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 (http://www.bol.ucla.edu/~sharnett/ITMvariance.html)

[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.

/images/graemlins/crazy.gif Irieguy

[LinusKS]

Aleo, thanks for your response.

I read the following paragraph

[ QUOTE ]
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.

[/ QUOTE ]

to mean that variance - or the standard deviation - is highest around 50% ITM. If that's wrong, let me know.

I avoided math in college. I'm finally getting punished for it.

[LinusKS]

It still looks like there's a connection, based on that list. It looks like (I didn't count them) out of the first twenty or so the variance ranges from about 2.8 up to 3.4. In the last twenty there are no results below 3 at all. I don't know if that means anything. Perhaps it's statistically insignicant.

[LinusKS]

I'm not sure variance is meaningful in a sample of two - certainly it's not meaningful in the sense of variance that I was using - the amount the actual results differ from the mean. If you have two results, you don't really have a meaningful average. Although the more your two results differed from each other, the more "variance" you'd have.

[eastbay]

[ QUOTE ]
I'm not sure variance is meaningful in a sample of two - certainly it's not meaningful in the sense of variance that I was using - the amount the actual results differ from the mean. If you have two results, you don't really have a meaningful average. Although the more your two results differed from each other, the more "variance" you'd have.

[/ QUOTE ]

I thought that's what you'd say. It's actually irrelevant to the point I'm trying to make.

But, after thinking about the first question, think about this one:

You play 1,000,000,000 tournaments. Your ITM is 50%. What's your variance?

eastbay

[KJ o]

[ QUOTE ]
But, after thinking about the first question, think about this one:

You play 1,000,000,000 tournaments. Your ITM is 50%. What's your variance?

[/ QUOTE ]

Unknown, since it depends on your ROI. And I'm not sure it's interesting either... That same question with ROI instead of ITM would, IMO, be interesting.

[eastbay]

[ QUOTE ]
[ QUOTE ]
But, after thinking about the first question, think about this one:

You play 1,000,000,000 tournaments. Your ITM is 50%. What's your variance?

[/ QUOTE ]

Unknown,


[/ QUOTE ]

Correct. And for the same reason as 2 tournaments.

[ QUOTE ]

since it depends on your ROI.


[/ QUOTE ]

Eh. Sort of.

[ QUOTE ]

And I'm not sure it's interesting either...


[/ QUOTE ]

Agree.

[ QUOTE ]

That same question with ROI instead of ITM would, IMO, be interesting.

[/ QUOTE ]

Not very, IMO. I tend to worry about things which make me more money. I don't see how these questions do that.

eastbay

[Irieguy]

Quote:
--------------------------------------------------------------------------------


That same question with ROI instead of ITM would, IMO, be interesting.


--------------------------------------------------------------------------------



"Not very, IMO. I tend to worry about things which make me more money. I don't see how these questions do that."

Yes, I agree. It was in my coming to the conclusion that I don't care about variance in this way that I stumbled upon what was an epiphany for me. I would be interested to know what you think of my quantum variance quandry posted on this page.

Irieguy

[KJ o]

[ QUOTE ]
Not very, IMO. I tend to worry about things which make me more money. I don't see how these questions do that.

[/ QUOTE ]

I worry about both risk and return. The latter is of course more interesting in a poker setting, but the former is very important if risk of ruin is even the slightest concern.

Would you play $100k SnG with a 1% ROI? You'd make a cool $1k/tourney, which would be great. Chanses are you'd also go broke soon unless you are *extremely* rich to beign with.

[PrayingMantis]

Some thoughts with regard to this discussion:

First, I generally agree with eastbay. This could be an interesting theoretical debate, however, at the point where it is now, I can not see how it helps a good player in gaining even one more $1 in real-life. I'll explain.

Basically, this whole variance and SD issues, when they relate to SNG enviroment, are relevant pretty much _only_ in some psychological aspects of BR management. For instance: player X has $600 BR. Should he play $100 SNGs? Well, even with the most perfect knowledge of his ROI and ITM at this buy-in, we cannot answer this question properly, without knowing specifically his thinking and mentality as a poker player. Notice, for instance, that if he's going to play only one $100 SNG now, he's risk of ruin is 0. If he's strong enough mentally to play this one game, and if he loses it to go down playing $20 SNGs, for instance, then there's no problem IMO, playing this one $100. However, if he's a player that can't handle going down, and he's going to try another $100 if he loses, and then another, UNTIL HE BUSTS, than clearly his ROR is starting to get significant. However, even some crazy player will probably understand that if he's only left with $100, he better play a lower-buy in, and so on and on. So what's his "real" ROR? You have to know his exact behaviour in every specific point, and as in other areas of poker, you can never know that. Nobody knows that.

