re the variance thread blowup NL
 

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.

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

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.

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

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)

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

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.

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

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???

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

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.

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

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.

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

Can someone summarize?

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

[ 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.

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

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*.

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

Sorry, I'm trying *really* hard not to be confrontational.

I'd *really* like to get over this issue, so that there can be some rational discussion about the original question, because as long as there's disagreement on this point, we're not even existing in the same universe.

I would love for somebody from the math forum to come over here, but I can guarantee 100% that they're going to agree with me.

In fact...we're both gamblers. Care to place a bet on it? If so, lets settle the details on the bet before we get somebody over here to check the thread out. I'd be willing to put $100 on them agreeing with me, with the condition that we manage to attract a regular poster who appears to have a pretty good idea of what they're talking about.

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

[ 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

[/ QUOTE ]
You know, I really think this sentence represents the key to the entire disagreement.

Why should we account for how much that money represents with respect to the table stakes?

Table stakes are arbitrary. I can play any level I want. The real effect I want to see is how my bankroll is affected.

To be able to make any sort of valid comparison between HUSNG and cash, you *have* to be able to compare different stakes directly, because late in a $100 sng, your $100 stack is no longer worth 75BB. It's worth 10BB. You've said that this situation is exactly the same as a situation comparing a 100BB $100 stack against a 10BB $10 stack. I vehemently disagree, and have given all the mathematical proof that I possibly can.

You know what?

I'm talking in circles again. I cannot possibly restate this in any way that is more clear than the ways I have already stated it.

So, do you accept the bet or not?

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

Whether there's a bet or not, when picking an appropriate judge, would it be best to query the probability forum, or the science, math, and philosophy forum?

I'm thinking probability.

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

[ QUOTE ]
Can someone summarize?

[/ QUOTE ]

Honestly, no, I can't.

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

I can't either. This thread is an attempt to convince the only really active poster on the topic that he's making a mathematical mistake in his comparisons, so we can actually move on and discuss the original topic, which was comparing HUCASH and HUSNG variance.

In other words, about three million posts on a fairly small (but very important) detail.

So far, it's been a resounding failure.

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

What I understand :
omgwtfnoway says that playing $10 stacks at NL100 is lower variance than $10 stacks at NL10, because the st dev in big blinds will be higher at NL10. It really doesn't make any sense imo and I agree with TNixon you have to think in $ here, that's what is relevant.

Also I'm not sure I understand the point in your calculations where you push at 60% equity different stack sizes, since the deeper the stacks the less often these situations will happen and you have to take this into account.

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

I am a little interested in what we are actually looking for here, finding out which HU gametype has the higher standard deviation per hour when the SnG buy-in is 100 times greater than the big blind in the cash-game.

But I wasn´t too interested and was hoping the guys would find this out for me. This seems far away at the moment, so I will try and help (read: make it worse [img]/images/graemlins/laugh.gif[/img]).

First I wish to dispute that a shortstack will have smaller variance than a big stack if the blinds are the same.

I can understand how one may think it is so (I did it myself not long ago). One would think that a 10 big blind stack would have a smaller variance than a 100 big blind stack because it is limited how big pots it can create. With a 10BB stack the pot can maximum be 20BBs, but if you had 100BBs you could have made the pot bigger because you have more to throw into the pot.

But the thing is there are situations where a small stack would create a bigger pot than a big stack would. When stacks are smaller than 10BBs, for example, it is common to open-raise all-in. If your stack is 9BBs, you are effectively making a 9BB raise! And the opponent is not more scared of calling than he would have been if stacks had been 100BBs and the raise only 3BBs!

Such a huge open-raise would never take place with a big stack. With a big stack you would have bet 3BBs and if the opponent calls one of you will often take it down by making a 4BB bet on the flop winning a 6BB pot instead of the 18BB pot in the shortstack-case.

There are of course situations where a big stack would create a larger pot too, and this roughly cancels out the larger pots of the shortstacks, making both stacks creating roughly the same average pot.

