I wanted to make a post about variance for my 1k post. I know i have just 500 posts but this thread can be a direct answer to the SneakyFerret thread. So i think it's better to post this now than in a few months.

First, i apologize for my bad english

I come from video games. I played CounterStrike at "international level" for 7 years in differents french team (famous: Goodgame (same team as elky) and aAa). One movie @ me: http://www.youtube.com/watch?v=KC4LkmfLqHY .
I have discovered poker end of 2006. I've started to read lots of poker book before moving to realmoney with my first deposit of 50$. In 2007, i tried lots of poker rooms and all possible style like cashgame, sng, tourney. Since september 2007, i've choosen my style: STT => It's fast, no tilting, and clear for me. On january 2008, i cashout all my BR and start again with 50$ on february on absolute. I moved on pokerstars on april in order to make decent volume (8+ tablings). Now i'm playing 16 tables on 27$ and i hope i will maintain at least a 5% ROI.

After this few word about me, go back to the main subject: the variance.

There is some mathematics formula for calculate variance, standard deviation, confidence interval.
You can find all of them on wikipedia.
As you know (or not), variance is the average of the distance between (median and result)² of a data's sample.
Standard deviation is the square root of variance.
Let see all these concepts in a real case.

This is my real data (16$ BI):
2,149 11%

ITM:
* 288 first
* 259 second
* 305 thrid
* And so 1297 not ITM

16$ prizepool:
* 1st: 67.5$ so +51.5$ = +3.21875 BI
* 2nd: 40.5$ so +24.5$ = +1.53125 BI
* 3rd: 27$ so 11$ = +0.6875 BI

ROI is +11% so +0.11 BI.

Variance:
(288*(0.11-3.21875)²+259*(0.11-1.53125)²+305*(0.11-0.6875)²+1297*(0.11-(-1))²)/2149=2.329570 BI

squrt = square root

standard deviation sDev = squrt(Variance) = 1.526292

Confidance intervals for "N" games played
You are:
* 68% sure that you ROI is on the interval [ROI-(sDev/squrt(N)); ROI+(sDev/squrt(N))]
* 95% sure that you ROI is on the interval [ROI-2*(sDev/squrt(N)); ROI+2*(sDev/squrt(N))]
* 99% sure that you ROI is on the interval [ROI-3*(sDev/squrt(N)); ROI+3*(sDev/squrt(N))]

If you plan to play 1000 games per month (a decent volumes), what is your expected ROI ?
* at 68% you can say your ROI will be on [+6.17%; +15.83%] that's means an deviation of 4.83% !!
* at 95% you can say your ROI will be on [+1.34%;+20.7%], so a deviation of 9.66% !!

Now let's see another example:

This is another real data (27$ BI):
542 6%

ITM:
* 67 first
* 67 second
* 76 thrid
* And so 332 not ITM

27$ prizepool:
* 1st: 112.5$ so +85.5$ = +3.1667 BI
* 2nd: 67.5$ so +40.5$ = +1.5000 BI
* 3rd: 45$ so 18$ = +0.6667 BI

ROI is +6% so +0.06 BI.

Variance:
(67*(0.06-3.1667)²+67*(0.06-1.5)²+76*(0.06-0.6667)²+332*(0.06-(-1))²)/542=2.189293 BI

standard deviation sDev = squrt(Variance) = 1.479625

Confidance intervals with 1000 games:
* 68% => [+1.32%; +10.7%]
* 95% => [-3.4%; +15.4%]

So a 6% winning reg at 27$ can finish a month @ -3.4% or +15.4%.
The deviation is 9.4% it's a + or -150% on his expected winnings.

To get a deviation of 1% in this case, we need to increase the number of games played by 100.
That's 100 000 games !

I hope some people will understand now the brutal nature of the variance in STT.

I want to thank 2+2, all the 2+2 STT communities and specially a thanks to:
AMT, staffy, luisgallo, little john, Scotty_12, darinvg, sippin_criss, simplicity8, 7castle, juandadi, dave_w11, eurythmech, emperor_norton, drzen, NJD77 and wobuffet for the quality of their posts.

nice post...LOL at you having the 1k post ready but being at 500 posts and then deciding to publish it anyway at 518!!!

Theoretically we should adjust our bankroll requirements accroding to our ROI and therefore dsquared.

I always thought the 100BI was a too conservative approximation and it might work for somebody barely beating the games but certainly not for whoever is decently winning at low stakes and wants to move up.

Also I would assume that the tails are a bit bigger if stakes are higher because of the extra variance given by the field (unless you table select).

If I could decipher any of the math in the post I'm sure I could appreciate it more, however, I'm pretty dumb.
I also must really suck at math because I thought 1k = 1k, apparently 518 = 1k tho. Or maybe that's just a French calculation

TY for posting, I'll try and re-read it later.

Very good post...I personally always liked these kind of posts.

