Variance revisited HUCASH vs HUTRN

[jay_shark]

Correct Indiana , I forgot to square the mean term .

All of this boils down to how much you buy in for in relation to the blinds . If you're playing a 0.5-1 game but you buy in for $1000 against another player who has you covered , then clearly you'll experience more variance in this game than if you had bought in for $100 .

I fully believe that if you buy-in for at most 75 times the amount of the BB , then you'll experience more variance in a stt sng .

[TNixon]

Just came across this tidbit in another thread, from one of the authors of Professional No Limit Hold'em.

[ QUOTE ]
For example, say you have $100. If you play the $100 as a 50bb stack in a $1-$2 game, your variance will be higher than if you play the $100 as a 200bb stack in a $.25-$.50 game. However, playing a $100 stack in a $1-$2 game will still be lower variance than playing a larger stack in a $1-$2 game.

[/ QUOTE ]

When you start a $100 sit-n-go on full tilt, you start out playing with a 75BB stack that is basically worth $100. At this point, any variance you experience in chips is equivalent to variance in dollars at a cash table if you were to buy-in for 75BB rather than the max 100.

As the tournament progresses, your stack is still worth the same $100, but the blinds have increased. Isn't this therefore a higher-variance situation? The swings in your bankroll are directly related to your winrate, but your winrate is dependent on the variance of the individual hands inside the tournament, just like a cash table. In fact, losing a big pot or two greatly increases the chances that you'll lose your entire buyin, even though you still have chips left, due to the pressure from the blinds.

Another train of thought: If "high variance" means that the results hit the outside ends of the spectrum more frequently, then lets say we only have $100. In this case, the the chance of going broke (the ultimate bad result of variance) playing a $100 sit-n-go would actually be greater than the chance of going broke playing $100 at a 0.5/1NL table, would it not? With a 60% winrate, you have a 60% chance of doubling up, and a 40% chance of going broke. On the cash table, there's a very wide possibility of results. At one extreme, you can lose your $100 and be broke. At the other extreme, you can double (or triple, or quadruple, or whatever, if he keeps rebuying), but most of the likely results over the same number of hands that you would play in a sit-n-go lie in the range between losing your buyin and gaining a buyin, for example, losing half your stack, or gaining 25%, or whatever. The chance of going broke here seems like it would have to be significantly less than 40%, I would think.

Call me crazy, but it seems pretty obvious that in this situation, the SNG variance is going to be higher than the cash variance.

[ QUOTE ]
whereas in cash you win a few big pots and break even or lose a little in most pots. So in cash your results are swingy and spread deeply around the mean.

make sense?

[/ QUOTE ]
Not really, because you win and lose those same big pots in SNGs, with what I would guess is an even greater frequency, due to blind pressure. You start out playing $100 75BB stacks, but can very quickly end up playing $100 10BB stacks. Losing a big pot in a SNG isn't that much different from losing a big pot at a cash table. In fact, it should be a more painful blow in a SNG, because there's an increased likelihood that the rest of your chips will follow, as you are forced to gamble more and more. In a cash game, you can reload, removing this pressure, and although that does increase the maximum swings, it should reduce the *average* swings, should it not?

The *only* difference in absolute $ variance between the two, as far as I can tell, should be the ability for both sides to reload in cash, and as far as I can figure, that ability should have the effect of reducing the average swings, not increasing them, because there's less pressure to gamble. (Assuming, of course, that gambling on marginal situations is by default higher variance than not gambling, which only makes sense.)

[omgwtfnoway]

[ QUOTE ]
Just came across this tidbit in another thread, from one of the authors of Professional No Limit Hold'em.

Quote:
For example, say you have $100. If you play the $100 as a 50bb stack in a $1-$2 game, your variance will be higher than if you play the $100 as a 200bb stack in a $.25-$.50 game. However, playing a $100 stack in a $1-$2 game will still be lower variance than playing a larger stack in a $1-$2 game.

[/ QUOTE ]so small stack poker has inherently less variance than big stack poker.

[omgwtfnoway]

[ QUOTE ]
The only thing I, and most of the people in this forum want to know about variance is how it affects our bankrolls.

[/ QUOTE ] yea me too, but it only makes sense to measure this effect in terms of a percentage or proportion of your bankroll.
take two husng players, a $5 player and a $50 player, they both have a winrate of 57%. by your definition, the $50 player experiences higher variance because he has larger monetary swings. you're wrong, since variance in husng is a direct function of winrate they have exactly the same variance relative to the buyin/bankroll which is what's important.
[ QUOTE ]
As the tournament progresses, your stack is still worth the same $100, but the blinds have increased. Isn't this therefore a higher-variance situation?

