re the variance thread blowup NL

[TNixon]

First of all, I want to apologize for getting carried away. I did feel like you were attempting to talk down to me through the entire thread, in a way that seems to be extremely common to students taking their first class on a topic, where they automatically assume they know everything there is to know, and that they can't possibly have made any mistakes.

Because of that, when you made what appears to be a very simple but fundamental mathematical error, I blew up in a very unbecoming way.

When two people disagree on a topic where there can only be one correct answer, and no amount of discussion seems capable of convincing either side that they're wrong, things are certain to get very heated unless the discussion is simply brought to an end.

Which is why Leader closed the variance thread. We reached a point where absolutely no good could possibly come of it.

This is why discussions of religion are typically nothing but big flame-fests. Both sides "know" they're right, and won't be convinced otherwise.

We have a very similar case here. I know I'm right, not about the relationship between variance at HUSNGs and HUCASH, because the jury is still pretty clearly out on that, but about the variance relationship between shortstack and deepstack poker. You also know you're right.

In short, I'd like to apologize for being such a jackass about it.

But there is a fact here somewhere, and we can't possibly *both* be right.

On my drive home, I think I managed to formulate an explanation of the issue at hand in a way that should be very straightforward, so I'm going to give it one last shot.

Because I, after all, am right, despite your unwillingness to accept it. [img]/images/graemlins/smile.gif[/img]

Whether my belief that I'm right is correct or not, this should either convince you, or provide the logical hole that you need to poke a hole in it all. So, without further ado:

The discussion is shortstack vs deepstack poker, and which is higher variance.

I will freely admit that I'm not completely up on the terminology and the definitions (formulas I can do with ease, but it's very possible that I'm going to call things something other than what they are), so I'd like to cover that first.

First of all, I'm going to be making comparisons based on the std. deviation. If variance is just the square of the std. deviation, then any comparisons will still hold true, as long as direct numerical comparisons aren't made. For example, if you have two calculated standard deviations, and std. dev A is higher than std. dev B, you can say with absolute assuredness that the variance of situation A is higher than the variance of situation B.

Forgive me if this seems too basic and elementary, but there have already been far too many points of confusion between us, if everything is laid out from square 1, then there can be no misunderstanding.

The reason I'm going to be using standard deviation rather than variance is that standard deviation is going to be measured in units that we can work with. It is highly likely, however, that I will mistakenly say "variance" when I really mean "standard deviation". Just assume that whenever I say variance, I really mean std. dev.

So, the std. deviation can be measured in many ways. In a cash game, a very convenient way to measure it is in BB/hundred hands, because that's very relevant to the play of a cash game. You could also measure it in BB/hand (or BB per orbit). Those would be valid measures, they would just require different formulas, and the numbers would be different. The point here is that those are all just different ways of measuring the same thing, in different units, in the same way that 12 inches is still 12 inches, even if you happen to prefer calling it a foot. But you can't measure the std. deviation of SNGs in BB/100, because that doesn't even make sense for HUSNGs (well, actually it does, and gives a good way to make direct comparisons between the calculated variance at SNGs vs cash games, assuming you have an average number of hands per SNG, but more on that later), so that can't possibly be the only valid measure. A more appropriate measure for SNGs would be in buyins/game (or buyins per hundred games, or whatever, it doesn't matter exactly). The point here is not to decide on a measure for SNGs, but to point out that you can measure the std dev however you want to, in whatever units are appropriate to the situation. This is going to be important, because we want to compare two situations, and BB/hundred hands is not sufficient, because the value of the big blind in question is going to be different, so they won't be directly comparable.

So, having established that, lets move on to the situations in question, the situation is this:

Player A is perpetually pushing a 100BB stack in on every hand, against an opponent who always has at least 100BB. If player A loses, they rebuy to 100BB, and if they win, they take money off the table, so that they begin every hand with 100BB.

Again, sorry if it seems I'm being pedantic. I am, but I feel it's necessary, to ensure that there are no possible points of misunderstanding.

Player B does the same thing, but with a 10BB stack instead of a 100BB stack.

So, both players are playing for their entire stack every hand, and they each have a 60% chance of winning.

