Variance revisited HUCASH vs HUTRN

[TNixon]

Ok, I know I've asked this numerous times, but never really got a satisfactory answer.

Common wisdom says that the swings in HUCASH will be bigger than in HUTRNs, that the variance in cash is bigger. Intuitively this makes sense. It is very easy to prove mathematically that variance goes down in freezeout tournaments as the number of entrants is reduced, and although I don't know how you'd go about proving it, it does make sense that variance goes up on cash tables as the number of participants goes down.

So, heads-up should be the lowest variance form of freezeout tournaments, and the highest variance form of cash, which would lead to the belief that cash is higher variance than tournaments.

What I can't figure out is why this would be true.

In regular tournaments, there are situations where a +chipEV situation could be -EV, because of the payout structure. In a winner-take-all setting, these situations never occur, and chipEV is exactly equal to realEV.

There are differences between the two, of course, but the differences ought to cancel each other out.

In tournament play, if you lose a big chunk of chips, the player can't just leave and take away the opportunity to win them back, but in cash play you can reload, making it that much easier to get them back.

In tournament play, the blinds grow, but what that really means is that you're make higher variance plays as time progresses, and in cash play, you can afford to be more patient, and play like you would during the early stages of a tournament.

Please, somebody explain. Not understanding this bit of conventional wisdom (while also being unable to completely buck it in my mind) is bugging the crap out of me. [img]/images/graemlins/smile.gif[/img]

The only possibility I can really think of is that heads-up is a meeting point of sorts, where the high variance of tournaments meets the low variance of cash tables, and heads-up just happens to be in the middle, with the variance being equal.

Obviously I've got too much time on my hands, to think about this issue as much as I have.

[img]/images/graemlins/smile.gif[/img]

[jay_shark]

Lets start with some definitions of what variance means .

It is defined as Var(x) = E(x^2)-E(x) , where x is a random variable denoting the number of buy-ins .

Say you're a consistent 60% winner . In other words , you win 60% of the time independent of tilt or any other psychological factors that may come into play .
Your variance is :


Var(x) = 1^2*0.6 +(-1.05)^2*0.4 - (1*0.6 - 1.05*0.4) = 0.861

Your standard deviation would be sqrt(0.861)=0.9279... .

If you're a 70% winner , your variance is going to be lower .

Var(x) = E(x^2)-E(x)
Var(x) = 1^2*0.7 + (-1.05)^2*0.3 - (1*0.7 - 1.05*0.3)=0.64575

It should be clear that the better player you are , the less variance you will experience playing heads up .

For tournaments it's a bit tricker to quantify since the probability you win a second round match is not the same as the probability you win a first round match . For simplicity , if we assume that the probability you win each match is 60% , independent of each round , then the probability you win a 4 player tourney is 0.6*0.6=0.36

Var(x) = E(x^2)-E(x)= 3^2*0.36 +(-1.05)^2*0.64 - (3*0.36-1.05*0.64) = 3.5376

It should be clear now that your variance increases as you increase the number of players in the heads up tourney .

Ask if you're not clear about something .

[Indiana]

It is defined as Var(x) = E(x^2)- E(x)^2

FYP jay shark

Also, it should be clear that HUSNGs are <<<<<variance than HU cash. As far as showing this mathematically...In HUSNG your outcome is a binomial random variable so the variance is a direct function of the mean so its easy to write down on paper.....In cash, your variance is not a simple function of the mean(because your outcome is a from a normal sampling dist'n) and should be estimated from actual data using the normal distribution.

But at the end of the day its pretty common sensical...to experience high variance in HUSNGs you need to have a very very low win rate because you will need to have lots of bad runs (like losing 5-6 in a row which is rare for a .7 winner) 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?

[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]

I did find this while browsing around some people's blogs:

http://wtfnoway.blogspot.com/

Search for Variance, there's an entry about the differences between sit-n-gos and cash games.

As far as I could tell, it basically boils down to "variance at SNGs depends only on your winrate, while variance at cash depends on a lot of other factors, because you can lose partial stacks".

While it is true that variance at SNGs depends purely on winrate, that winrate depends on all of the same things that your winrate in a cash game does.

A tournament is a series of heads-up hands, just like a cash session. And since there are no chipEV/EV differences in a heads-up tournament (and people almost always overestimate those differences even in situations where they are fairly significant), plays that are correct on a cash table should also be correct in a tournament.

The only differences are that you can't rebuy, and that the blinds increase over time, but both of these things mean you're making higher variance plays with smaller effective stacks, which would tend to lead me to the conclusion that HUCASH is *lower* variance than HUSNGs, not higher.

Maybe the difference is that there's sort of an auto step-down mechanism built into tournament play, because you can't rebuy?

On a cash table, if you lose half your chips (or take half the other player's chips), there's probably going to be a rebuy, and further hands are again risking a full stack instead of just half a stack. In a tournament, you're basically now playing the equivalent of a lower buyin where you have double the max buyin.

Only not really...because of the blinds, it's more like playing a higher buyin but buying in short (while your opponent buys in even shorter), which is, again, a higher variance situation than having 100BBs at lower stakes. So it's not really an auto-stepdown mechanism, but an auto-stepup into a small effective stack situation.

Obviously I'm either very confused, or conventional wisdom really isn't all that wise.

[TNixon]

[ QUOTE ]
It should be clear now that your variance increases as you increase the number of players in the heads up tourney .

[/ QUOTE ]

That is absolutely clear, but there seems to be a bit of confusion. Everywhere I've said "tournament", think "sit-n-go". I'm not talking about multi-table heads-up tournaments, but single-table ones, just the normal one-on-one.

Sorry if I was unclear.

Anyway, conventional wisdom states that variance at HUCASH is greater than HUSNGs. That's the part I'm trying to understand, because it doesn't really make any sense to me no matter how I approach it.

[jay_shark]

Ohh thx for clearing this up . I'll try to explain it without heavy math .Ohh and by the way , there is less variance playing hucg's then there are playing husng's ; but, it may get tricky if we make the assumption that cash game players are better than sng players .

Suppose you play a $30 heads up match (sng)and your expected win rate is 60%. Likewise , suppose you play with a $30 buy in where the blinds are 0.4 and 0.2 . Note that this is equivalent to playing a $30 sng with starting chips of 1500 and initial blinds at 10-20 . The only difference is that the blinds remain fixed . Now , if we play this cghu until one player loses his buy-in , then it should be clear that the probability you win his buy-in is greater than 60% . This may be equivalent to winning a husng two thirds of the time .

Now what you're saying is true provided that a $x sng is equivalent to a cash game with (x/75)BB and (x/150)Sb.