A Question About Variance and ITM

[LinusKS]

Ok, it's obvious if you never won, you'd have zero variance.

And if you always won - you always got first - you'd also have no variance.

But is there a relationship in the middle ranges?

In other words, if you graphed variance to ITM, would it look like a bell-shaped curve, or something else?

[Eder]

Well....after 100%+ ROI since i started 11$ Sng I've bin OTM last 8 ...variance is fact of life...hope I can ride this out before I need to get the drywall repaired lol...anyway it is inevitable and cards go dead as fast as they slap you in the head...nothing to be done but tilt a bit I guess...as far as bell shaped thats prob the shape of my brain atm

[dethgrind]

I wrote a perl script to figure this out. The format is this:
1st 2nd 3rd stdev

where stdev is expressed as a multiple of the buyin, ie 3.61 at $10+$1 means the standard deviation is $36.10. This was a while ago and it was sloppy, so this might actually be variance, and it could be wrong. Check the numbers.

Here is a list from .10 to .16 in increments of .015:

0.1 0.1 0.1: 2.80
0.1 0.1 0.115: 2.80
0.1 0.1 0.13: 2.80
0.1 0.1 0.145: 2.79
0.1 0.1 0.16: 2.79
0.1 0.115 0.1: 2.84
0.1 0.115 0.115: 2.84
0.1 0.115 0.13: 2.83
0.1 0.115 0.145: 2.83
0.1 0.115 0.16: 2.82
0.1 0.13 0.1: 2.88
0.1 0.13 0.115: 2.88
0.1 0.13 0.13: 2.87
0.1 0.13 0.145: 2.86
0.1 0.13 0.16: 2.85
0.1 0.145 0.1: 2.92
0.1 0.145 0.115: 2.91
0.1 0.145 0.13: 2.90
0.1 0.145 0.145: 2.88
0.1 0.145 0.16: 2.87
0.1 0.16 0.1: 2.95
0.1 0.16 0.115: 2.94
0.1 0.16 0.13: 2.92
0.1 0.16 0.145: 2.91
0.1 0.16 0.16: 2.89
0.115 0.1 0.1: 3.02
0.115 0.1 0.115: 3.01
0.115 0.1 0.13: 3.01
0.115 0.1 0.145: 3.00
0.115 0.1 0.16: 2.99
0.115 0.115 0.1: 3.06
0.115 0.115 0.115: 3.05
0.115 0.115 0.13: 3.04
0.115 0.115 0.145: 3.03
0.115 0.115 0.16: 3.01
0.115 0.13 0.1: 3.09
0.115 0.13 0.115: 3.08
0.115 0.13 0.13: 3.06
0.115 0.13 0.145: 3.05
0.115 0.13 0.16: 3.03
0.115 0.145 0.1: 3.12
0.115 0.145 0.115: 3.10
0.115 0.145 0.13: 3.09
0.115 0.145 0.145: 3.07
0.115 0.145 0.16: 3.05
0.115 0.16 0.1: 3.14
0.115 0.16 0.115: 3.12
0.115 0.16 0.13: 3.11
0.115 0.16 0.145: 3.09
0.115 0.16 0.16: 3.06
0.13 0.1 0.1: 3.23
0.13 0.1 0.115: 3.22
0.13 0.1 0.13: 3.21
0.13 0.1 0.145: 3.19
0.13 0.1 0.16: 3.18
0.13 0.115 0.1: 3.26
0.13 0.115 0.115: 3.24
0.13 0.115 0.13: 3.23
0.13 0.115 0.145: 3.21
0.13 0.115 0.16: 3.20
0.13 0.13 0.1: 3.28
0.13 0.13 0.115: 3.27
0.13 0.13 0.13: 3.25
0.13 0.13 0.145: 3.23
0.13 0.13 0.16: 3.21
0.13 0.145 0.1: 3.30
0.13 0.145 0.115: 3.29
0.13 0.145 0.13: 3.27
0.13 0.145 0.145: 3.24
0.13 0.145 0.16: 3.22
0.13 0.16 0.1: 3.32
0.13 0.16 0.115: 3.30
0.13 0.16 0.13: 3.28
0.13 0.16 0.145: 3.25
0.13 0.16 0.16: 3.23
0.145 0.1 0.1: 3.42
0.145 0.1 0.115: 3.41
0.145 0.1 0.13: 3.39
0.145 0.1 0.145: 3.38
0.145 0.1 0.16: 3.36
0.145 0.115 0.1: 3.45
0.145 0.115 0.115: 3.43
0.145 0.115 0.13: 3.41
0.145 0.115 0.145: 3.39
0.145 0.115 0.16: 3.37
0.145 0.13 0.1: 3.47
0.145 0.13 0.115: 3.45
0.145 0.13 0.13: 3.42
0.145 0.13 0.145: 3.40
0.145 0.13 0.16: 3.38
0.145 0.145 0.1: 3.48
0.145 0.145 0.115: 3.46
0.145 0.145 0.13: 3.43
0.145 0.145 0.145: 3.41
0.145 0.145 0.16: 3.38
0.145 0.16 0.1: 3.49
0.145 0.16 0.115: 3.47
0.145 0.16 0.13: 3.44
0.145 0.16 0.145: 3.41
0.145 0.16 0.16: 3.38
0.16 0.1 0.1: 3.61
0.16 0.1 0.115: 3.59
0.16 0.1 0.13: 3.57
0.16 0.1 0.145: 3.55
0.16 0.1 0.16: 3.52
0.16 0.115 0.1: 3.63
0.16 0.115 0.115: 3.60
0.16 0.115 0.13: 3.58
0.16 0.115 0.145: 3.56
0.16 0.115 0.16: 3.53
0.16 0.13 0.1: 3.64
0.16 0.13 0.115: 3.61
0.16 0.13 0.13: 3.59
0.16 0.13 0.145: 3.56
0.16 0.13 0.16: 3.53
0.16 0.145 0.1: 3.65
0.16 0.145 0.115: 3.62
0.16 0.145 0.13: 3.59
0.16 0.145 0.145: 3.56
0.16 0.145 0.16: 3.53
0.16 0.16 0.1: 3.65
0.16 0.16 0.115: 3.62
0.16 0.16 0.13: 3.59
0.16 0.16 0.145: 3.56
0.16 0.16 0.16: 3.52

