as well? I can't see any of this really applying to HU LHE where variance should be quite low for a player with a high winrate. And isn't caculating a "true winrate" for HU NL kind of pointless since you are always playing vs different opponents who you have a different edge against? There is no way to measure the variance of your opponent quality, you could play 50k hands vs players you have a large edge on and then 50k hands vs players who have an edge on you. This will surely result in what looks like monster 'variance' when infact your opponent is the biggest variable.

I hope it doesn't sound like im trying to bag on the OP because thats not my intention at all, I just found the post very thought provoking and the material confusing to me.
You and pushing are both correct in your assertions.

first, pushing said you're hardly the same player after 2.4 million hands. I couldn't agree more. My gut says that intuitively your winrate will be higher, but I also think there is a significant chance it could be lower (you get too comfortable with your stakes, you start picking up bad habits etc).

As for what you (chester) said, you are also correct. You can never pin down your winrate with pinpoint accuracy, but you certainly can get a great idea of what your winrate is over a large sample. After a huge sample (2+ mil) your aggregate sum of edges vs your opponent is going to reach some reasonable mean (of course based on your game selecting methods).

My OP isn't supposed to be the end-all beat-all method of calculating winrates, but it certainly brought me much closer to a solid understanding of variance in HU cash.

As for LHE, I don't think the OP applies at all (sorry). I'd certainly be interested in seeing how that pans out.



All,

What you really should take from this post:

1. Importance of game selection
2. Importance of really beating your current stakes in order to smooth variance (step down if you're not crushing).
3. How to see where you are and how you are progressing with statistics.
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Old 12-31-2007, 09:44 AM   #23
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Re: (Theory) - Amazing Bankroll/Variance Post

This whole thread makes me kind of wonder why someone would play HU cash instead of FR cash were you can comfortably multitable with much less variance.
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Old 12-31-2007, 09:49 AM   #24
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Re: (Theory) - Amazing Bankroll/Variance Post

Quote:
Originally Posted by black666 View Post
This whole thread makes me kind of wonder why someone would play HU cash instead of FR cash were you can comfortably multitable with much less variance.
Winrate.

I don't think any other form of HE comes close in terms of maximum sustainable winrate.

Also keep in mind that I can make sure 100% of my opponents are bad. You can never say this at FR or even 6max.

I don't know FR SD ranges off hand, but the spreadsheet should work for FR.

I know it's true for 6max and I'm assuming it's true for FR (please let me know if I'm wrong here) but the ratio of sustainable Winrate to Standard Deviation is much higher in HU cash.
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Old 12-31-2007, 12:41 PM   #25
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Re: (Theory) - Amazing Bankroll/Variance Post

nice post.

one quick thing to think about -- there is a variance of variance in heads up that is tough to quantify. when you are tilting an opponent, your variance skyrockets. i'm not sure if there's a way to quantify this or if it also converges over a certain number of hands.

on a different note, i think people think too much of variance as quantifiable sklansky dollars -- the type you can see in pokerEV. when most people think of variance, they usually think of how cards play against cards. they think things like "i had a flush draw and two overs but they didn't beat his pair."

what they usually don't think of is the percentage that one line works against another. for instance, maybe you generally fire 2 times in a reraised pot x% and he chooses to float y% of the time.

i think people underestimate this line versus line variance (because it's tough to see) and overestimate the card versus card variance (because it's easy to calculate).
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