[/ QUOTE ]as the blinds rise the individual tournament often becomes subject to higher variance but not because the stacks are shorter. the variance may increase because your effective edge becomes smaller. please don't forget that we're speaking of variance in husng in totality, it is purely a function of winrate and is totally independent of any individual hand/situation which you may or may not think has high variance.
[ QUOTE ]
he swings in your bankroll are directly related to your winrate, but your winrate is dependent on the variance of the individual hands inside the tournament

[/ QUOTE ]winrate is generally independent of variance of individual hands in a cash game. two players may have identical winrates but nonidentical variance or vice versa.
[ QUOTE ]
With a 60% winrate, you have a 60% chance of doubling up, and a 40% chance of going broke. On the cash table, there's a very wide possibility of results. At one extreme, you can lose your $100 and be broke. At the other extreme, you can double (or triple, or quadruple, or whatever, if he keeps rebuying), but most of the likely results over the same number of hands that you would play in a sit-n-go lie in the range between losing your buyin and gaining a buyin, for example, losing half your stack, or gaining 25%, or whatever.

[/ QUOTE ]typical std deviations for hu cash games like in the hundreds of ptbb/hundred hands. this is more than a buyin of deviation per hundred hands. after one hundred hands at std dev of only 100bb/100 you can expect to have gone broke 34% of the time starting from a $100 bankroll. this example uses an UNREASONABLY SMALL std deviation from the winrate.
[ QUOTE ]
Call me crazy

[/ QUOTE ] ok, you're crazy

[TNixon]

[ QUOTE ]
so small stack poker has inherently less variance than big stack poker.

[/ QUOTE ]
You might want to reread that.

If we were talking about a $100 100BB stack compared to a $20 20BB stack, you would be correct.

Since we're actually talking about a $100 100BB stack that turns into a $100 10BB stack (which is what happens in HUSNGS), here's the relevant part of that quote

[ QUOTE ]
If you play the $100 as a 50bb stack in a $1-$2 game, your variance will be higher than if you play the $100 as a 200bb stack in a $.25-$.50 game.

[/ QUOTE ]

[ QUOTE ]
as the blinds rise the individual tournament often becomes subject to higher variance but not because the stacks are shorter.

[/ QUOTE ]
Yes, because the stacks are shorter. Read the quote again. I really don't understand the difficulty here. If your $100 is worth 10BB, and my $100 is worth 100BB, your swings are obviously going to be bigger than mine, because I will have many hands where a couple dollars are exchanged. You won't have *any* hands where anything less than $10 is exchanged, and you will exchange $100 *far* more often than I will.

[ QUOTE ]
winrate is generally independent of variance of individual hands in a cash game. two players may have identical winrates but nonidentical variance or vice versa.

[/ QUOTE ]
Your winrate in a HUSNG is *directly* related to the "variance" of the game itself. (I'm going to start using variance in quotes whenever I'm talking about the size of the actual swings in your bankroll, to try to avoid confusion about a term that is apparently being used to mean a few different things)

If you think of the 1500 tournament chips as your bankroll, when you swing up high enough, the tournament is over and you win. When you swing down low enough, the tournament is over and you lose.

Winning or losing a $100 sng is very much like sitting down with $100 at a cash table. You can play a high-variance style in either, but decisions that are correct at a cash table are also correct in a HUSNG. The main difference is that at SNGs, you only win or lose in increments of one buyin.

[ QUOTE ]
after one hundred hands

[/ QUOTE ]
Unless you're playing a much different crowd than I am, many HUSNGs are over well before the hundred hand mark.

If you're going to make comparisons, at least make them equitable comparisons.

[ QUOTE ]
at std dev of only 100bb/100 you can expect to have gone broke 34% of the time

[/ QUOTE ]
Which is less often than you go broke playing the single sit-n-go. Which lends support to the idea that cash is lower variance than sngs. What percentage of the time do you double your $100? Less than 60% of the time? What exactly are you trying to convince me of again?

[ QUOTE ]
this example uses an UNREASONABLY SMALL std deviation from the winrate.

[/ QUOTE ]
Fine. Come up with a percentage from a fair std deviation, and a reasonable winrate, since I think we both agree that 10BB/100 hands is not even remotely comparable to a 60% winrate at SNGs.

[ QUOTE ]
take two husng players, a $5 player and a $50 player, they both have a winrate of 57%. by your definition, the $50 player experiences higher variance because he has larger monetary swings.

[/ QUOTE ]
Fair enough. This is obvious and elementary enough that I didn't actually think it was necessary to point out that when I'm talking about getting specific $ amounts, it would be relative to a specific bankroll amount, though.