So, this is a psychological issue, rather than mathematical one. I'd say that for certain players (strong enough mentally, and good players) it makes perfect sense to go up in buy-in every time they have 12 buy-ins (just a number), and try their "luck", if they know that if they "fail", they can go back down. I think Daliman's story is an example for a player who wasn't strong enough menatally to go down in buy-ins, although he's no doubt a great player (I'm not judging him for that, of course, and I'm not sure he even had to go down. It's him to decide). But basically, No SD, variacne or "ROR" calculations will change this crucial aspect of poker, from the perspective of the player.

So, when we speak about ROR and variance in a very abstract way, we do not take in consideration many much important aspects, that are mainly psychological. Because we speak about variance and ROR as if an SNG player is a kind of a random numbers generator, while in reality it is quite the opposite, and a player has a huge control on how he handles situations of losing, or winning, X games in a row. The way he treats his BR, and the risk vs. reward he's willing to take, is something that is at least important as speaking stricktly about "pure" statistical data (of course, there are theories in economics, for instance, that deal with all this, I'm not an expert or anything).

Another importat point, that I think is missing here, is that variance is very much dependant also on the style of play (of the specific player and the field he's against). Think about this: 2 players can have the same exact ROI%, and same ITM%, and still their SD could differ, because of the specific distribution of their finishes.

So all in all, I think a strong SNG player, with strong mentallity and no serious ego-problems, can know pretty little about variance and SD, and still kill the games. I don't see how knowing more about his SD could "improve" his game, excepy only in some extremely minor way (taking a decision that gives him same _exact_ ROI as other decision, but with lower variance. Is this really "improvement"? Can you even think of such an example in an SNG?)

Please flame. /images/graemlins/grin.gif

[eastbay]

[ QUOTE ]
[ QUOTE ]
Not very, IMO. I tend to worry about things which make me more money. I don't see how these questions do that.

[/ QUOTE ]

I worry about both risk and return. The latter is of course more interesting in a poker setting, but the former is very important if risk of ruin is even the slightest concern.

Would you play $100k SnG with a 1% ROI? You'd make a cool $1k/tourney, which would be great. Chanses are you'd also go broke soon unless you are *extremely* rich to beign with.

[/ QUOTE ]

Sure. But these types of questions were answered in full eons ago.

eastbay

[AleoMagus]

Well, strictly speaking, 50% ITM would be the highest SD possible provided that your results were all 1st and OOTM. Higher or lower ITM% could easily yeild more variance practically speaking.

Actually, now that I think about it, in order to really get a 50% ITM, I'd bet you'd have a relatively low SD in comparison to some. The reason for this is that I think in order to get 50% ITM you would be sneaking into 3rd a lot with a shortstack instead of taking some risks to go after 1st. So 50% ITM would yeild a lot of 3rd place finishes (I suspect) that might otherwise be 1sts or 4ths. 3rd and OOTM is smaller in terms of variance from the mean so your SD might go down.

I should also make it clear however that this is NOT what you want. You want a 40%ish ITM with lots of variance and a nice healthy ROI. Higher ROI figures will usually give you more variance because that means more 1st place finishes and 1st place finishes deviate from the mean quite a bit.

Regards
Brad S

Edit. One other important thing.
Almost all SD results that you will find in a realistic sample of SNGs will be close. Closer than you might expect. A completely average player with absolutely uniform finish results for example will have an SD of about $16+ in $11 SNGs. The best players who finish ITM 45% and get ROI 40% will be getting about $19. I have said in the past that almost any SNG winner who wants to immediately estimate his SNG SD can just take the net profits for a second place finish in their particular SNG level and use that. IT will be close every time (usually a bit too big but not by much).