You ran the math, determining the std. dev for both a 10BB stack and a 100BB stack. I have no reason to believe those formulas are incorrect, so I'll assume that they're valid. A recount of the results of your calculations:

10bb case.
hero's equity in this pot is (.6)(20bb)=12bb for a net of 2bb. so u=12bb/hand.
3/5 times hero actually wins 20bb. the other 2/5 he wins 0bb.
var=3*((20-12)^2)+2*((0-12)^2)=192+288=480
std dev=21.9bb/hand

100bb case
hero's equity in this pot is (.6)(200)=120 for a net profit of 20bb. u=120bb/hand.
3/5 times hero wins 200bb. the other 2/5 he wins 0bb.
var=3*((200-120)^2)+2*((0-120)^2)=48000
std dev=219bb/hand

So the std dev for player A is 21.9bb/hand, while the std dev for player B is 21.9bb/hand.

But we can only make direct comparisons using these units if the value of the big blind is the same. You can't directly compare 3 feet against 36 inches, because the units are different. To be able to make a direct comparison, and say that 3 feet is equal to 36 inches, you have to convert one of the values.

We're going to consider two cases. In the first case, both players are playing a 0.5/1 cash table, so player A's units are equal to player B's units, and we can make the direct comparison. In this case, player A has $100, with a std deviation of 219bb/hand, and player B has $10, with a std deviation of 21.9bb/hand. 219 is greater than 21.9, so player A, with 100BB, will have a higher variance than player B.

But in the second case, lets say that player B (with 10BB) is instead playing on a $5/$10 table. In this case, player B is still playing with 10BB, but they are playing with $100 instead of $10. The formula for std deviation, represented in bb/hand is identical. So player A still has a std. dev of 219bb/hand, and player B still has a std. dev of 21.9bb/hand.

But because the value of the big blind is different ($1 vs $10), we cannot immediately say that player A's variance is higher than player B's without making a unit conversion first. In this case, saying player A's 219bb/hand is greater than player B's 21.9bb/hand would be akin to saying that 36 inches is greater than 3 feet. That statement may or may not be true, but you cannot say one way or the other until the units are the same.

So, to be able to compare the numbers directly, to be able to say which player has a higher variance, we need to make a unit conversion.

Fortunately, there's a very simple unit conversion we can make here. Since we know the ratio of big blinds per dollar, we can convert from big blinds per hand into dollars per hand.

For player A, the ratio is $1 per big blind. Therefore, the std dev of 219bb/hand is equal to $219/hand.

For player B, the ratio is $10 per big blind. Therefore, the std dev of 21.9bb/hand (multiplied by $10/bb, the bb units cancel, leaving $/hand), is equal to $219/hand. (21.9 * 10)

So, in case 2, player A and player B have the same variance. Which, logically, has to be true. Both players are playing for $100 every hand, with a 60% chance to win. The fact that they are playing different tables stops being an important factor as soon as the money gets all-in. $100 is $100, no matter how you look at it. They both have different std deviations when that deviation is measured in BB/hand, but that's only because the value of the big blind is different. A simple unit conversion shows that their variance is actually equal.

So, your claim that case 1 and case 2 are identical is quite clearly false, and you have fallen victim to the bane of every chemistry student on the planet: a unit error.

If you need further proof of this, calculate the variance in terms of dollars, rather than big blinds, in both cases. You can get all the values you need (expected value, etc) in dollars rather than big blinds, if you know what the blinds are, and calculating that way should lead you immediately to the conclusion that the variances in case 2 are identical, even though 1 stack is 10BB and the other is 100BB.

Now all we've shown here is that the variance of a $100 stack is identical when pushed every hand, no matter how big the blinds are, while the actual point of discussion is that in this case, where the blinds increase, the shorter stack will have a higher variance than the deeper stack, which has not yet been shown.

You surmised that my arguments in this direction would be:
[ QUOTE ]
i know what your arguments are going to be:
1)we're less likely to enjoy a 60% edge on our opponent playing for 10bb than we are for 100bb
2)we're not going to be allin for our stacks as often for 100bb than we are for 10bb

[/ QUOTE ]

I'm not sure that I would agree with #1, but #2 would definitely be a huge focus, and you said that you did agree that we would be allin for stacks more often for 100bb than for 10bb.

The problem with your counterarguments is that they were mathematical arguments based on calculations involving values that were not represented in the same units, all built on a foundation of this assumption that a 10bb stack would always be lower variance than a 100bb stack, no matter what the actual values for the blinds were. If you redid the same calculations, converting the units properly first, the results would be vastly different.