"* at 95% you can say your ROI will be on [+1.34%;+20.7%], so a deviation of 9.66% !!"

So given 11% at 16s people can run realllly differently...but in ~99% they are at least profitable per month (1000 games) if undersood it good.

What is interesting imo...the higher buy-in the less "top heavy" structure and placing ITM is a LITTLE more important than 1st place. It is because the increasing rake.

"16$ prizepool:
* 1st: 67.5$ so +51.5$ = +3.21875 BI
* 2nd: 40.5$ so +24.5$ = +1.53125 BI
* 3rd: 27$ so 11$ = +0.6875 BI"

"27$ prizepool:
* 1st: 112.5$ so +85.5$ = +3.1667 BI
* 2nd: 67.5$ so +40.5$ = +1.5000 BI
* 3rd: 45$ so 18$ = +0.6667 BI"

It is 1,6% difference in 1st place payout..hmm. It can be even more at higher stakes (55+5$ for sure)...but I am too lazy now to count it

Nice post.

Nice post k4b4l. Scary stuff really. Variance sucks big time. It makes me chuckle when I think about how frustrated I used to get when I was 2-tabling $11 games on VC years ago, and would be at my wits end that I had broken even for 200 games.

Interesting analysis, but beware that using 2.1k games to derive your hypothesised ROI for your next 100k games (to reach your std dev of 1%) obviously has some flaws.

And you don't like my posts? :|

Oh, and I can say with 99.975% confidence that 518 < 1k posts!

damn, i thought about you aussie rhino but i forgot to write your name, i'm updating the thread

edit: i can't

That's ok mate

That's exactly why I didn't put names in my 1k post!

And by pure random chance, it said posts: 1,000 - what were the odds of that? ;p

And I enjoy your posts too...

Interesting post.

I didn't do the math, but I guess you're assuming that your 11% ROI is your true winrate, and from here you calculate your confidence intervals. Although it is the maximum likelihood estimate of your winrate using your ~2K data points, it is my understanding that most winning SNG players fall closer to 3-5% max winrate? I think I read that somewhere. Might not be true at the lower levels though.

So my point is that you can make an adjustment to your results if you know what the distributions look like for most players. The adjustment is done using bayes theorem, and there is an example of how this is done in the book 'mathematics of poker'. I do not know all the details, and I'm not a big math guy. But it's worth knowing that your sample may be skewed by variance itself, from your actual true distribution.

Cheers

Nice video, I really miss CS watching that. I went from CS to poker in 2003.

I'm no good with poker math like ICM and variance, but from what I seen in practice your numbers seems pretty good. I actually think I hit both top and bottom during my latest endeavor at $16.

ban op for confusing me this early in the morning





















jk.....good job, good post, good luck sir

Hahahaaa k4b4l from aAa, no way I could have imagined that
I remember watching videos from your team back in the CS 1.5 days, especially aAa summer 2002 frag collection (or something close to that), that was a nice one. Yeah, by the way time to come out with the dreaded truth : I'm also french.

Your post is really good and was definitely needed in this forum, if improved just a little it would deserve to be a sticky or go into the FAQ, along with a link to ROI simulator that can also show you the likelihood of a downswing (and it's scary even if you're a very good player). As I like math stuff applied to poker, I was planning to make a post like this one someday but you beat me to it.

Congrats on you 1k post in advance and good luck for your future poker adventures.

I always like maths posts, nice post k4b4l. I briefly played competitively in a first person shooter (obv not to the same extent you did), the trigger finger sure helps for multitabling. You should have changed your location to say "Posts: 1,000", that would have fooled em.

Both payouts look exactly like 50/30/20 to me.

11% is certainly a sustainable winrate for k4b4l at the 16s - judging from the content of his posts. I agree there is a nice section in 'mathematics of poker' about making adjustments based on bayes theorem and attainable win rates. In the past I have tried a similar method, using a normal distribution to model the range of expected roi's for a random player. For example, at the $16s I would centre the normal distribution around -6.25% roi (the rake), and make the end taper off at ~15% roi which would be about as high as you would expect. But the problem with that, is that someone who reads 2+2 is not just a "random" player, if they are following general 2+2 strategy, then bayes theorem would actually suggest that they are more likely to be a winning player. But you still bring up a good point - if your win rate is higher than what is expected to be possible you should make adjustments, because unless have studied a ridiculous amount of ICM, chances are you aren't going to be better than the best players at your level.

k4b4l, i used to play high level CS too. i remember playing vs you and elky, not sure if u played at the same time, with roman and co. i played as ssj, or cAWLEY most likely ud remember me as. playing against aAa with Noneed, c4u, lastchance/landed maybe?
After that i got bored of no good uk teams, so played WoW, then started poker how was your transition? Did u feel the skills you learned from CS helped poker, i felt that when playing cs i dedicated alot of time, so i used that with poker, i used alot of time to try and be the best, as that was in my nature from CS.