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variance in 6max LHE

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DrivingDog

09-23-2007, 11:41 PM Post subject: variance in 6max LHE

#1 (permalink)

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I've begun fooling around with some calculations of variance and found some results that I thought were quite interesting. I'd be curious to hear what other people think.

The following table is based on a player with a standard deviation of 17 BB/100 * (standard deviation is a measure of the variability of your winrate for you non-stats-nerds. PT displays it in the 'session' folder under 'more detail' - i use 17 BB/100 because it happens to be my SD). The table shows the percentage chance that a person will experience a swing of at least that many BBs for 1k, 10k, and 50k hands. (if that doesn't quite make sense to you the examples below should clear it up)

Swing +/- ... L (1k) ..... L (10k) ..... L (50k)
25 ............ 33.4 ........ 44.0 ........ 47.5
50 ............ 16.9 ........ 42.1 ........ 44.8
100 ........... 3.3 ......... 30.8 ........ 39.7
200 ......... <0.01 ....... 13.1 ........ 30.1
300 ......... <0.01 ........ 4.2 ......... 21.5
500 ......... <0.01 ........ 1.6 .......... 9.5
1000 ....... <0.01 ..... <0.01 ......... 0.4

where Swing+/- is the swing (in BB) greater or less than what a player would expect to win for a session of that many hands, and L is the Likelihood (in %) that they will have a swing at least that large for that number of hands. The distribution is symmetrical - i.e., you are equally (33.4%) likely to experience either a 25BB upswing or a 25BB downswing per 1k hands with a standard deviation of 17BB/100.

E.g., if you're a break even player, you expect to win on average 0 BB over 1k hands. You will have a 16.9% chance of winning at least 50BB over 1k hands, and a 16.9% chance of losing at least 50BB. If you're a +2 BB/100 player, you expect to win 20 BB over 1k hands. You will have a 16.9% chance of winning at least 70BB (20 BB expected winnings +50 BB swing) over 1k hands and a 16.9% chance of losing at least 30 BB (20 BB expected -50 BB swing).

I find three things interesting and/or surprising about these numbers:

1) Swings of +/- 200 BB or more over 1k hands are extremely rare. In fact, if you played 1 million hands (which would take close to 3 years if you play 1k hand a day), you should experience on average one or fewer +200 BB swings and one or fewer -200 BB swings per 1k hands. This suggests that if you do experience one of these major downswings, you should step back and re-evaluate your game. And if you experience a similar-sized upswing you should probably thank the poker gods (or your opponents).Variance exists for small samples, but in correspondingly small amounts.

2) Swings of +/- 100 BB over 10k hands are not that rare at all. In fact, over 10k hands, you're 30.8% likely to experience such a swing (one way or the other), for a total of 61.6% of the time. In other words, such swings will happen (one way or the other) about three out of every five times.

3) Swings of +/- 500 BB over 50K hands are not unheard of. On the contrary, one or the other will happen about once every 5 sets of trials. Thus, if you are a +1BB/100 player (who expects to win 500BB for every 50k hands you play), there is about a 1/10 chance you will win 1000 BB (500 expected +500 swing)and about a 1/10 chance you will break even (500 expected -500 swing) for any given set of 50k hands. If you play 50k hands a month, that's about one swing either way per year.

*Note that this table assumes a standard deviation of 17 BB/100 hands. For smaller SDs, larger swings become progressively less likely (i.e., there is less variance) and the converse is true for larger SDs.

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arborman

09-26-2007, 09:51 PM

#2 (permalink)

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Join Date: Aug 2007

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I recently (last week) went through a 140 BB downswing that took the spring out of my step. Some (30-40%) of it was crappy play and a bit of tilt after losing some of the bigger pots - everyone knows the story. Much of it was the traditional 'winning small pots and losing big ones' that feeds into most downswings. Those ~3000 hands brought my average winrate down to ~5BB/100 at that limit, so I can't really complain much, but downswings can be brutal to the psyche.

DrivingDog

09-27-2007, 11:35 AM

#3 (permalink)

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Yeah it takes a lot of self-discipline to not make a downswing worse by tilting. I think i'm getting better at that but still need to work on it somewhat.

"You can fool some of the people all of the time, and those are the ones you want to concentrate on." (George Bush).

Xanadu

09-27-2007, 05:59 PM

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In actual practice, a swing of 200BB in 1k hands is far more likely than the statistics you show would make it appear.

Basically, by using this kind of analysis, you are employing the Central Limit Theorem, which from a most basic level states that the distribution of the means of any sample sets from any distribution will have a limit approaching the normal distribution for a large n, where n is the number of sample sets.