It is true that all your calculations were in BB/hand, but BB/hand is simply not the same measure for player A as it is for player B in case 2. You can't mix the numbers directly without converting one of the numbers into units that match the other first. Two values, both expressed in the units of big blinds per hand, are not actually expressed in the same units if the values of the big blinds are different. The results of your equations simply don't mean anything, because there's a unit mismatch.

So lets try to fix that, and see what happens. The results are actually extremely surprising.

Here's your counterargument to #1:
[ QUOTE ]
let's find out how small an edge we'd have to have in the 10bb case for our variance to approach 48000 as it is in the 100bb case.
assume our edge is p
var=5p*((200-120)^2)+5(1-p)((0-120)^2)=48000
320p+720-720p=48000
the solution to this is p<0; you'd have to be drawing less than dead to have the same variance as the 100bb case.

[/ QUOTE ]

Actually, now that I'm *really* looking at this (before, I only needed to focus on the fact that you were comparing units improperly, because that made everything else invalid), that equation is wrong in a number of ways. First of all, the right side of the equation is the variance of of the 100BB stack, 48000. Since you're trying to determine at what edge value the 10BB stack's variance would be equal to the 100BB stack's variance, you need to actually use the formula for the 10BB stack's variance instead of the one for the 100BB stack's.

Which is really
5p*((20-12)^2)+5(1-p)*((0-12)^2)

The second problem is that because of your unit error, the right side of this equation is incorrect. For case 2, the left side is going to generate a number in units of ($10^2)/(hand^2), and the right side is in units of ($1^2)/(hand^2), because we've already established that BB/hand for the 100BB player is equivalent to $1/hand, and BB/hand for the 10BB player is equivalent to $10/hand. You can't just use the units of BB/hand. BB/hand is different on both sides of the equation, because the value of BB is different.

So after converting the right side from ($1^2)/(hand^2) to ($10^2)/(hand^2) by dividing by 100, the real formula is:

5p*((20-12)^2)+5(1-p)*((0-12)^2) = 480

Which boils down to 320p + 720 - 720p = 480.

Which is actually very close to what you figured, leading me to believe you did use the right formula on the left side (the 10bb stack), but simply typed in the wrong one. Right reduction, wrong equation.

But, as you pointed out, this is not solvable. So lets perform a sanity check, and and see if we can use the same method to determine what "edge" it would take in order for the 100BB stack's variance to match the calculated value. The result *should* be p = 0.6, correct? Let's see:

No unit mismatch, so we've got:

5p*((200-120)^2)+5(1-p)*((0-120)^2) = 48000
...
5p * 6400 + (5-5p) * 14400 = 48000
...
32000p + 72000 - 72000p = 48000

This one is unsolvable too, which leads me to believe that there's a *very* serious problem with the equations here. We can't calculate what the 100BB stack's edge needs to be in order for his variance to match the variance of the 100BB stack with an edge of 60%!?!?

I can't answer this one. I don't variance formulas handy, so I have no idea what's gone wrong here, I just know that something obviously *has* gone wrong, because we can't calculate something that we already know.

I believe your counterargument for #2 simply falls victim to the unit error. Hopefully by now you do understand why the actual size of the big blind *does* matter in all these calculations, and why there's a unit error at all (on one side of the equation, you're using feet, but on the other side of the equation, you're using inches)

On one side of the equation, you're using a variance of 202.5bb/h, and on the other, you're trying to calculate an the point where the variance is equal, but one side's bb/h is equal to $10/h, and the other side's bb/h is equal to $1/h.

The calculation you used:

[ QUOTE ]

var=(10/4)(avgpot^2-1)
to achieve variance less than 202.5 the avg pot will have to be:
202.5=(10/4)(avgpot^2-1)
avgpot=9.13bb

[/ QUOTE ]

Would be more properly written as:

2.25 = (10/4)(avgpot^2-1)

Because the left side (the 10BB stack's variance) is expressed in terms of $10^2/h^2, but the right side is going to end up in $1^2/h^2.

Obviously making this change, though, is going to result in an absurdly small average pot, which makes absolutely no sense whatsoever.

So, lets run the sanity check again.

[ QUOTE ]
var=10p((X-(p*avgpot)^2)+10(1-p)((X-(p*avgpot)^2)
where X=avgpot in the first term and 0 in the second term.