Just look at the 30BB Cap-games on Full Tilt, they have about the same average pot as non-cap-games.

Obviously when stacks get tiny, the average pot must go down as the stacks are getting so small at some point that even if both players were all-in in all hands, the average pot would be smaller than the average pot of a normal game. I don´t know exactly where the stacks start decreasing the average pot, but I would guess around 7BB stacks.

Then here is how I think we should approach our problem:

1) We simplify and assume there is no rake and no fees.

2) We simplify further and assume all players are equally skilled.

3) We assume you can play the same number of hands per minute in an SnG as in a cash-game.

4) Since stacks don´t influence the standard deviation in cash-games as long as stacks are greater than 7BBs, we make it easy for ourselves and assume the cash-game stacks are always 9BBs and either go all-in or fold (easy to relate too and not too unrealistic).

5) Let´s say both players will go all-in in the SB with their 70% best hands and call all-in in the BB with their 40% best hands.

This means the BB will win the SB 100% - 70% = 30% of the time. The SB will win the BB 0,7 * (1-0,4) = 0,42 = 42% of the time. And they will compete for an 18BB pot 0,7 * 0,4 = 0,28 = 28% of the time.

Pokerstove says the SB has a 44,6% chance of winning the 18BB pots and the BB thus has a 55,4% chance of winning. I am ignoring the small chance of a draw because it is so rare.

6) Let´s set a specific monetary size on the games. It is NL100 and a $100+0 SnG.

7) We know that in a $100+0 SnG, you will either win $100 or lose $100 after having played a number of hands. So the "variance" (used very unmathematically here) of it is $100.

8) Run a simulation of a hand in a cash-game where the possible outcomes of a hand are according to the probabilities pointed out in step 5. Note the amount player A (just choose one of them) won/lost.

Then do the same for the next hand and add/subtract what player A won/lost to/from his gain/loss from the last hand.

Then do the same for next hand a summarize the gain/loss from all three hands.

Keep doing this until player A has either won or lost a total of $100, then note the number of hands it took.

9) Repeat step 8 (note the number of hands it took). Do it again and again until you feel you have done it enough to get a fair sample. Then calculate the average number of hands it took.

10) If a particular type of HU SnG lasts more number of hands on average than the number found in step 9, it has less variance.

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

Ok, I came up with one more way of stating the problem that isn't saying something exactly the same way that I already have.

In fact, this is how I'd want to frame the question for anybody coming over from the probability forum to settle a bet.

The following question is asked:

"I have a $1000 bankroll, that I don't mind replacing if I go broke, so I want to play it very agressively, but I'd still prefer *not* to go broke if I can avoid it, while keeping in mind that I want to maximize value by playing agressively.

I've narrowed my choices down to two:

Choice A: Playing full buyins at 100NL.
Choice B: Buying in short at 1000NL, for $100 or 10 big blinds.

Which of these 2 choices is going to be lower variance? (and please don't try to talk me out of playing above my bankroll. I've already stated that I want to play agressively. Just tell me which of these two choices is lower variance)"

What is your answer, and why?

My answer is Choice A is obviously lower variance, with a smaller chance of going broke, as long as you don't play a supper-aggresive style where you're playing "optimal" push-or-fold.

Your answer, if you're going to be consistent with everything you've already said, should be that choice B is lower variance. You even "proved" mathematically that choice B would be lower variance, despite the fact that you would be getting the chips all-in more often as the 10BB stack, because the average pot size in choice A would have to be less than around 9BB for choice A to be the lower-variance choice.

Is that your final answer?

If so, do we have a bet? I'm even willing to give you 2 to 1 odds. Your $50 to my $100, or your $100 to my $200.