This is to say that SNGs played in any realistic fashion will yeild very similar amouts of variance. So why doesn't it feel that way? Because some are big winners and some are big losers but some are playing a break even kind of game. Foe those whose results are not netting them a big profit or loss either way, that fluctuation can feel huge because it can make a player go from moderate winner to moderate loser over even large samples. Players who are getting 30%+ ROI tend not to worry so much about variance because even the big negative swings will usually still net them a slight profit or at worst only a small loss. (unless of course they are living off their profits. Then it can feel pretty bad too to make $50 in a month)

[rachelwxm]

This is an interesting topic, here is my thoughts, flame and discussions welcome.

First, generally people are negative about standard deviation since it creates random fluctuations in you bank roll. Is this really that bad? Let’s think about two imaginary scenarios:

Scenario 1. Let’s say your ITM is 39% and equally divided into 1st 2nd and 3rd. For 10+1 the average ROI=2 and STD=18.12.
Scenario 2. Now if you increase the 1st finishes, let’s say all your ITM is first. Now ROI=8.5 and std=24.5. Now your STD increases, but you sure love the second Scenario better. So the std alone is not a good indicator of how you do.

Thought 1.
It does not make sense to think standard deviation alone without ROI. In fact if you want to use std, ROI/std is better measure since the two cases gives 11% and 34.6%. This is pretty much like Sharp Ratio in statistical arbitrage.

Now go back to real life. I category two types of players in SNG games. Most of the people follow the strategy from Rock-Maniac transition, but I constantly see people even in 10+1 games play very loose and aggressive early on try to build a big stack lead by out playing weak opponent. Let’s use Scenario 1 to describe the first type player and for second type, let using
Scenario 3. ITM 30%, 1st 22% 2nd 4% 3rd 4%. ROI=2 and STD=20.91.
We see that reduce the ITM from 39% to 30% hurt the result a lot, you need to have 22 of 30 1st finishes in order to get the same ROI while the ROI/STD is 9.5% less than the first scenario. That’s probably why there are more people play tight early on.

But as long as your play falls into one of those two catogories, STD does not have much meaning (as least not negatively). As we see in scenario 2, higher STD is welcome while fixing you ITM. To see another scenario, assuming your ITM=39%, you have only 1st and 3rd finishes, to match the same STD, you need to have a break down of 1st/3rd 15/24 std=18 and ROI is only 1.3!! /images/graemlins/shocked.gif

I think one of the main reason people care about ROI/STD is trying to see the leverage effect. For a good strategy, if you have same ROI but much less STD, your back roll allows a much bigger leverage effect. Of course, everybody like to see a linear increase of bank roll, but the discreteness of the payout (you have only 4 payout choices 39/29/9/-11, you cannot be paid exactly 4$ every time you play although that will be ideal!) couples with the fact the more or less correct strategy has very similar STDs make STD a very weak indication. I would be surprised to see two guys with similar ROI have drastically different STD.

So overall, if you play rock-maniac strategy, I don’t think STD is very revealing nor is ROI/STD.

I am sure there are errors in my post, let me know what you think.
/images/graemlins/smile.gif

[AleoMagus]

[ QUOTE ]
So all in all, I think a strong SNG player, with strong mentallity and no serious ego-problems, can know pretty little about variance and SD, and still kill the games. I don't see how knowing more about his SD could "improve" his game, excepy only in some extremely minor way (taking a decision that gives him same _exact_ ROI as other decision, but with lower variance. Is this really "improvement"? Can you even think of such an example in an SNG?)

[/ QUOTE ]

You are right, of course

... but some of us are big nerds and like to talk about these things anyways. /images/graemlins/grin.gif

And the same goes for the bizzare, absolutely irrelevant preflop whole table AA examples! And all the other weird never-will-apply-to-any-game-ever theoretical conversations!

One of these days the probability forum is going to get jealous.

Regards
Brad S

[PrayingMantis]

[ QUOTE ]
... but some of us are big nerds and like to talk about these things anyways

[/ QUOTE ]

Don't get me wrong, I enjoy thinking about "non-practical" matters quite a lot. And your particular thoughts about SD and SNGs are very interesting, IMO. But I feel there are people who might think that some of this could have a significant impact on their decisions, regarding the play in a specific game, and their BR management, if they could understand SD and ROR "fully". And I thought I could help by pointing out it's more of a theoretical, speculative debate.

/images/graemlins/laugh.gif