Hello,

I recently had a thought about variance in poker. Like in every website people tell you that increasing your winrate will lower your variance. But isn't that incorrect?

The greater your winrate is the lower chance is that you lose money but overall variance will remain the same imo.

Can someone tell the math behind this?

Variance depends on the game, not the winrate, the winrate may, depending on the sample size, depend on the variance

Some time ago I posted this in another thread:

https://forumserver.twoplustwo.com/1...ghlight=zkesic

Rusty corrected me with this:

those graphs are not really interesting for what variance is telling you. if you substracted the expectation from the second and third graph, which is (p-0.5)*n, then you would have graphs to compare. variance for a binomial distribution like the coinflip is:
p*(1-p)*n
so you can see that it is 0 for p=0 and p=1 and the maximum is 0.25*n for p=0.5. so it is actually influenced by winrate, but again for that example you see that for a 51% winrate variance is .2499*n so it only gets meaningful for greater variations in winrate.

Variance and trend (in this case win rate) are completely different concepts. While they can be related in the real world, in skeptical they are in poker. I think what most people really mean when they say "you have a lower variance with a higher win rate" is just a crude way of saying you have a lower chance of a persistent or deep losong streak when your win rate is high, which is true

Generally, the higher the win/loss rate, the lower the variance. This is seen in the typical MTT variance calculations/simulations and generally in cash game variance graphs compared to win rates.

One can play in a way that lowers variance and so have a lower variance even when it lowers the win rate. A higher win rate also helps.

When one is short in roll/chips perhaps, lowering the variance makes it more likely one will survive. The cost is a lower win rate down to losing but maybe in a tournament (at least the ICM).

In the technique, the different stack sizes need a different range and a way of playing, but that's not in itself about the variance.

The lower stacks might decrease the variance of the bankroll because of less money/chips involved but might increase the variance because the bigger money goes in lighter (more variance), though more often (less variance perhaps).

I tried to calculate the topic with this calculator:

https://www.primedope.com/tournament...ce-calculator/

It gives us the standard deviation and the standard deviation is the square root of the variance.

I used these specs:



I calculated with 20 different ROI's from -90% to 100% and but all results to excel.



Surprisingly the result was: bigger ROI means bigger variance!!!

Any comments on this?

Is there something on real life poker the variance calculator doesn't take account?

When poker players use the term "variance", they mean "swings". More specifically they mean downswings. But you can also have large upswings.
A game like PLO is high variance. You can have awful downswings, but you can also go on incredible heaters that are pretty much impossible to have in NLH (a lower variance game).
If you bink the Sunday Million, you have a ridiculous ROI and are way above expectation (because you don't expect to win it every time you play, obviously). Therefore a big win is an example of running (way) above EV. It's an example of 'good' variance.

Your variance just depends upon your standard deviation. i.e. How much your results vary from session to session. If you tend to win a little or lose a little, you're in a low variance environment. If you have large swings, both up and down, you're in a high variance game.

For an STT you can calculate the theoretical statistical variance given buy in, fee, prize structure, ITM placing distribution and your ROI. I did that for a 9 man sit’n’go with a 9 – 1 entry and assuming equal relative placing when ITM with a 50-30-20 prize structure. The variance increases with the ROI up to an ROI of about 55% and then starts to decrease.

Different parameters will result in different results but it appears that variance is not monotonic with ROI at least for these kinds of tournaments.

Mathematically variance is a measure of how smooth results are (how much above or below the overall mean each individual result is.)
I think the answers already given above do clear up these issues.

As ArtyMcFly says most times posters tend to use variance as a term for swings and not its mathematical meaning (or perhaps imo also sometimes just as another word for luck).

As ohly says for coin-flips variance = p*(1-p)*n and it reaches a max when your roi is 0.0%, ie, breakeven. But as statmanhal mentions for larger field tournaments with typical itm structures the maximum variance will be above the mid break even point, for the very large fields (with typical payout structure) the max mathematically possible variance will be with with a very high roi.

If you always won, or always came 2'nd, always 3rd, etc, or always out of the money then your results have a zero variance and if plotted they would then look like a ruler straight line on the plot.

It is possible to always lose if you intend to but not really possible to always come first so your results plot will typically be a bit 'bumpy', the measure of the bumpiness is variance.

The good thing about knowing the variance is that you can use it to predict how likely you are to be above or below a roi value after X games (although you also need to know the mean. With an approx. mean you can get an approx. answer).

For tournament with field sizes above HU typically your variance will actually go higher if your roi goes higher (unless you have some crazy massive value of roi). If you do get better the amount your variance goes up is pretty small compared to changing field sizes though.

The approx. var per game for different sized tournaments (all approx and with a flat finish distribution, a bit of rake and 0% roi measured in Buy-ins squared) are:

6 max STT per game variance: 1.98
9 seat STT per game variance: 2.1
18 seat per game variance: 3.91
27 seat per game variance: 5.3
45 seat per game variance: 7.1
90 seat per game variance: 11.5
180 seat per game variance: 23.8

If you play 90's and get a bit better your variance may rise to 12.0 perhaps, this is not much when you look at the variance the break even 180's have.

