A Discussion of Variance

[mosuavea]

It didnt help losing nearly 2 buyins with a full house old timer [img]/images/graemlins/wink.gif[/img]

I would say I was down at most 8 buyins at the 100s and only a couple (3-4 at most) at the 200s. I can confirm this with PT in a little bit.

To my credit, I would move down if I felt the need eventhough I am properly rolled to sustain losses like that.

[Pokey]

Let me start by apologizing if anybody took offense at the comments from my post. I should clarify that whenever random numbers are involved, "never" is an ugly word. Of course, putting it all in caps makes it even uglier. [img]/images/graemlins/grin.gif[/img]

I did some math to try and figure out how things looked. If you want to follow along, you can play around with the winrate calculator that I used.

I'm sure we've all looked at our likely winrates at some point in the past; here, I turn that on its ear. Assuming a standard deviation of 50 BB/100 (which Grunch assures me on his mother's sacred honor is absolutely 100% accurate for ALL SSNL players [img]/images/graemlins/wink.gif[/img] ), plug that number in. Then, put down 20,000 (or whatever) for the number of hands. Now put in a theoretical winrate and figure out the odds of break-even or less. Remember that since this is a two-sided distribution, the odds of being below the bottom threshhold is one-half of 100% minus the "confidence interval." The upshot of this all is:<ul type="square">[*]With a winrate of <font color="blue">8.2 PTBB/100</font>, there is a <font color="blue">99% chance</font> over any given <font color="blue">20,000 hand</font> stretch that you do better than break-even.[*]With a winrate of <font color="blue">5.8 PTBB/100</font>, there is a <font color="blue">95% chance</font> over any given <font color="blue">20,000 hand</font> stretch that you do better than break-even.[*]With a winrate of <font color="blue">4.5 PTBB/100</font>, there is a <font color="blue">90% chance</font> over any given <font color="blue">20,000 hand</font> stretch that you do better than break-even.[/list]<ul type="square">[*]With a winrate of <font color="blue">5.2 PTBB/100</font>, there is a <font color="blue">99% chance</font> over any given <font color="blue">50,000 hand</font> stretch that you do better than break-even.[*]With a winrate of <font color="blue">3.7 PTBB/100</font>, there is a <font color="blue">95% chance</font> over any given <font color="blue">50,000 hand</font> stretch that you do better than break-even.[*]With a winrate of <font color="blue">2.9 PTBB/100</font>, there is a <font color="blue">90% chance</font> over any given <font color="blue">50,000 hand</font> stretch that you do better than break-even.[/list]That leaves us with a few possibilities. First, you might be a fantastically winning player who just hit the "variance lottery" and had a 1-in-1000 or 1-in-10,000 downswing. We have hundreds of players at 2+2, and they play tens or hundreds of thousands of hands per year each; someone's bound to have a flukey outrageous unbelievably bad downswing. Maybe you're the one. Secondly, it's possible that your winrate isn't very stellar. Given a 50,000 hand breakeven stretch, that's FAR more likely to happen with a barely winning player than with a strongly winning player.

Ultimately, I was -- as you pointed out with much more politeness than I deserved -- quite wrong to say "never." However, in the realm of probability, what we're seeing in these graphs is highly unusual. I don't doubt you at all, but I don't think that your case undermines the basic principle that I was trying to convey.

[4_2_it]

Pokey,

I hate when you back up your reasoning with mathematical processes and formulae that I am cannot refute because of my non-math background [img]/images/graemlins/smile.gif[/img]

Even if OP was a 4.2PTBB winner, does a 90% confidence interval mean that there is a 10% chance that he will have a 20k hand swing that could negative? I took statistics about 20 years ago and got a C so my recollection on confidence intervals and the like is fuzzy at best.

My take is this. Conventional wisdom around here is that someone needs at least 20 buy-ins to handle the downswings at a given level. If I read the charts correctly, this looks like a 10 buy-in downswing at its worst point.

Why are we so quick to automatically blame bad play or tilt (though discounting them totally would be foolish) when these charts are clearly in line with what we hold to be conventional wisdom.

I have had swings of 8 buy-ins at NL $200, some of which was bad beats and some just bad play. You can't separate bad beats and tilt totally when analyzing this (or any)chart. We have no way to know if 80% of his losses were bad beats or tilt-related. It is impossible to play optimally at all times and even if you did you could still get sucked out on 8 times in a row.

I would say thank you to OP for sharing this info because I think it generated a lot of good discussion (that I hope continues).

[epdaws]

[ QUOTE ]
Pokey,

I hate when you back up your reasoning with mathematical processes and formulae that I am cannot refute because of my non-math background [img]/images/graemlins/smile.gif[/img]

[/ QUOTE ]

Damn straight. This broadcast news major is miles behind your math mastery, Pokey. What do you do for a living, anyway? [img]/images/graemlins/smile.gif[/img]

[Pokey]

[ QUOTE ]
I hate when you back up your reasoning with mathematical processes and formulae that I am cannot refute because of my non-math background [img]/images/graemlins/smile.gif[/img]

[/ QUOTE ]

Gah, sorry! I keep forgetting that while we're all geeks, we're geeks in different ways. [img]/images/graemlins/grin.gif[/img]

OK, quick crash course in probability:

When you are about to play a hand of poker, the outcome on that particular hand is unknown in advance. Pretend that there is a gigantic barrel full of "outcomes," and every hand you draw an outcome from the barrel at random. Some of those outcomes are giant losses, some of those outcomes are small losses, some of those outcomes are small wins, some of those outcomes are giant wins. Of course, the small wins and small losses are much more common than the giant wins and giant losses. Let's assume that, if we looked at EVERY outcome in the barrel, the "distribution" of those outcomes was a standard normal distribution:

This image shows a "standard normal" distribution. The expected outcome is 0, but the actual outcome is randomized. In this distribution, 34% of outcomes are between the average and one standard deviation below average; another 34% are between the average and one standard deviation above average. That means a total of 68% (about two-thirds) of all "outcomes" will be the average outcome plus-or-minus one standard deviation. Looking at the graph we see that about 95% of all "outcomes" are the average level plus-or-minus two standard deviations, and over 99% of all "outcomes" are plus-or-minus three standard deviations. Strange outcomes can happen, but they're unusual.

In other words, we assume that the probability of a particular outcome is a decreasing function of how far that outcome is from our "normal" outcome. The very middle of this distribution will be your average winrate per hand; the farther from this winrate a particular outcome is, the more unusual (read: rare) that outcome will be. So, if we play this one hand of poker, we might get a huge win, we might get a huge loss, but we're most likely to get our usual winrate.

Of course, we don't play just one hand in poker; we play oodles of them. Each hand is another draw from this random distribution, independent of the prior draws. That means that drawing one "outcome" and having it be, say two standard deviations below average, is FAR more common than drawing two "outcomes" and having them together be four standard deviations below average: each of these events happens only one time in twenty, and since they are independent, that would collectively happen only once in 400 sets of two draws.

The more outcomes we draw, the more our average outcome "converges" to the overall average of our distribution. (In poker terms, the more hands we play, the more closely our average winrate should resemble our "true" winrate.) Of course, the volatility means we'll never converge completely; our only question is how fast do we converge? In poker, the answer is "not that fast."

To finally answer your question: the "confidence interval" on the webpage posted above indicates the width around zero of outcomes we're considering. If we choose 90%, then we're looking for the band around our proposed BB/100 winrate such that there is only a 5% chance of seeing a winrate greater than the "Maximum BB/100" and only a 5% chance of seeing a winrate less than the "Minimum BB/100." If we choose 98%, then the odds that we'd see a winrate greater than the "Maximum BB/100" over however many hands we typed in would be only 1%, and the odds that we'd see a winrate less than the "Minimum BB/100" over that many hands would also be only 1%.

The numbers I reported were the odds that a player with a true winrate of the specified level would see a breakeven (or worse) stretch over the course of the specified number of hands. So, referring back to my previous post, if your TRUE winrate was 8.2 PTBB/100 and your standard deviation/100 was 50, then if you played 20,000 hands of poker you'd have a 99% chance of winning money. Conversely, 1% of the time you'd break even or lose money under those same conditions.

There! Either you now understand it or your head popped like the people in the movie Scanners. Either way, my work is done here. [img]/images/graemlins/grin.gif[/img]

epdaws: I teach college-level economics for a living. That should explain why I'm so longwinded and pedantic all the time. [img]/images/graemlins/grin.gif[/img]

[4_2_it]

ARRRGGGGHHHH! HELP ME! (NSFW?)

[mason55]

Pokey -

That post ties together all the winrate/SD math better than any other post I have ever seen.

Thank you, and well done.

[epdaws]

[ QUOTE ]
epdaws: I teach college-level economics for a living. That should explain why I'm so longwinded and pedantic all the time. [img]/images/graemlins/grin.gif[/img]

[/ QUOTE ]

I like the longwindedness, poke. Your posts are generally filled with substance and good humor, and they make this forum quite enjoyable. That econ prof background sums it up... [img]/images/graemlins/grin.gif[/img]

[mosuavea]

Pokey,
Two excellent posts. The post you made that comment in was also excellent. I was planning on making this post in that thread but thought it would get buried and also thought it could generate enough discussion to merit its own thread, which it has.

Ive dealt with my share of confidence intervals in college and have to tackle them again during my current set of MBA classes, but that was an excellent crash course on them and thanks for that! I agree that it unusual which is why I posted the graphs. I know that I am not an upper tier player that could sustain an 8 PTBB/100 win rate as I advance, I dont have the time or the energy to devote to it, nor does the vast majority of the players in SSNL if they actually looked at their skill without bias.
Poker is a hobby for me and I enjoy playing. It gives me a little extra income where I can do things I wouldnt be able to do otherwise.

I do see myself as a winning player though and even through this stretch, can sustain a decent winrate, enough to keep me happy coupled with bonuses/rb .

I think stuff like this can happen, however rare it may be. I knew when I posted I would hear the you arent skilled enough to play that level or you moved up to fast, but I can take it and figured the discussion would be good. People can make their opinions however they choose and I would never take offense to them. This is what prompted me to post, not to bash your thoughts.

Thanks for the contributions, always enjoy your posts.

[dardo]

Please, correct me if I'm wrong ...

** (all is per 100 hands)

-- 50NL - results for a good player

Winrate 10 PTBB - SD 50 PTBB

Winrate 10$ - SD 50$

-- 1/2 Limit - same

Winrate 2.5 PTBB - SD 10 PTBB

Winrate 10$ - SD 40$

Where's the difference?

regards,

dardo