[/ QUOTE ]
If we plug in 0.5 for p, we should come out with avgpot = 8.99, which was the average pot for the 10BB stacks, when playing optimally? Correct?

using A instead of avgpot, because it looks prettier and makes things simpler

202.5 = 10p((X-(p*avgpot)^2)+10(1-p)((X-(p*avgpot)^2)
... (using A instead of avgpot, because it looks prettier)
202.5 = 10p((A-(pA)^2)+10(1-p)((0-(pA)^2)
...
202.5 = 10p(A-p^2A^2) + (10-10p)(-p^2A^2)
... setting p = 0.5
202.5 = 5(A-.25A^2) + (10-5)(-.25A^2)
...
202.5 = 5A - 1.25A^2 - 1.25A^2
...
202.5 = 5A - 2.5A^2
...
2.5A^2 - 5A + 202.5 = 0
...

Please check the above reduction, because the result we're about to get is a little strange.

Using the quadratic equation, where a=2.5, b = -5, and c = 202.5:

A = (-b +- sqrt( b^2 - 4ac ))/2a

The only thing we need to calculate here is b^2 - 4ac:
(-5)^2 - 4(2.5)(202.5)

25 - 2025

-2000

The fact that this value is negative leaves us looking for imaginary solutions.

Yet again, our sanity check failed, indicating a severe problem with either the equations we're using, or the method we're using to try to reach a solution.

All questions of unit conflicts aside, your equations don't seem to be able to solve problems that we already know the answers to, which should be a HUGE warning flag.

[TNixon]

Duh.

I'm a retard.

The second equation (counterargument to #2) does in fact lead to an imaginary solution, but both equations in counterargument #1 are very easily solvable.

This is why I'm trying to spell out every single detail, so it's easy to poke holes when necessary.

So, with the unit conversions correctly applied:

320p + 720 - 720p = 480.
-400p = -240
p = 0.6.

Indicating that the 10BB's edge needs to be exactly the same as the 100BB's blinds's edge. Which is exactly what I would expect to see, since they're both putting in the same amount every hand ($100).

And the sanity check:

32000p + 72000 - 72000p = 48000
-40000p = -24000
p = 0.6, exactly what it was supposed to equal.

All I can say is...DURR.

But, your original equation was unsolvable because you didn't convert units correctly.

Going to look over the equations in #2 again, to see if I did something equally stupid, leading to imaginary solutions.

[TNixon]

Actually, if you already read the section on the sanity check for #2, you know that it is so completely horked up that it's a little bit scary.

As a super-lame excuse, you used one variable (p) to mean different things in different equations, and that seems to have tripped me up. More incoming shortly.

Deleting the offending section for now. Yes, it was that bad. [img]/images/graemlins/smile.gif[/img]

Or maybe I won't delete the offending section, because the editing time has expired. Ah well. I don't mind leaving my temporary stupidities on public display, I was just going to try to save any future readers a bit of confusion while reading through something that obviously doesn't make sense.

Still more incoming, because the nonsensical result of an absurdly small average pot still indicates that there's something wrong somewhere. (At first I thought I might have converted units the wrong way, but I've double-checked that a number of times now)

[TNixon]

After quadruple-checking my unit conversion, I realized that I did multiply the wrong way, so

[ QUOTE ]

var=(10/4)(avgpot^2-1)
to achieve variance less than 202.5 the avg pot will have to be:
202.5=(10/4)(avgpot^2-1)
avgpot=9.13bb


[/ QUOTE ]

should really be

20250 = (10/4)(avgpot^2-1), which leads to an average pot of 90, which is exactly what I expected to see, meaning that for a 100BB $100 stack to equal the variance of a 10BB stack, with that monster winrate, he would have to be playing average pots of 90bb.

Which is basically equivalent to push-or-fold. So the result here is that if you're playing push-or-fold with $100, and your opponent plays optimally, your variance will be exactly the same whether you're sitting at a 0.5/1 table or a 5/10 table.

So, once again, your math has fallen victim to unit errors, and my math has fallen victim to my own stupidity.

[HokieGreg]

This is the worst drama-bomb in the history of 2+2.

What would be more fitting for this particular forum than a HU 4 Rollz Challenge???

