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  #41  
Old 08-31-2007, 06:04 PM
Leader Leader is offline
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Default Re: A plea to omgwtfnoway (re the variance thread blowup)

[ QUOTE ]
this thread is so tl;dr its insane.

[/ QUOTE ]

qft. I suggest no one say anything even close to the line in this thread because, if you think I'm going to go back and try to figure out if you were provoked, you've lost your mind.
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  #42  
Old 08-31-2007, 06:06 PM
TNixon TNixon is offline
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Default Re: A plea to omgwtfnoway (re the variance thread blowup)

[ QUOTE ]
tbh it would make more sense for 100BB+ HUCASH to be lower variance than a sng with the buyin equal to 100BB in the cash game.

you have 75BBs to work with and it decreases as time goes by.

If you forget about rake, playing a $100 + 0 HUSNG (1500 chips) would be equivalent to playing $100HU with a BB of $1.33 (=blind of 15/30)then it goes up to $3.33 (blinds of 25/50) and even up to $13.33 (blinds 100/200).

[/ QUOTE ]

This is exactly why it's so important to establish once and for all that a $10 10BB stack is so very different from a $100 10BB stack.

Which is something that omg's math actually proved in the other thread, after the appropriate conversion from one set of bb/hand units to the other.

But that's the point of contention here. That bb/hand is just bb/hand, and you can compare any 2 values without taking into consideration what the blinds actually were when the calculation was made.

I've used up every way I can think of to try to explain why that is false, though. If the distance/second bit wasn't convincing, then probably nothings else can possibly be.

If I thought it would do any good, I would go back through omgs math, and compute everything in $ instead of big blinds, which would clearly show that the $100 100BB stack (at 0.5/1) is much lower variance than the $100 10BB stack (at 5/10).

But how can any mathematical argument possibly work on somebody who refuses to admit that one calculation of bb/hand is not necessarily directly comparable to a separate calculation of bb/hand, if what you're trying to determine is the actual effect on your bankroll.

Still, I'm a little bit surprised that there are two people out there who disagree strongly enough to throw out insults, but not strongly enough to try to pick up what should be the easiest $200 of their lives. I mean, if they're that sure they're right, it's free money, right? Not that I haven't thrown out my fair share, but I'm perfectly willing to put my money where my mouth is. omgwtfnoway already gave me all the formulas I need to prove almost conclusively that the correct choice in the $1k bankroll question is choice A, playing for full buyins at 0.5/1 over 10BB buyins at 5/10.

[ QUOTE ]
none of these formulas take into account the variance of variance. we've all seen people go absolutely crazy esp in heads up.

[/ QUOTE ]
One thing at a time...one thing at a time.

It would be a good idea to get a consensus on variance before even considering second-order effects. Not that any such consensus seems likely anytime soon, if ever. [img]/images/graemlins/smile.gif[/img]
[ QUOTE ]

qft. I suggest no one say anything even close to the line in this thread because, if you think I'm going to go back and try to figure out if you were provoked, you've lost your mind.

[/ QUOTE ]
lol.

Fair enough. I am trying to be a good little boy in this one. [img]/images/graemlins/smile.gif[/img]

Sorry about the verbosity and the long series of posts everybody, but obviously I'm trying every possible approach to prove something that *should* be obviously true to anybody who thinks about it for just a minute.

Obviously I just type too fast, or I wouldn't be able to put nearly as much up in the short periods of time that I have been.

:/
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  #43  
Old 08-31-2007, 06:17 PM
TNixon TNixon is offline
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Default Re: A plea to omgwtfnoway (re the variance thread blowup)

LAST CHANCE ON TAKING ME UP ON THE VARIANCE BET. I'm laying 2-1 odds, for a max of $100 of your dollars ($200 of mine). The proposal is earlier in the thread. Let me know if it's buried too deep in other crap and you need a refresher.

If I take this to the probability forum, and they shoot me down, you will have lost your chance at a free $200, if you really feel that strongly that I'm off my rocker here.

The offer stands open until the end of the day today. Tomorrow I cross the forum line, into a group of people that hopefully have more experience running variance calculations and comparisons than most of the people in this one (myself certainly included. I didn't even *know* the formulas until silly errors in my earlier sanity checks forced me to go looking definitions and formulas and stuff up on the internet)
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  #44  
Old 09-01-2007, 01:45 AM
LordMushroom2 LordMushroom2 is offline
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Default Re: A plea to omgwtfnoway (re the variance thread blowup)

[ QUOTE ]
when i tried shortstacking (20bb) as a learning process for nl (over 50k hands), my SD was 23bb per 100. when i played fullstacked, 100bb, over 300,000 hands, my SD was 38bb per 100. these might be PTBB but the comparison stands nonetheless.