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

It's unfortunate that the original topic was locked. I am wondering what affect the game structure has on the HUCG vs HUSNG variance question. A tourney has an obvious endpoint, it is over when someone loses all of their chips or 1 buyin. However a cash game has no defined endpoint. Say HUSNGs average 50 hands per match (just guessing), the most each player can win or lose is one buyin at the end of the match. In theory a cash game player can win or lose 50 buyins in the same 50 hands. More to the point, a cash game player can have multiple buyin swings within those 50 hands and still win or lose just one buyin as the end result. Does this affect variance?

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

Tnixon , I'm actually from the math forum and I agree with Omg .

Nothing you're saying is really adding up . Perhaps I still don't understand what you're trying to convey .

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

Let me summarize what I think is going on .

Take player A who is playing with 20BB's in a game where the BB is $a . Player B is playing with 100BB's in a game where the BB is $b .

If a=b , then player B will experience more variance .
If a>b or a<b, then player A's variance may or may not be higher than player B's .

If we talk about variance per BB , then clearly A will experience more swings . If we talk about variance per $ , then B will experience more swings .

This is analogous to a player winning 3 BB's playing a 10-20 limit game or a player winning 2 BB's playing a 20-40 game . You earn more money per BB in the former game , but you earn more profits in the second game .

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

you guys can theorize all you want, but i'll give you experience. 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.

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

jay_shark, so you would recommend to somebody that playing $100 at 0.5/1 NL is *higher* variance than buying in for $100 at $5/$10?

Can you please justify your answer?

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

[ QUOTE ]
If a=b , then player B will experience more variance .

[/ QUOTE ]
Absolutely.

I have not questioned this statement even once.

[ QUOTE ]
If we talk about variance per BB , then clearly A will experience more swings . If we talk about variance per $ , then B will experience more swings .

[/ QUOTE ]

Thank you. This is exactly what I've been trying to say all along.

Except that my point is that the only variance we actually care about, if we're trying to answer real questions, and make comparisons, is the $ variance. Variance per BB is simply not a useful measure if you want to compare two different stakes. You have to make some sort of conversion somewhere if you want to be able to compare 0.5/1 and 5/10 with stacks consisting of a different number of big blinds. Otherwise you're comparing feet to inches, without taking into account that they're not the same measurement.

Are you *sure* you agree with omg? I'm saying exactly the same things you seem to be. He has been disagreeing all along, and quite vehemently, I might add.

[ QUOTE ]
you guys can theorize all you want, but i'll give you experience. 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 ]
Presumably at the same limit? 100NL in both cases? So you're buying in for $20 short, and $100 fullstacked?

Once again, I have not said even once that a short stack is higher variance than a deep stack *at the same limits*.

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

[ QUOTE ]
If we talk about variance per BB , then clearly A will experience more swings . If we talk about variance per $ , then B will experience more swings .

[/ QUOTE ]
Thinking about this some more, I honestly do not believe you can even draw the comparison when measuring variance in bb/hand.

I really don't know how to explain this any better than I already have, but the two measurements are simply not the same. A has a variance of Xbb/hand, and B has a variance of Ybb/hand, but you cannot directly compare X against Y and say that A has a higher variance. It's the same as trying to say that 36 inches is greater than 3 feet.

In a measurement like meters/second, the units are both fixed measurements. The length of a meter will never change, and the length of a second will never change.

So, if you have two values that are expressed in meters per second, you can directly compare the numbers to see if one is higher than the other. The numbers are in the same units.

In Xbb/hand, bb is not a constant measurement, like meters or seconds, or any other standardized measurement. It's a variable measurement that depends on the stakes. So you cannot directly compare 2 bb/hand values without taking into account what the bb is for each value.

Lets say we wanted to measure speed in distance/second rather than meters per second.

If I tell you that one person moves 10 meters in 1 second, and another person move 10 feet in 1 second, then in the units we're measuring with, both people are moving a distance of 10 per second.

But nobody in their right mind would argue that these two people are moving the same speed, because they're both moving "10 distance per second". In this case, "distance" is not a fixed measurement. You can't even compare the results in a way that makes since without knowing how long "distance" is in each case.