[TNixon]

Finally, if after all that, you're still not convinced that the unit of measurement "big blinds per hand" is not a constant measurement like "miles per hour" (since the length of a mile never changes, and the length of an hour never changes, this unit of measurement never changes, but big blinds per hand depends very directly on the value of the big blind)

then try this.

Variance doesn't have to be measure in BB per hand. You have all the information you would need in both cases to calculate variance in dollars per hand.

If you compute that, you shouldn't need any further convincing that a comparing a $100 100BB stack to a $10 10BB stack is *very* different from comparing a $100 100BB stack to a $100 10BB stack.

I may be stupid, but the numbers cannot lie.

[omgwtfnoway]

i said i wasn't going to post in this thread again but i just can't help myself:

i normalized the variance of each game to bb to simplify the calculations as much as possible, so that the only variable that differed between the two examples was stack size. i showed that all other things being equal (bankroll, stakes, etc) that deeper games have higher variance, surely you won't dispute this.
in terms of variance it came down to this:
short stack < deep stack (the units on both sides of the equation are bb/hand because this was the simplest way to solve the problem)
remember that ALL OTHER THINGS BEING EQUAL, VARIANCE IS HIGHER THE DEEPER THE EFFECTIVE STACK

then you found a "unit conversion error" and did this:
(short stack)*$10 !< (deep stack)*$1 and concluded that
short stack > deep stack
you broke the fundamental theorem of algebra. both sides of the equation are bb/100, you can't decide that a bb is $10 on one side and only $1 on the other side and claim to have found a unit error.
if you want an inequality to hold you must perform the same operation to both sides.

[cwar]

Can someone summarize?

[TNixon]

[ QUOTE ]
ALL OTHER THINGS BEING EQUAL, VARIANCE IS HIGHER THE DEEPER THE EFFECTIVE STACK

[/ QUOTE ]
This is true. But in the second case, not all things are equal.

The blinds are bigger for the 10BB stack than they are for the 100BB stack.

[ QUOTE ]
did you break the fundamental theorem of algebra? both sides of the equation are bb/100, you can't decide that a bb is $10 on one side and only $1 on the other side and claim to have found a unit error.

[/ QUOTE ]
They're not equal to begin with, which means you're using the wrong numbers to begin with, even before you attempt to solve the equations. Algebra only comes into the picture once the two sides of the equation are actually equal. You're starting out with equations that aren't equal.

Basically you're starting out with something like 36in = 3ft, ignoring the units altogether, and saying that 36 inches is obviously not equal to 3 feet, because 36 != 3.

And I *can* decide the BB is $10 on one side and $1 on the other side because *those are the rules of the situation we're trying to draw a comparison from*.

One guy has $100 on a $5/$10 table, and the other has $100 on a $0.5/1 table.

You can't draw conclusions from a situation if you're not even taking the specified situation into account.

Seriously, just do the variance calculations in dollars rather than big blinds for case 2, where player A is $100 on a 0.5/1 table, and Player B is $100 on a $5/$10 table.

If you do that calculation, everything should become very clear.

[TNixon]

From your PM to me:

[ QUOTE ]
the next point is that since you're talking about two different games (different blind levels) you have to normalize the variance accordingly.

[/ QUOTE ]
Why are we even arguing? You just said exactly what I've been trying to say all along. That you can't use the same calculation to compare a $100 100BB stack vs a $10 BB stack that you can to compare a $100 100BB stack to a $100 10BB stack.

You've been saying the entire time that those two situations are identical, in that the variance is the same in both situations, with the 10BB stack being lower variance than the 100BB stack, *but you're not taking into account the fact that you need to normalize the variance numbers* in your calculations. The way you normalize them is by converting one unit to another, not by just waving your hand and saying "10BB = 10BB, no matter what the big blind actually is"

bb/hand when the BB is $1 is not the same unit of measurement as bb/hand when the BB is $10. You can't just make a direct comparison between numbers expressed in the two units without performing a conversion.

[ QUOTE ]
as such, once you find the std dev in $/hand you should account for how much that money that represents with respect to the table stakes. this means dividing by the size of the big blind

[/ QUOTE ]
If you've got the std deviation in $/hand, why in the world do you want to convert it back to BB/hand?

We want real $ comparisons to make against our bankroll. If you convert it back to BB/hand then you just have to refigure the actual blinds you're playing to figure the *real* impact on your bankroll. We don't care how the variance relates to the table stakes. We care how it relates to our *bankroll*.