[Irieguy]

First I think we have to define how we measure variance in SNGs. In ring game play, profit/loss is expressed in dollars per hour and so is variance.

But with SNGs, we use ITM and ROI to express our profitability (though we usually convert that to dollars per SNG, at least I do.)

So, what makes the most sense would be to express variance in dollars per SNG. If your ITM was 100%, your dollars per SNG win rate would still have some variance, depending on your proportion of 1sts, 2nds and 3rds. If your ITM rate was 0%, you would have no variance. But as soon as your ITM rate was greater than 0, your variance would be large.

Variance is going to be a function of the magnitude of diffences between results. So, if you think of possible SNG results in terms of magnitude, the possibilities are: -1, +1, +2, and +4 for PP's payouts(I'm ignoring the vig to make the numbers whole numbers). This means that your variance is affected most significantly by OOTM finishes, and 1sts. Here are the variance implications as they relate to your question:

1. For a given ROI%, variance increases as ITM% decreases.
2. For a given ITM%, your variance increases as your ROI increases (because of a greater number of 1sts).

So, the shape of the graph is not bell-shaped. I'm not sure what shape it is, or if ROI changes would change only its position, or its position and shape. But I bet eastbay, aleo, rachel, or one of the other stat whizzes knows.

Irieguy

[LinusKS]

Deth, I'm not sure I understood what you did here (perhaps because I'm not that bright - I don't even know what Perl is), but it looks like you're saying variance increases as ITM goes up - at least from 30% ITM to 48% ITM.

Is that about right?

[LinusKS]

Defining what you mean by variance is important.

The kind of variance most players are interested in is negative variance, in other words, what they want to know is, "Assuming I'm a winning player, how often can I expect to lose?"

The problem with looking at it like that, though, is that it leads to a circular response (a truism): "Assuming you're a winning player, the more you win, the less you'll lose."

That doesn't strike me as a very helpful way of looking at the problem.

