The quantum variance quandry

[Irieguy]

Linus' post got me thinking quite a bit about variance in SNGs, and I have an idea as to why it's so hard to talk about.

We borrow the concepts of hourly rate and variance from live game play. It's helpful in that context to know how much you are earning, and how big your swings will be. If, for example, you earn $50 an hour and your standard deviation is $150/hr, you have some useful information. You know how much you will be up after a significant amount of play, and you know that 97.3% of the time or so, you will win up to $500 or lose up to $400 per hour.

But that technique doesn't work for SNGs, nor is it helpful. With SNGs, we win and lose in quanta. Either -1 unit, +1 unit, +2 units, or +4 units (minus vig). It's still helpful to come up with a win rate per SNG that will not be one of those quantum digits because we can average the number over a significant number of SNGs and calculate how much we can expect to make. That's helpful.

But, is it helpful to know what the limits of what we can win or lose per SNG or per hour? We already know that answer. 100% of the time we will either lose 1 unit, win 1 unit, win 2 units, or win 4 units. You can extrapolate those quanta to hourly numbers too by simply multiplying by how many SNGs per hour you play. But you know before you start a session what your possibilities are for winning and losing.

So variance in that regard is a useless measure for the SNG player. What we really need to know is, "how many of these things can I lose in a row?" Or, "what's the maximum number of SNGs I can play and still have a negative ROI if my ITM is X%?"

The answers to those questions measure our functional variance, and fortunately we have plenty of discussions to refer to regarding those questions. But, the bottom line is this:

The higher your ITM%, the lower your functional variance will be.

I really think that's the answer to Linus' question.

Irieguy

[Irieguy]

Smiley gives a nice example of why our conventional use of variance and standard deviation does not apply to SNGs.

He lists his results in the $20 SNGs and has a SD of $38.45. Now, mathematically, that means that he can expect to win or lose more than about $77 in 1/3 of the SNGs he plays. But really, his chance of losing more than $20 or winning more than $80 is clearly zero. (again ignoring vig).

So it seems like our standard methods of figuring variance don't work in the SNG setting.

Irieguy

[AleoMagus]

I thought a lot about this once also and even posted a similar question in the probability forum. here

I've since stopped caring and actually think that the usual conceptions of SD and variance work pretty well once the sample gets big enough.

The quantum nature of SNG results does change things a bit, but not so much as you'd expect. Limit Hold'em games actually give quantum results too, we just don't see it that way very often. Your betting is all in multiples of the Blinds and these are your 'quanta'. We know that after a lot of hands though, we could be up or down just about any amount and so the more conventional ring game SD seems to make sense. You could attatch a SD to the return on each bet you make in a limit game however, and the result would definitely be something weird that is not an actual possibility just as most SNG SD figures are not an actual possibility.

Where it starts to become meaningful is after lots and lots of bets... or lots and lots of SNGs. I actually prefer to look at my 100SNG SD when I'm thinking about results. In fact, I'm starting to think about 100 SNGs as 1 SNG these days. Perhaps this is unclear.

I guess what I mean to say is that if we knew we played some game where a whole bunch of stuff happened and after all that 'stuff' we could be up or down anywhere from -3000ish to +10000ish, we would not really be too sceptical of the SD figures. they would make sense. Well, all that 'stuff' is my latest 100 SNG sample.

Hope this does something to answer the question. Any thoughts?

Regards
Brad S

[PrayingMantis]

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I actually prefer to look at my 100SNG SD when I'm thinking about results. In fact, I'm starting to think about 100 SNGs as 1 SNG these days.

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One can actually look at his/her whole poker career as a single bet. [img]/images/graemlins/grin.gif[/img]

But you make very nice points. I agree that the "quantum" aspect of SNGs is the same for any kind of poker, only with different "quanta", be it BB, dollars pounds or cents. Money, in essence, is not something that can be endlessly divided into smaller elements.

[rachelwxm]

I just post on the original Variance thread, I though I might as well move it here. [img]/images/graemlins/cool.gif[/img]
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!!

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.

[dethgrind]

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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.


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I get different values for STD.

Case 1:
variance= .13(39-2)^2 + .13(19-2)^2 + .13(9-2)^2 + .61(-11-2)^2 = 325
STD = sqrt(325) = 18.03

Case 1:
variance= .39(39-8.5)^2 + .61(-11-8.5)^2 = 594.75
STD = sqrt(594.75) = 24.39

Of course I only checked your numbers because they disagree with my list [img]/images/graemlins/smile.gif[/img]

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I would be surprised to see two guys with similar ROI have drastically different STD.

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I wanted to see how accurate this statement was. Looking at ROI's of .18 on my list (which has 1st/2nd/3rd finishes ranging from .10 to .16), the lowest variance was 2.89 and the highest was 3.61. For 10+1, this means STDs of $17 and $19 respectively. So by this example, I'd say it's damn accurate.

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

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Could you summarize again why you don't think ROI/STD is important? I'm not sure I follow.

[AleoMagus]

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I would be surprised to see two guys with similar ROI have drastically different STD.

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I would be surprised to see two guys with drastically different SD. Period (actual people with realistic results I mean)

Regards
Brad S

[rachelwxm]

dethgrind,

Thanks for checking the numbers. I generate 100 data points and use excel function stdev(), it uses unbiased estimate of standard deviation (dividing the total variance by N-1 not N). So you might argue that there might have a small common shift in all the number I provided, but the whole idea is still there. [img]/images/graemlins/smile.gif[/img]

As far as ROI/STD is not important. I know this is a pretty bold statement. A lot of people might have different opinions regarding this. I think there are two important factors determine the bank roll fluctuation (losing stream consideration see my thread on this):
1. the distribution of your finishes, 1st, 2nd etc
2. the autocorrelation of your results

I tend to believe most of the winning players at SNGs have similar pattern in terms of 1. And 2 is not captured in std calculation per game. If you have positive autocorrelation in your SNG result, in order to take into account this, you need to calculate std per n game for example. And if your games is not correlated, then

STD(n game)=sqrt(n) * STD(1 game)

But this is in general not true in practice. The existence of positive autocorrelation plays an important role in your losing stream or bank roll consideration. (see the theoretical analysis and some off the real example in my thread) And ROI/STD (per game) does not capture that.

That’s just my thoughts.

[dethgrind]

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And if your games is not correlated, then

STD(n game)=sqrt(n) * STD(1 game)

But this is in general not true in practice.

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Is this really true? I don't have enough of my own data to agree or disagree, I'm just not sure I believe it. Is this an effect of playing worse when on a losing streak and better when on a winning streak? My intuition tells me that if I always bring my A game to the table, my games will pretty much be uncorrelated. So STD per game will be an accurate and useful tool, even if it is about the same for every player.

[rachelwxm]

You can think of this intuitively. Let's assume your win 50% of time and your pnl is either 1 or -1. If you do simulation, your std per game is 1. STD(n game)=sqrt(n) according to Random walk theory from stochastic calculus.

But if your result if auto correlated, another word, you will likely lose money if you lose last time because of tilt or other effect, then if you look at pnl for every two games, the STD is greater than sqrt(2). This is because the consecutive games are not independent any more.

Hope this helps.