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 .

[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.

[The Yugoslavian]

I think theoretically cash is slightly less variance than HU STTs for the reasons explained above (they are quite persuasive).

HOWEVER

I think that in cash games much more so than tournaments, TILT really really adds to one's variance. I don't even mean mega-monkey tilt. In any tournament-based format, it's not only psychologically easier to figure out when you're tilting and to stop playing but also, the breaks in playing one contest and another helps one subconsciously IMO. Your subtle tilt or monkey tilt will calm down in those 30 seconds where in a cash game you're liable to do any number of retarded bully plays.

Does this even make sense? Meh, I do think there is something there, however. And that it could very well explain why in practice, nl cash ends up being higher variance.

Yugoslav

[PartysOver]

deep play and constantly reloading is definetly a factor

[omgwtfnoway]

TNixon, that's my blog you quoted. unfortunately the logic in that post isn't sound and i have a better understanding of variance now.

you're making a common mistake in that you assume small stack poker is necessarily high variance. the results of large field turbo tournaments have high variance because of the number of players and the skewness of the payout structure; your largest tournament cashes will have values extremely far away from your expected value in the individual tournament.

in hu nl cash games you similarly have individual results that are very far away from expected value. A player with a solid 10ptbb/100hands edge on his opponent will frequently play pots for 200+bb if they are both playing aggressively.

Based on his winrate, the player would expect to win .1 bb on average from that 200bb pot. Do you see why this will cause high variance?

so large field tourneys and short handed nl cash games have this in common: most of the individual results fall very far away from the expected values based on winrate. This is the source of variance.

small stack poker is actually very low variance. when you play husng you can only win or lose one buyin at a time. assuming you are a solid 60% winner your results can only deviate from your expected value (roughly 15% roi) by at most .85 buyins.

contrast this to the examples where you win a large field tourney (for 100+ buyins) or stack your opponent in a 500bb+ cash pot.

small stack, hu, freezeout poker is the lowest variance form of poker.

[derosnec]

does rake have an impact on this topic?

for example, at 50nl HU cash, PT shows me paying 2.5% rake, whereas in HU SNGs, the rake is 4.5%.

[TNixon]

[ QUOTE ]
small stack poker is actually very low variance.

[/ QUOTE ]
I'd like to see some logical or mathematical backing to this statement, because in my mind, it's very clear that playing 5BB stacks is much higher variance than playing 100BB stacks, assuming an equal stack value. Playing a $100 stack at 0.5/1 blinds is clearly going to be lower variance than playing a $100 stack at 10/20 blinds, but that's exactly the situation a tournament puts you in, playing your $100 stack for larger and larger blinds.

Open-pushing with A5 is a high variance move. I'm never going to open-push with something like A5 when I have 100BB, but that is a common situation when effective stacks are small compared to the blinds. The only reason you make the move in the first place though, is because the blinds are so big compared to the stacks that the risk vs reward ratio is in your favor, whether your opponent calls loose (in which case your hand has a positive EV), or they call tight (in which case you're buying the blinds enough to be freerolling for probably a 30%-ish chance to win when they finally do call).

[ QUOTE ]
assuming you are a solid 60% winner your results can only deviate from your expected value (roughly 15% roi) by at most .85 buyins.

[/ QUOTE ]

Since you don't sound a complete idiot, I'm going to assume that you're *not* trying to say what it sounds like you're trying to say here, that the swings due to variance will be less than a single buyin for a 60% winner. (which is obviously not true)

But if you're not trying to say that, then I don't have any idea what you really are trying to say. [img]/images/graemlins/smile.gif[/img]

Also, the examples from large-field tourneys are pretty irrelevant. It is clear and obvious that the more entrants there are in a tourney, the higher your variance will be. So heads-up sit-n-gos are clearly the lowest variance of all the freezeouts, but the fact that adding more people to a tournament increases the variance has absolutely no bearing on whether heads-up cash is higher or lower (or the same) variance as heads-up sit-n-gos.

[ QUOTE ]
A player with a solid 10ptbb/100hands edge on his opponent will frequently play pots for 200+bb if they are both playing aggressively.

[/ QUOTE ]
You seem to be claiming here that players will get their stacks all-in much more frequently in cash games as opposed to tournaments.

I don't have a huge amount of experience at heads-up cash, but from what I've played and what I've seen and heard, this is not true.

In fact, I would argue that the reverse is true, that the chips get all-in more frequently and with weaker hands in a tournament, because of the pressure from the increasing blinds.

And every tournament is for a full stack. If you're running particularly badly so that you can't find any hands worth putting chips after, you don't get off easy and just lose a third or a half your stack. You lose it all, due to the pressure of the blinds, or more likely, you end up flipping coins for your full stack.