The textbook definition of variance, I think (please feel free to offer a different one) is something like, "How much your actual results will differ from the mean."

If you look at it like that, a graph of a set of results (expressed as profits over time) that looked like this:

_____________x
__________x
________x
_____x
___x
_x

would have a very low variance.

A graph like this

_____________x
_____xx_____x
___x___x___x
_x______x_x
_________x

would have a much higher variance.

In the first, the actual results look almost exactly like the mean, in the second, the results depart significantly from the mean.

By that definition, a very low ITM will still have a very low variance, since only a handful of your results will differ from the mean. A graph of the results will look very smooth, with only a few small breaks in the line.

I'd argue that looking at it this way makes sense, since it's streaks of wins and losses that create variance, not individual results.

It makes particularly little sense to look at variance in terms of individual wins and losses when it comes to tournaments, because there are only four possible pre-set results to each trial.

[AleoMagus]

By it's strict definition, Variance is a measure of how spread out a distribution is. It is the average squared deviation of each number in a sample from it's mean.

Standard deviation is far more often used as a measure of spread in SNG poker. It is the square root of variance.

You are correct in saying that similar results will yeild a smaller standard deviation. In this way, someone who finished in the money an extremely large number of times, AND who placed (say) 2nd almost always would have an extremely small standard deviation.

Practically speaking, however, as your ITM rises, so also will your SD. The reason players would probably never see a lessening of their SD as results improved is that it is probably not possible over a significant sample to improve results that much. Many consider 50% ITM to be an upper bound and to really see the lessening of variance again you would probably have to be doing way better than that.

As far as the initial question, that is actually a very complicated question.

One way to answer it I guess would be to define a sample size (for example, 2 sngs because it's really easy) and then consider all of the possible combinations of outcomes. ie... for a 10+1 sng

1,1...100%ITM...SD=0
1,2...100%ITM...SD=10
1,3...100%ITM...SD=15
2,2...100%ITM...SD=0
2,3...100%ITM...SD=5
3,3...100%ITM...SD=0
1,O....50%ITM...SD=25
2,0....50%ITM...SD=15
3,0....50%ITM...SD=10
0,0.....0%ITM...SD=0

Now, I guess that you might want to either leave the 0 SD spikes in, but if you are just talking about average %ITM correllation to SD, you might want to average them. Assuming they can correctly be averaged without any weightling on certain finish places (a big assumption which isn't true) then this would look like...

0%ITM.....SD=0
50%ITM....SD=16.7
100%ITM...SD=5

So, now do this for every combination of whatever sample you desire (100SNGS?1000?) and you will sort of have an answer. A huge amount of work to be sure. No doubt there is a simpler way that some math whiz could tell us about.

And then, there is the fact that this might not really answer your question properly because certain finish combinations are no doubt, more likely than others.

Who knows... that's a start anyways.

Regards
Brad S

[dethgrind]

Variance can't be expressed as a function of ITM. Irieguy's post is pretty on the money. Variance is determined by the specific breakdown of 1st, 2nd, and 3rd finish probabilities. So no, variance doesn't necessarily increase as ITM increases. In fact, as Irieguy pointed out, variance decreases as ITM increases if you keep ROI fixed.

Here is the list again, sorted by increasing ITM. The format is this:
ITM ROI 1st 2nd 3rd: variance

the list

[eastbay]

[ QUOTE ]
Ok, it's obvious if you never won, you'd have zero variance.

And if you always won - you always got first - you'd also have no variance.

But is there a relationship in the middle ranges?

In other words, if you graphed variance to ITM, would it look like a bell-shaped curve, or something else?

[/ QUOTE ]

I'll answer with a question which should make it clear:

If I play two tournaments, and place in one of them, my ITM is 50%. What's my variance?

eastbay

[Irieguy]

Thanks Linus for forcing me to rethink this issue. I now realize that I have absolutely no clue on the topic.

I'm going to play with some numbers and look through some statistics resources, but I'm pretty sure the precise answer to your question is much to complex for me to figure out. I hope somebody smart can save me the trouble and explain this in terms I can grasp.

[img]/images/graemlins/crazy.gif[/img] Irieguy