Increasing your roi really does protect you well from downswings though, playing 180's with 1% roi will leave you very likely to see massive swings, if you have a 20% roi these are much, much less likely. With a 1% roi in HU as the variance is so small you won't be likely to see big swings. But per game with 1% per game you still earn the same it's just that the 180's are effectively much more of a gamble in the short term.

Variance per game doesn't change with speed, ie, normal, turbo, hyper but with hypers you may play 10 times the amount of games per hour compared to normal and so your per hour variance with hypers is 10x as high.

I think the reason the plots you posted seem so much smoother as the roi goes from break-even to 51 to 55% w-r is not really that the variance changes (as it is HU it will go down slightly see ohly's post) but also for two other reasons.
(i) as your w-r goes up your chances of downswing drop very strongly (on the first plot you do see a big swing).
(ii) for the higher w-r plots the 'y' axis scale/range changes from -60 to +10 to -200 to 1200, this compresses the individual result changes in the visualized plots.

The variance will drop due to the rise of roi and this also will make the plots smoother but the big thing is the swing present on the first one. The 'bobble' in the line as the results arrive is pretty much the same.

You are right in that with 55% w-r a lower BR would be needed although 10BI even for a 51% w-r is very tight. (Note: also if you always spent your excess above the 10BI then you would still go bust even with a 55% roi - or any sized roi if you play enough games .)

Seems like there's a difference between tournaments and cash games here. In tournaments, your ROI is capped at both ends since there's only so much you can win/lose over a given sample.

Obviously someone losing in every single tournament has no variance as does someone who is winning in every single tournament. If the loser in the first case starts to win a little his variance increases and if the winner in the second case starts to lose a little his variance also increases. These bounds don't seem to exist in cash games though unless you want to start to factor in something like bankrolls. I'm not really seeing how someone winning more is necessarily taking on more or less variance.

I think in some ways it is a bit more complicated for cash games as there are more different ways to play, it is not as constrained as tournament due to then having fixed payouts. Even in tournaments it is possible to win more and your variance go down though, eg, if in 9 seat STTs you play an icm style to more often come in 3rd than a wild go for the win approach you could win more but drop your variance.

I still think for cash games that often increasing your bb/100 will also increase your bb/100 variance although it is definitely possible to be playing in a way that the opposite would occur.

As your variance is proportional to the squared number of bb you play so a good player will more often be at a table with 200bb stack depth instead of 100bb. Stacking off with any of these 200bb hands will have 4x the variance of the 100bb depth ones, this would be one way the variance increases.

I suspect if a player is making 1bb/100 as a full ring super nit and improves by playing a slightly higher vpip it is likely that the variance of the bb/100 increases.

However, if a maniac again wins at 1bb/100 but improves by tightening up a little I suspect the variance seen would reduce.

Most of the nits hands would be folded and so these results are very close to the mean (The mean is 1bb/100 or per hand 0.01bb). This player somehow earns this 1bb/100 average in the actively played hands. Each of these active hands with have a high variance whereas the others are almost zero.
Variance of a folded (non blind) hand is (0.0 - 0.01)^2 = 0.0001

It will depend on how large a pot for the other active hands but let's say they are often +20bb or -19bb, the variance of these hands is very big eg, (20.0 - 0.01)^2 = 399.6 or (-19.0 - 0.01)^2 = 361.38

I think most of the variance would be coming from these active hands and if somehow this player gets a little better so goes to 2bb/100, perhaps the active hands are now 21bb and -19bb pots so now these are (21.0 - 0.02)^2 = 440.16 or (-19.0 - 0.02)^2 = 361.76.

The overall variance will have gone up.

It is quite hard to give reasonable examples and I might also be wrong here, I always play tournaments and so I haven't thought much about how cash variance works.

I suspect typically most players if they increase w-r in bb/100 also increase their variance but I may be completely wrong here and style of play would matter a lot.

I agree with these points. I play a low variance nitty style in cash, that somewhat limits my potential winrate. I could play a bit looser pre-flop and bluff more post-flop, and that could increase my theoretical winrate by a couple of bb/100 maybe, but I'd also be at risk of having much more frequent losing sessions (from bluffs getting called, or making the 2nd best hand too often).
It's generally the case that the LAGs have the biggest heaters (when they are running well), but they also have the most extreme downswings. (See: RedBaron's old graphs).
To minimize variance, you just press the FOLD button more often. Your stack (or bankroll) can't fluctuate if you don't put money in the pot.

In short, it's not really that high winrates create more variance, as per OP. It's more that to achieve a high winrate, you have to take more risks, which means you have greater fluctuations, both up and down.

Yeah I guess I mostly agree, I think average internet player who is trying to get better and especially average 2+2er probably has fundamentals down enough that increasing winrates means finding good bluff and semibluff spots and image building more which means increased variance. Poker players in general though (including live fish), play too loose/passive on average, but its tough to tell overall if correcting their play to tight/aggressive decreases variance (due to tightness) or increases it (due to aggressiveness). Might be a wash. I think I'm OK in general with saying increased winrate slightly increases variance in cash games as long as your pointing out that its not something mathematically forced, and that variance isn't the same as being downswing prone or losing streak prone.

where did you get that excel sheet?

I made it.