But that is *exactly* what you're doing when you try to directly compare variance numbers computed at bb/hand, where the bb is different for the 2 values. The comparison doesn't even *make sense*.

But making that comparison is the only way you can ever say A has a higher variance than B.

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

Glad we cleared this up .

I also agree that it's important that we talk about the units that are being measured . Variance/bb and Variance/$ are two different things just like BB/h and $/h are two different things .

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

I just know that if I say the same things in a sufficiently large number of different ways, something is eventually going to click, and a great big monstrous

DUUUUUUUUHHHHH

will be heard around the globe.

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

So do you still think you agree with omg? He's been saying the units don't matter, and that comparing a $100 100BB stack vs a $100 10 BB stack is the same as comparing a $100 100BB stack vs a $10 10BB stack. He even accused me of making simple algebra errors when I modified his equations to take into account the fact that the units were different.

These two situations are obviously very different, and the equations he used to "prove" that a 10BB stack is always less variance than a 100BB stack completely ignored the fact that the value of BB was different in the case we were talking about, and wouldn't even acknowledge that the actual value of the BB did matter when trying to compare two bb/hand values.

And what's your answer to Mr. Aggressive Bankroll Play question about which way to play to minimize his chances of going broke? Presumably you agree that choice A is going to be lower variance?

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

[ QUOTE ]
jay_shark, so you would recommend to somebody that playing $100 at 0.5/1 NL is *higher* variance than buying in for $100 at $5/$10?

Can you please justify your answer?

[/ QUOTE ]

To make things easier , it is recommended that you reduce the former game to having $10 at 0.5/1 . This is true if we assume that the win rates do not change .

Now we have $100 at 0.5/1 and $10 at 0.5/1 . The following is true :

1)You will experience more variance /$ playing with a stack of $100 .

2)You will experience more variance /BB playing with a stack of $10 .

Do you agree with these statements ?

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

[ QUOTE ]
To make things easier , it is recommended that you reduce the former game to having $10 at 0.5/1 . This is true if we assume that the win rates do not change .

[/ QUOTE ]
omg.

I thought we making progress here, but you are now making the same mistake omg has been making all along.

By trying to make the problem easier, you've completely changed the result. $10 at 0.5/1 *is not the same* as $100 at 5/10. They are both 10BB stacks. But you cannot simply change the value of the big blind and expect to come out with the same result.

[ QUOTE ]

1)You will experience more variance /$ playing with a stack of $100 .

2)You will experience more variance /BB playing with a stack of $10 .


[/ QUOTE ]

And no, I don't agree with those statements. First of all, I'm not sure exactly what you mean when you say variance/$ and variance/BB. The variance is going to be measured in bb/hand or $/hand, depending on whether you use blinds or $ for the calculation. "Variance per dollar", or "Variance per big blind" doesn't really even make sense.

But in any case, after your modification of the stacks, the $10 stack will have a smaller std. deviation measured in bb/hand, so the $10 stack is always lower variance if you match the value of the blinds. In fact, as omg correctly pointed out, in this case, the $100 stack had 100 times the variance of the $10 stack, in all situations, *if the blinds are equal*. The $10 stack has a lower variance whether it's measured in bb/hand or $/hand. (and because the bb happens to = 1, the calculated values for bb/hand and $/hand will be identical, both for the $100 stack and the $10 stack)

But by modifying the stacks and blind levels *you have changed the problem*. The answer is different if you solve the original problem, rather than the "easier" version.

Correction: variance is measured in bb^2/hand^2, std deviance is measured in bb/hand. I do mix these two terms fairly often, which is bad, but it should still be clear what I'm referring to, and they are directly related, so it's not a fatal error.

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

I think you have a few screws loose .

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

Ok, back to square one. Jay_shark, is your answer to the $1000 bankroll question choice B?

Do *you* want to take me up on the bet? Obviously you've disqualified yourself as a judge, even though you're from the math forum, but I would have wanted a chance to see the quality of previous postings before agreeing to you as a judge anyway.

