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.

Great post, nice example.

I would love to play around with the spreadsheet.

I second that

Link to the file: http://www.savefile.com/files/1292691

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?"

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.

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.

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

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!

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

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.

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.