[/ QUOTE ]

I think thatīs PTBB because I saw someone posting that he had a standard deviation of almost 100 big blinds per 100 hands.

Can some of the difference in shortstack and big stack standard deviation be explained by you having a smaller skill-advantage over your opponents when playing full-stacked (because you were playing at higher stakes), or would any such effect be insignificant?

Why big stacks have higher variance than shortstacks.

I have always thought that the size of the average pot dictated variance, but now I can see how it doesnīt.

Say there are two (non-poker) games. In one every pot is $10 and in the other the pot is $0 50% of the time and $20 50% of the time. Which one has the greatest variance?

I donīt have a way to calculate it mathematically, but by putting it in a poker context, I can illustrate it. Say player A is playing $5 HU SnGs and player B is alternating between playing play-money HU SnGs and $10 HU SnGs. Both have $150 bankrolls and both make $2,5 per hour.

The variance in player Bīs case must be higher because his risk of ruin is greater because he is in reality playing only $10 SnGs and we know the risk of ruin is greater when playing $10 SnGs instead of $5 SnGs all else being equal.

As risk of ruin is calculated using bankroll, winrate and standard deviation, and we know bankroll and winrate are equal, the reason for player Bīs higher RoR must be a higher standard deviation.

What was the point of all this? When you play with a big stack in a cash-game, the size of the pots will vary to a greater degree from the average pot than it would if you were playing a shortstack, which means a big stack experiences higher variance than a shortstack. And my statement that stacks didnīt matter on variance is false.

The standard deviation of a normal SnG per 100 hands

Thanks to the guy I quoted, we have the standard deviaton per hundred hands in a full buy-in cash-game. Letīs try to find the standard deviation per 100 hands of SnGs.

This will vary depending on how many SnGs you can play in 100 hands and how big your winrate is.

To have something tangible to work with, letīs say you are playing a type of SnG, which lasts 50 hands on average, the fee is 5% and your winrate is 60%.

Since it lasts for 50 hands, we get to play 2 SnGs per 100 hands. Letīs see what the standard deviation for those 2 SnGs is (warning, I almost always mess up the math):

Var=(2^2)*0,6*0,6+[(1-1,05)^2]*0,6*0,4*2+(-2,1^2)*0,4*0,4-(2*0,6*0,6-0,05*0,6*0,4*2-2,1*0,4*0,4)^2
Var=(4*0,36)+(0,0025*0,48)+(4,41*0,16)-(0,72-0,024-0,336)^2
Var=1,44+0,0012+0,7056-0,1296
Var=2,0172
Standard deviation=1,42

If the 38PTBB/100 was at NL100, its standard deviation would be $76/100.
If the buy-in of the SnG was $100+5, its standard deviation would be $142/100.

This would suggest the variance is much higher in SnGs. Now let the slaughter of my math begin. [img]/images/graemlins/laugh.gif[/img]
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  #45  
Old 09-01-2007, 04:43 PM
TNixon TNixon is offline
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Default Re: A plea to omgwtfnoway (re the variance thread blowup)

Ok, I think I've got all my ducks in a row now, and I'm *very* disappointed nobody was willing to take me up on the bet, because it would have been such easy money. [img]/images/graemlins/frown.gif[/img]

There are two comparisons under consideration here. We want to compare the variance over 10 hands of a 100BB $100 stack vs a 10BB $10 stack (Case A), and then a 100BB $100 stack vs a 10BB $100 stack (Case B).

I'm going to borrow from the math conveniently provided in this thread: http://forumserver.twoplustwo.com/sh...page=&vc=1

Which proved that the variance of a short stack is lower than that of a big stack.

Only I'm going to take a slightly different tack, without making any assumptions about our player's edge, and see how it comes out.

The assumptions for the 10BB stack are that the player and the opponent are playing optimal push-or-fold poker, which gives the variance formula:

p is our player's "edge", the percentage of hands he will win
A is the average pot size, which for this situation is 8.99 big blinds, but we're going to leave it as a variable for a minute.