I'll give you the same 2 to 1 odds that I'm offering omg.

If you're willing to put money on it, here's my proposal: We find three or 5 judges that both sides can agree on from the probability forum (I expect people frequenting that forum to be slightly more familiar with this sort of math than a more general math forum), and offer them a cut of the winnings if necessary to get people to actually participate. Each of us takes one side of the "Minimize variance with a $1000 bankroll" question that I already posed, and writes a short summary piece on why we believe our advice to that player is correct. Then give the judges discussion time, and have them vote. Majority wins the bet.

Up for it? Anyone? I reserve the right not to take 20 people up on this, but I'll take the first one or two who are willing.

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

[ QUOTE ]
I think you have a few screws loose .

[/ QUOTE ]
And I think both you and omg are missing points that are so mind-numbingly obvious that I have no clue how we're still not understanding each other here.

And I was so sure the "distance per second" bit was going to work, too, because the comparison was so obviously wrong to anybody with any sense at all, but so clearly the same as trying to compare two numbers measured in bb/hand.

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

Tnixon , I actually post most frequently in the probability forum but it's understandable if you don't find me credible .

You may want to ask BruceZ from the probability forum if you need assurance .

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

It doesn't take 2thousand word posts to explain this. It's quite simple actually.

Shorter stacks reduce someone's edge. Everyone should know this. The smaller someone's edge is, the higher the variance.

This is true in bankroll situations. If I have a small edge (1ptbb/100), I am more likely to have big bankroll swings aka high variance.

How do shortstacks effect variance? Quite simple again.

If I am playing a donk 20bb deep, let's assume I will win 60% of the time. However, if we are 1000bb deep, I will win much closer to 100% of the time, reducing variance.

Now, this "variance" is not measured in big blinds. It is measured in buy ins. In a husng, stacks start at 75bb= more luck and higher variance.

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

type wars

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

This whole flame war *has* to be the most verbose in 2p2 history? amirite???

Yugoslav

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

[ QUOTE ]
This whole flame war *has* to be the most verbose in 2p2 history? amirite???

Yugoslav

[/ QUOTE ]

qft

tnixon- i would suggest that you try to condense each series of 5 posts into one post. this thread is so tl;dr its insane.

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

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).

I skipped some blind levels but you get the idea. If you play with higher blinds with the same buyin I think your variance will be higher because most of your edge is postflop and you take alot of postflop play out of the game by playing short stakes. So I think you end up getting your money in with a lower edge, therefore higher variance.

So in a sng you are effectively playing that same amount of money with higher blinds. (it being a freezeout may affect the variance, but i'm not sure in what way).

Variance depends on your edge then. Against most opponents your edge will be higher the deeper you are. (an exception is if your opponent folds way way too much when effective stacks are short)

So that leads me to believe that HUCASH has lower variance in terms of $swings in your BR.

That said there are some things that will add more variance to HUCASH games.
[*]playing with 200BB+ stacks.

Sometimes you can get pretty deepstacked in HUCASH and that is going to add a TON to your variance.
[*]playing a good opponent for long periods of time

This is usually a good thing to avoid, but if you do it your variance will skyrocket.


So I think if you play HUCASH and refuse to play effective stacks greater than 100BBs your variance will be lower than a HUSNG player.

But I think most of the posters here that play HUCASH don't shy away from playing the 200BB and 300BB stacks that might develope. That is why I think their variance is higher and why it seems that way to use at the HU forum.


Feel free to disagree with me and point out flawed thinking with explainations. I'm not going to be sticking my ego into this debate, but I will offer my thoughts.

Re: A plea to omgwtfnoway (re the variance thread blowup)
 

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

in a sit n go, you play other people. if you tilt, then your variance will increase. if they tilt, well you might see them again, you might not

cash games are different because your variance will increase if you tilt and if you tilt other people. a lot of players think lags have higher stand dev because they are hitting every edge and this is partly true. what's also true is that people get annoyed and do stupid stuff v lags and that too accounts for more variance