An Exploration Of Variance

[vbnautilus 48]

The last few weeks I have experienced a streak of bad luck, the kind that just keeps getting worse when you think it can't. I startedkeeping records to try and understand if my perception was a cognitive distortion or if it really was an objective string of unlikely events. After convincing myself of the latter, I decided to do a little exploration of poker variance. You might be interested in this for poker reasons, or you might just read it to feel good that you are not as nerdy as I am. Either way I thought this might be useful here. I found it very helpful to visualize how this kind of randomness works. Basically, the question I am asking here is: how many hands do you have to look at before you can reliably find the "expected" outcome? In other words, when we see poker odds -- for example the little numbers that come up on ESPN when someone is all in tell us that AA is an 87% favorite over 72 -- those are calculated over thousands and thousands of hands. Obviously the smaller the sample, the more likely you are to see a result that deviates from the expected winning percentage.. but how large does your sample need to get before you approach the expected outcome over a series of observations of the same size? I wrote some code that will take any two poker hands and simulate outcomes over many many iterations. The data I will show here were done over 10,000 iterations. Once we have 10,000 outcomes -- that's 10,000 wins or losses, we can then look at various little subsamples - epochs I will call them. The code will take an epoch size (say we look over 10 hands) and slide that window forward one hand at time to see what winning percentage we get with each 10 hand window. We then do the same with 20 hand epochs, or 100 hand epochs, etc. That way we can see winning percentage varies with the size of the epoch that we sample. The first matchup we will look at is the extreme AhAd versus 7c2s. Over 10,000 iterations AA won 87.4% of the time. The longestlosing streak was 5 in a row and the longest winning streak was 51 in a row. Imagine losing with your aces to 72 5 times in a row? The following image shows the results for the epochs of different sizes from 10 to 500. You can see when you look at the small, 10 hand samples (blue) you get many cases where AA loses 60% or 50% of the time. By the time we reach the 200-hand samples (yellow) things have settled down quite a bit and rarely deviates more than 5% or so from the expected outcome. Here's another one, this time AhKd versus TcTs. This one is really all over the place with 10 and 20 samples. Plenty of times with 10 samples it went 100% one way or the other (longest streak 15 for TT). Again, though, things do settle down with several hundred iterations. In other words, its quite typical to get 10 or 20 hand segments that deviate far from expectation, but it's very rare to find a 100 hand segment that deviates. I know the principle here is obvious to anyone with an understanding of statistics, but the specifics of how the variance relates to scope are interesting to see.

[Flushgarden 0]

That was interesting, thanks for posting.Sometimes you really do feel like something is working against you...but this is a good reminder that variance can go to extremes that might seem impossible to us. At the same time though you still cant pass everything off as variance and you still often need to examine your own game and make sure you're really getting your money in the middle in the right situations.

[Frez 0]

Nice work.In March my KK was a net loser for me, winning only 43% of the time. It happens over short runs, like you say. Just stick to the game and things will work out over time.

[jmbreslin 0]

Quite fascinating, thanks.It is reassuring that as long as you keep getting your chips in the middle with the best hands, bad runs won't last very long...and then you'll be due for a nice run to make up for it. Getting your chips in with underdogs is a different story...

[BudBundy 0]

Good read , thanks.

[GabeTheKid 0]

Quite fascinating, thanks.It is reassuring that as long as you keep getting your chips in the middle with the best hands, bad runs won't last very long...and then you'll be due for a nice run to make up for it. Getting your chips in with underdogs is a different story...

I think that statement is flawed. If you lose with AA 3 times in a row, the chances of you losing with them the next three times are the exact same as they were to lose with them the first six times given you already lost 3. The fact that it would actually be 6 times losing with AA in a row disqualifies once you lose the first 3 times and the actual probability that it's going to lose 6 times in a row (or 3 more times) is the probability that AA loses three times. Just because you lost 3 times doesn't make it more likely for you not to lose the next one.

[Abbaddabba 0]

That, and not everyone can be so blessed as to get all their money in with aces against offsuit non-connectors preflop every time.

[cubbybri 0]

Potato, potato..... The point of this (I think, or I hope) is that if you are making the right decisions over the long term, things sort themselves out EVEN THOUGH previous outcomes have no affect on the current action.A visual display on variance to keep the spirits up to those that think their pocket Aces are cursed.Semantics more than anything(maybe some just trying to be smarter or knitpicky) but I think jmbreslin does get the point of the post.Aces versus anything will show the same trend over time. This was just the most extreme example.

[vbnautilus 48]

Potato, potato..... The point of this (I think, or I hope) is that if you are making the right decisions over the long term, things sort themselves out EVEN THOUGH previous outcomes have no affect on the current action.\

That's right. The "long term" comes pretty close to what you'd expect it to be over a few hundred hands; but when you look at a small sample you very often deviate considerably from expectation. The fear is that the long term is so long that one will not be likely to have a reasonable distribution of wins and losses over one's lifetime. I no longer think that's the case.

[Actuary 3]

it's a essentailly a normal dist. (see Normal Approximation to the Binomial)So you might not get your share, but most likely will get within a rangeBut most likely, won't exactly your share of wins.

[Abbaddabba 0]

That's right. The "long term" comes pretty close to what you'd expect it to be over a few hundred hands; but when you look at a small sample you very often deviate considerably from expectation. The fear is that the long term is so long that one will not be likely to have a reasonable distribution of wins and losses over one's lifetime. I no longer think that's the case.

The edge that a typical person has in a game, if it exists at all, is tiny relative to that example.The smaller the edge, the more meaningful the gap between expected wins v actual wins is.

[jmbreslin 0]

Thanks, cubby, that's exactly what I was getting at.It's reassuring because it means we're not necessarily doing something wrong when we're in the midst of a bad run (although we might be, and it is important to review hand histories for leaks). It's easy to lose confidence during a bad run but I personally find that this explanation of variance helps put things into perspective.

[vbnautilus 48]

it's a essentailly a normal dist. (see Normal Approximation to the Binomial)

This is a bit nit-picky but it does raise some interesting statistical issues: the values here are probably not normally distributed -- this distribution appears to deviate in a couple of important ways from a normal distribution, the most obvious way due to the skew induced by the ceiling, especially in the case of something like AA vs 72. Additionally, from eyeing thevalues, the distribution appears to exhibit high kurtosis, meaning a disproportionate amount of the variance is dueto infrequent extreme deviations from the mean (a few really bad streaks). Things I'd be interested in quantifying in the future,especially kurtosis. My aim here though was not to characterize the shape of the distribution, but only its width (variance).

[Actuary 3]

it's approximately Normal in terms of the Z-score.I'm not sure about the precise shape; but to the extent in approxmates, it would have to be close in shape, obv.Whether the actual generations you ran are "Normal Shaped" is subject to the RNG used.I'm speaking Real Distribution, not Sample.

[CobaltBlue 662]

This thread reminds me a little of this one: http://www.fullcontactpoker.com/poker-foru...c=98324&hl=

[Zach6668 513]

This thread reminds me a little of this one: http://www.fullcontactpoker.com/poker-foru...c=98324&hl=

lololololololol

[aim786 0]

lololololololol

stfu donk

[vbnautilus 48]

What am I missing here?