If you want to express that as a percentage of the bankroll rather than a dollar amount at a specific point in a specific bankroll, feel free. But you haven't done that, either. All you've done is say "cash is higher variance" without giving any sort of backing logic or math, while I've attempted half a dozen times to give logical trains of thought that provide evidence that cash is not actually higher variance, bits of logic that you haven't actually even attempted to poke any holes into other than by basically saying "you're wrong", and making absurd claims like "shortstack poker is lower variance" while completely ignoring the actual situation under discussion, which is *clearly* a higher-variance situation. If the swings at cash are going to be bigger than the swings at SNGs, then lets see some numbers.

When I say something like:
[ QUOTE ]
The only way to measure that in a way that I care about is in $. If you want to compare the variance between HUSNGs and HUCASH, the only meaningful comparison in this context is in real dollar amounts.

[/ QUOTE ]
It should be painfully obvious that there has to be an actual bankroll amount (in dollar amounts) that goes with those variance swings, and that a $5 swing in a $50 bankroll is not the same as a $5 swing in a $50 bankroll.

Even if I don't have a complete grasp on all the underlying mathematics (but I do program computers for a living, and I'm pretty damned good at it because I have a strong sense of logic, and I did make it through some fairly advanced math courses before I dropped out of college, so I probably grasp more than you seem to be giving me credit for here)...

I do actually understand the basics, and frankly I'm getting just a little bit tired of being talked down to.

Maybe we could tone the assumption of superiority down a little? Because you're starting to sound more and more like a student taking his first statistics course. Which, from what I gather, is exactly what you are.

[omgwtfnoway]

ok now i'm convinced you're a gimmick

[HokieGreg]

hi mom

[TNixon]

Why do I suddenly have the sinking feeling that I've been attempting to carry on a discussion with a troll?

You got me. Nice one.

[TNixon]

Ok, so in order to determine if there is actually something to glean out of this, or if you've just been attempting to feed me a line completely unjustified by logic (which I have tried to provide in as much abundance as possible, to give opportunities to point out holes in the underlying thought processes), answer one question for me.

Two players with the same bankroll size, skill level, and winrate.

Player A plays 0.5/$1 tables and always buys in for the max $100.

Player B plays $5/$10 tables and buys in for $100.

Which of these two players is going to experience the widest swings in his bankroll? Which player is least likely to go broke over, say, 100 hands?

If your answer is player B, be prepared to defend your answer with something slightly stronger than "You don't understand the mathematical definition of variance".

Oh, and yes, despite the fact that I typed this:
[ QUOTE ]
and that a $5 swing in a $50 bankroll is not the same as a $5 swing in a $50 bankroll.

[/ QUOTE ]
I do, in fact realize that a $5 swing in a $50 bankroll is exactly the same as a $5 swing in a $50 bankroll. Is this just further evidence of my lack of understanding, or do I actually get credit for a typo here?

Confrontationally yours,

[omgwtfnoway]

against my better judgment i'm going to continue this discussion, hopefully helping you to see that short stack poker is lower variance than playing with deeper stacks.

first we're going to try to clear up the definition of variance and standard deviation. when i said that you *have* to think of variance/std deviation in terms of the stakes you're playing, i wasn't just saying it in order to provide perspective for my argument, these are mathematical terms that are expressed in terms similar to the winrate. in fact, in the case of standard deviation, the units are THE SAME as the winrate. so for a cash game player measuring his winrate in BB/100hands, he will also measure his std dev in BB/100hands. Variance is the square of std dev so it will be measured in BB^2/10khands.
var(x)=sum((X-u)^2) where X is the individual result and u represents the mean or expected result.

as i understand it, the definition of variance that you're tacitly using is "how often you get stacked or stack your opponent." examining the difference between the two definitions should go a long way toward clearing up our argument. for what it's worth, i agree that you will put your whole stack in the middle more often the smaller your stack is, but that has less to do with variance than you think it does.

so basically variance is an approximation of how much you can reasonably expect your results to differ from your sklansky bucks.

now we'll take two hypothetical cash game hands. in both hands, hero gets all the money in as a 60-40 favorite. in the first hand the effective stacks are 10bb. in the second hand, 100bb. now we're going to run the board out 5 times and we're going to assume that the most probable outcome (that hero wins 60% of the hands) occurs. this will show the variance that is inherent in making these plays.

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 variance of the 100bb case is 100 times as large as in the 10bb case.

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

these arguments are valid, but irrelevant to the discussion of variance and i'll show you why.