V = 10p((A-pA)^2) + 10(1-p)((0-pA)^2)

Which reduces to

V = 10p(1-p)A^2

Now for the 100bb case, we assume that every pot is average, which leads to the variance formula:

V = 10p((A-pA)^2) + 10(1-p)((0-pA)^2)

Which should look familiar. The variance calculation over 10 hands will be exactly this formula anytime you assume that every pot is going to be equal to the "average" pot, which is an assumption that was made in both the 10BB jam-or-fold and the 100BB situations.

The reduction is again

V = 10p(1-p)A^2

The conclusion we can draw from this is that variance depends on both the player's edge and on the average pot size. Which seems fairly obvious, but it's nice to see that the math backs up things that are intuitively obvious.

Time to plug some numbers in.

In the 10BB case, the average pot is going to be 8.99 big blinds. But since we want to make direct comparisons, and because we eventually want to be able to compare our results against HUSNGs (where measuring by big blinds doesn't even make sense), so we're going to calculate variance in dollars.

In case A, the 10BB player has $10, and in case B, the 10BB player has $100.

The maximum variance will occur when p=0.5.

In case A (bb = 1):

V = 10(.5)(.5)(8.99)^2 = 202.05

In case B (bb = 10):

V = 10(.5)(.5)(89.90)^2 = 20205

Obviously, the 10BB stack in case 2 has a much higher variance than the 10BB stack in case 1, because he's playing $100 at a time rather than $10 at a time.

In case 1, since the formula for the 100BB stack is exactly the same, varying by edge and average pot, it should be fairly obvious that given the same edge, the 100BB player will have a higher variance unlesss his average pot is smaller than the 10BB player.

But it's safe to assume that the 100BB player is going to have some sort of edge over optimal jam-or-fold, since with 100BB, we're actually playing poker, not shoving all the time.

If we give the 100BB player a very reasonable edge of 10BB/hundred hands (again, using the formulas from the other post, but using a more reasonable number):

p = (A + .2)/2A;

Subsituting this into our variance formula for p:

V = (10A^2 - 0.4)/4

Solving for A:

A = sqrt( .4V + .04 );

So for case 1, (V = 202.05), the average pot would have to be just over $8.99 for the variance to match.

Which, again, is intuitively obvious. Since we already know that variance only depends on the average pot size and the edge, if a player with $100 is playing pots with an average of $8.99, and has the same edge, his variance is going to be exactly the same as a player with $10 who plays average pots of $8.99. In this case, we've given the 100BB player a very slight edge, so he gets the same variance as the $10 player with a *slightly* larger pot size.

At this point, to have any way of determining whose variance is truly higher, we need to know the average pot size. We know that if the 100BB player generally has an average pot size greater than 8.99 big blinds, then his variance will be higher than the 10BB stack playing optimal push-or fold. But how does 8.99BB compare to *real* average pot sizes?

A quick sampling of some of the currently running games at full-tilt (11 $2/$4 tables and 22 $1/$2 tables) shows that the average pot is 7.87 big blinds at 2/4 and 7.47 big blinds at 1/2, for an overall average of 7.6BBs.

Of course, there's a very wide range here, from 2.22BB to 18BB, but with an average of 7.6, it could reasonably be said that cash-table play is, on average, lower variance than that of a player playing 10BB optimal jam-or-fold.

Which is actually somewhat of a surprising result, because I was totally willing to accept the argument that 100BB stack poker would be higher variance than 10BB stack poker, even when the blinds were the same, but that does not appear to be the case in the real world.

On to case B, where the 10BB player has $100 instead of $10:

In this case, v = 20205, giving us an average pot of $89.90.

So to have the same variance of a player playing $100 at a 5/10 table, our 100BB player with $100 at a 0.5/1 table would have to be playing average pots of $90, even if (and this is very important) we don't give him *any* edge at all over the 10BB player. If we increase the edge at all, the average pot is going to have to be bigger and bigger to give the same variance as the 10BB player playing jam-or-fold. Without going through the trouble of using "real" edge numbers (such as 10BB/hundred hands) here is a demonstration of how the average pot will have to grow.

p = 0.5, A = 90
p = 0.55 A = 90.45
p = 0.6 A = 91.855
p = 0.65 A = 94.346
p = 0.7 A = 98.198

So, if the player's edge happened to be as high as .7 (which is really, really, unreasonably high), he would basically have to be pushing every single hand before his variance would be as high as somebody playing optimal jam-or-fold with 10BB.