The problem is that poker is discrete and has large tails in its distribution. Although the SD for a single hand may be rather small ... I am guessing somewhere between 1 and 3 BB, it is not uncommon to have massive data points on the tails ... -10 or more BB, +30 or more BB. It doesn't take all that much of a lucky run to have +200BB over 1k hands, and the odds of this happening are far higher than 0.1.

These statistics are probably getting fairly accurate for the largest swing shown of 1kBB at 50k or more hands. When doing this kind of analysis, you must keep in mind the limitations of the method you are using, and the assumptions and definititions. Coming up with meaningful statistics, even in what appears to be a straight forward and completely mathematical situation is not so simple as it appears, even for someone with a solid grasp of mathematics.

Fnord

09-27-2007, 08:59 PM

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Fnord is an unknown quantity at this point

I often wonder if a lot of blind battles in smaller games are a sick variant of the dollar auction.

http://www.youtube.com/watch?v=EBvge-c2_Xg
http://www.youtube.com/watch?v=w-Xz8aLuD7Y

KoRnholio

09-27-2007, 09:55 PM

#6 (permalink)

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Quote:

Originally Posted by Fnord

I often wonder if a lot of blind battles in smaller games are a sick variant of the dollar auction.

I never thought of it that way, but it's really quite true. Once semi-aware opponents see that you are betting and raising light, they often start to do the same, and it becomes almost 50/50 every time since so many showdowns are seen. Minus the 5% rake and it's bad for both parties.

Some days it feels like I've been standing forever, waiting for the bank teller to return so I can cash in all these Sklansky Bucks.

DrivingDog

09-28-2007, 09:00 AM

#7 (permalink)

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Quote:

Originally Posted by Xanadu

In actual practice, a swing of 200BB in 1k hands is far more likely than the statistics you show would make it appear.

The problem is that poker is discrete and has large tails in its distribution.

Interesting. How do you arrive at the conclusion that it's discrete? There are a lot more small pots and folded hands/blinds than large pots. I would think that the extreme tails of the distribution for single hands might be overrepresented because big pots do occur, but that by 100 hands you definitely approach normality due to the CLT. And it is blocks of 100 hands i am basing my analysis on.

(btw, 30 BB swings on a single hand require four players capping every street - iow, not happening).

"You can fool some of the people all of the time, and those are the ones you want to concentrate on." (George Bush).

DrivingDog

09-28-2007, 11:42 AM

#8 (permalink)

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My last 100 hands shows a mean profit of 0.14BB/hand with a SD of 1.7BB/hand (freq. distribution at bottom of this post, values rounded to nearest whole number). The biggest profit/loss in a single hand was 12BB, which was the only one >10BB. The distribution is not entirely normal but it certainly isn't discrete (which, if you mean the same thing by 'discrete' as i do, implies equal probabilities of all occurrences) - the center of the frequency distribution is strongly overrepresented because of all the folding preflop but otherwise there are only six values outside +/-2SD and two outside +/-3 SD, which is not that unusual for a normal distribution with a sample this size (for smaller samples a t distribution would be more appropriate).

More importantly, because we are dealing with the distribution of mean profits/losses for sets of 100 hands the assumption of normality seems quite plausible, and the problem of outliers becomes more or less insignificant, as these tend to be balanced between the two tails of the distribution.

So I would still contend that 200 BB swings are very rare over 1k hands. This is a mean profit of +/-20BB per 100 hands over ten sets of 100 hands. If you are playing like a complete fool it's certainly possible to lose this much but no matter how good and/or lucky you are it's pretty close to impossible to win this much.

Net BBs
won/lost... freq.
-4.............. 1
-3.............. 1
-2.............. 4
-1.............. 20
0............... 47
1............... 8
2............... 5
3............... 1
4............... 2
5............... 1
6............... 0
7............... 1
8............... 0
9............... 0
10.............. 0
11.............. 0
12.............. 1

"You can fool some of the people all of the time, and those are the ones you want to concentrate on." (George Bush).

DrivingDog

09-28-2007, 01:02 PM

#9 (permalink)

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...and my last 76 sets of 100 hands (7.6k hands total) had a mean 1.64BB profit/100 hands with a SD of 17.58BB/100 hands (raw data at bottom)

Below is the frequency with which the individual profits/100 hands fall inside a particular range of z-scores (collapsed across plus or minus - i.e., a z of -2.6 would go in the > 2.5 row), along with what is expected from a normal distribution.

z-score..............% of hands............% Expected
>2.5.....................0........................ ...1
2 to 2.5.................4...........................3
1.5 to 2.................11..........................9
1 to 1.5.................18.........................19
0.5 to 1.................29.........................30
0 to 0.5.................38........................38

Yup, looks pretty normal to me. It's good that you challenged me on it though because it wasn't necessarily a safe assumption.

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"You can fool some of the people all of the time, and those are the ones you want to concentrate on." (George Bush).

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variance in 6max LHE

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