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. obviously this can never happen so it's starting to look like there's inherently less variance the shorter your stack is.

now for the argument that not being allin as frequently in the 100bb case will lower our variance. note that i don't disagree that you will be allin less frequently playing for 100bb than you would for 10bb.
for this point look again at the equation for variance:
var(x)=sum((X-u)^2)
the fundamental difference in variance between the two cases can be surmised by looking at the x-u term. this term has limits placed on it by the game; it is limited by the effective stack of the two players.
let's take the variance equation to a generalized pot. let's assume that in both the 10bb and 100bb cases we're examining the variance over a sample of ten hands.
calculate variance assuming every pot we play is "average size":
var=sum((X-u)^2)
express u in terms of the avg pot size and equity in the pot: u=p*avgpot
var=sum((X-(p*avgpot/2))^2)
now we're going to need to compare the two cases individually. variance will be maximized in each case if both players are allin for their stacks with a random hand each time. however, i'm going to assume that both players are playing optimal jam-fold poker according to the solutions in mathematics of poker. as such over a large amount of hands each player's equity is .5. since we don't have a large number of hands i'm going to use 10hands (an even number, ldo) as a way to negate positional advantage).

now we have to find the average pot size.
at 10bb the sb will shove with 58.3% of hands. once he has shoved, the bb will call with 37.3%. (this information is on page 137 of MOP)
1-.583 of the time sb will fold and bb the pot size will be 1.5bb
(.583)(1-.373) sb will jam and bb will fold resulting in a pot size of 11bb
(.583)(.373) sb will jam and bb will call resulting in a pot size of 20bb.
avg pot=(1-.583)(1.5)+(.583)(1-.373)(11)+(.583)(.373)(20)
avg pot=.6255+4.0210+4.3492=8.99bb ~9bb
since both players have 50% equity in these allin pots, we expect the variance of this game to be generally high for reasons i don't care to go into. (it has to do with maximizing the (X-u)^2 term.)
var=sum((X-(p*avgpot))^2)
when p is .5 and avgpot is 9 the term u=4.5
for simplicity's sake we'll assume that all pots played are "average." half the time hero will win 9 chips and the other half he will win 0.
var=10(.5)((9-4.5)^2)+10(.5)((9-4.5)^2)=202.5

so this is the variance you will experience playing for 10bb if both players are playing optimal jam-fold over ten hands. if your equity in the pot changes in either direction the variance will decrease. try it for yourself if you don't believe me.

now let's look at the 100bb case. we're specifically looking for an equity in the pot (p) and an average pot size (avgpot) that will give us a variance lower than 202.5 (if we can't find values that satisfy this then we will have shown absolutely that you cannot experience lower variance playing for 100bb than for 10bb).
we're trying to find a minimum variance for this deepstack case. variance will be maximized at p=.5 so let's give our hero some sort of healthy advantage in the average pot in any effort to decrease variance. let's assume our hero enjoys the monstrous (and unreasonably high) edge of 50bb/100hands. he wins half a big blind every hand.
this edge is the difference between u and our stake of the avgpot. .5=u-avgpot/2
1+avgpot=2u
remember u=p*avgpot
1+avgpot=2p*avgpot
p=(1+avgpot)/2avgpot
so now, having assumed an absurdly high winrate for our hero, we can find our variance as a function of the average pot size for our sample of five hands.
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.
this is going to get really messy so i'm going to skip the algebra here and show the simplified variance equation, you can verify that i've done the algebra correctly on your own.
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
so if the average pot is just 9.13bb (a fraction of a bb higher than the 10bb case) the variance for our hero will be atleast as large as the maximum variance in the case where we're basically flipping for 10bb at the end of a husng. something interesting to note here is that we've confirmed that changing hero's winrate (from breakeven to .5bb/hand - a huge increase) had very little impact on variance as a function of the average pot.

so what we've found from all this is that as the average pot size (in bb) goes up, so does the variance REGARDLESS OF EITHER PLAYER'S EDGE. at the logical extremes of this you'll see that when stacks approach zero the limit of variance will become very small and as the stacks become deeper variance will increase because as the average pot becomes larger.

in order to confirm that the average pot will become bigger as stacks increase take the case where stacks are very small (as in less than 1bb; both players will be allin for an ante). as stacks go to zero the average pot will also go to zero.
when stacks become larger (say 10-100bb) each player will have enough money to post a full blind. the average pot must be still larger than this so we can safely say that the average pot size increases with increasing effective stack.
this last example is kind of silly but i didn't want to leave anything out.

on to your example:
adjust each stack to be in bb
player A plays w/ 100bb
player B plays w/ 10bb
the first part of my post above shows that player B will experience less variance and a lower standard deviation in bb/100hands than player A.

logical or mathematical enough for you?

not a troll,
wtfnoway