So, in case B, *even if his edge is only 50%, giving him the maximum variance possible*, the 100BB player's variance is clearly going to be lower than the 10BB player's variance, because he'd basically have to be playing optimal jam-or-fold to hit an average pot of $89.90.

And this is why you can't "simplify" the problem when trying to compare the variance between case A and case B by assuming that every 10BB stack is equivalent to every other 10BB stack, when comparing to a $100 100BB stack.

Attack at will, but this is not longer a matter of 'I say' vs "you say' anymore. The math is there, and if it is not correct, then point out where I've calculated something incorrectly. But if it is correct, then the people in this thread and the previous one who have tried to "simplify" the problem in a way that actually changes the result are the ones who have been lacking in understanding.

Conclusion: when throwing some real-world data into the mix, it seems that a player playing with 100 big blinds is *always* going to have a lower variance than a player playing with 10 big blinds, unless the 10BB player is playing lower stakes. (For example, a 100BB player at 5/10 would obviously have a higher variance than a 10BB player at 0.5/1)

A surprising result, but, as far as I can tell, a correct one.
*****************************************

The reason I've been fighting for so hard on this point, and from so many different angles, is this:

We want to be able to compare the variance between a $100 sit-n-go and a session at a cash table where you buy-in with $100. (something you simply cannot do if your variance calculations are in units of big blinds, by the way, which is why it's silly to calculate them that way for the intended purpose)

At the beginning of the cash table, you start with 100BB. At the beginning of the sit-n-go, you start with 50-100BB, and can advance to a point where your stack, even if it hasn't changed at all, is now worth 10BB, or even 5BB.

If every 10BB stack is equal when trying to compare variance (which is a statement that is so obviously false that I'm *still* flabbergasted that anybody could argue), then at the end of a sit-n-go, when you have 10BB instead of 75, you would be in a lower variance situation than you are in a cash table. And, in fact, since you started out with 75BB (which is a "smaller" stack than 100BB), you would be in a lower variance situation throughout the entire sit-n-go. If this were the case, then sit-n-gos would *obviously* be lower variance than a cash game. To the point that no further discussion is even necessary.

However, if, as I have just shown, there are significant differences between a $10 10BB stack and a $100 10BB stack (which, again, seems so mind-numbingly obvious that I'm amazed this discussion made it past "you can't compare variance values directly when the value of the BB is different"), then while it is true that your overall variance depends only on your winrate, that winrate will depend on the variance of the play of individual hands during the sit-n-go. And since we've already shown that a $100 10BB stack is higher variance than a $100 100BB stack (and that by comparison, a $100 75BB stack is also likely to be higher variance than a $100 100BB stack), you start the sit-n-go in a higher variance situation, and end it what is likely to be a *much* higher variance situation than a cash session that lasts as many hands as your sit-n-go does.

Of course, that doesn't necessarily mean that cash is higher variance than sit-n-gos, because there are other factors that affect the variance during a sit-n-go, for example, the fact that players can rebuy, and the fact that you're never playing for a double-stack or deeper in a sit-n-go, while you certainly can on a cash table.

But it does mean that there's actually a discussion to be had, and other factors to consider. If every 10BB stack compared to a 100BB stack is equivalent, then there's no discussion at all.

I'm going to ask BruceZ (a mod in the probability forum, who helped me get the final duck I needed in the right spot) to review this post and comment on it, but please let me know when it's safe to say "neener neener, I told you so".
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  #46  
Old 09-01-2007, 08:04 PM
LordMushroom2 LordMushroom2 is offline
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Default Re: A plea to omgwtfnoway (re the variance thread blowup)

I agree of course that a $10 stack at NL100 has a smaller variance than a $100 stack at NL1000. But I have a problem with one of your assumptions, and disagree that a $100 stack at NL100 has a lower variance than a $10 stack at NL100. The assumption that variance is dictated by only the average pot and the edge.

I would like to add (or rather replace average pot with) the spread in size of the pots, which is hard/impossible to find without some sort of observation.

I tried explaining why it must be so in my previous post in this thread under "Why big stacks have higher variance than shortstacks.", but I explained it pretty poorly.
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  #47  
Old 09-01-2007, 09:04 PM
olof86 olof86 is offline
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Default Re: A plea to omgwtfnoway (re the variance thread blowup)

wow, why would anyone care about this stuff ;0
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