browni, please just do some searching, rather than just posting about the way you think it should be. It has been conclusively shown (including the graphs) on this forum that variance is higher the more your winrate is based on SDV, especially AIEV. If you are a an aggressive nit or a short-stacker, you are often at the mercy of the deck with cards yet to come. If you chose those situations well, you will be ++EV, but you still will be playing for stacks with vulnerable hands often. Thus your graph will look like a richter scale.

This is hard to see live, as we don't track stack depth changes hand-by-hand like online trackers do. We track by session, where it is much less clear that the nit's profit came from one AA AIPF that held for a double up, and the LAG's came from a series of small-medium stabs.

Edit: garick beat me. Still leaving wall of text

Mpethy posted on this subject with some incredible analysis years ago. He was much better at math than any of us, and he was paid to analyze people’s databases back in the day.

One of the things he realized was that people that had lag stats actually had a lower standard deviation, which he initially found counterintuitive.

Meaning, 2 players, one with 18/16 stats and one with 30/25 stats that both had similar WR, the 30/25 usually had a lower standard deviation.

The resulting analysis showed that it has to do with how the lags played postflop, both realizing their equity better with premiums (standard today but odd lines back then, like checking 568 flops with AA), and also because of non-showdown winnings. These non-showdown winnings really help “smooth” out a winrate, which is one of the reasons people focus on “redline”. (This is the small to medium pot importance that garick is talking about)

TAGs rely heavily on expectation in a hands equity. LAGs showdown less and therefore dictate their own realized equity.

Meaning, a TAGs session results will be a straight line with the occasional heart beat jolt up or down. A LAGs session results will look like blades of grass going up or down a hill.

Words like sigma and square root of the mean can lose people, and I think detract from the argument and the scope of this forum. But the logic that Dwan actually has a lower sd in his WR because of hands like this makes sense to me.

Garicks and Avaritias last posts explain perfectly what I was getting at earlier and why people like GGs results can vary widely over 500-1000 hour samples. That makes guys like him who play uber tight think that sample sizes of 500-1000 hours mean nothing.

For people who play closer to my style which is semi LAG preflop and a lot less showdowns overall, sample sizes of 500-1000 hours mean a lot more and they vary a lot less.

Avaritia, to be clear, the standard error will decrease. The standard error is how likely the sample mean (observed winrate) varies from the true population mean (true winrate).

Otoh, the standard deviation (measure of swinginess), which measures how individual observations vary (our sessions), should be higher for LAGs.

It’s not. Unlike browni, I took 4 years of stats...that I’ve mostly forgotten...but I know the difference and understand what you are saying wrt standard error. It seems counterintuitive but standard deviation was lower for lag players, mainly because of small/medium pot non-showdown winnings.

Stated very simply (and this doesn’t even apply as much in 2019) a tag waits for Aces and wins a stack or loses a stack vs QQ. A lag plays many hands, smaller pots not larger ones (also counterintuitive to how most define lag) and wins/loses a few bbs at a time.

As garick said, this was extensively outlined by a near genius on the matter years ago. His name was mpethy if you want to play around with search. (The juice might be in our chat thread)

This doesn't mean anything at all. Variance is a property of a data set, not of an individual datum.

ETA: Variance does not exist and has no meaning until and less there is a mean from which to vary.

I should clarify, I’m not sure the same conclusion would be seen in databases today. What was once TAG vs LAG 18/16 vs 30/25 has now converged into 26/22. (Or whatever, you get the idea).

What lags were doing in 2007 were extreme exploits. I don’t think they apply as much in 2019 online poker which is unfortunately the only source we have for large data sets.

I definitely still think it applies to live poker though. There are very good players who are excellent at stealing orphan pots, for example. These medium risks for medium wins still smooth out variance over time.

Was this measured hand by hand? If so, that makes sense. But session to session, LAGs should see more variation between observations, and thus, a higher standard deviation.

This is meaningless. Decisions are not data points, hands are. It is just incorrect to say Lags observed win-rate converges to their true win-rate faster because they vpip more. It is totally possible that Lags have lower variance, but if true it is not for this reason.

What you just posted doesn't enforce your argument at all. Quote something specific because I may have missed it.

Are you trying to say that standard deviation of our bankroll increases as our sample increases, while the standard deviation of our win-rate decreases? What does that have to do with Lag vs. Tag?

Garick, please do not patronize me. Have I not in the past demonstrated any affinity for math and statistics? I am not saying there are not many things I don't know, or that there are not people more knowledgeable than me here, but I am probably more adept than 99% of the people in this subforum. If you want me to search for something you'll have to be more specific.

I'm not arguing that a red-line warrior won't have lower variance. I actually don't know but it seems plausible that they would. However it would NOT be simply due to playing more hands.

Short-stackers should not have higher variance than full-stackers though. Did mpethy's analysis say otherwise? This would be completely contrary to common sense, and it is trivial to show that the upper bound on variance is decreasing as a function of stack size.

This is plausible, but it's completely different than what cannabusto is saying. I'd really like to see mpethy's analyses.

I know the difference between standard error and standard deviation, too. What cannabusto is saying is incorrect. Standard deviation can not increase while standard error decreases with sample size held constant. I think he may be trying to say that sample size is not constant because Lags play more hands, but that's not correct either.

You're misunderstanding me. I'm no longer talking about a data set at that point. I'm talking about the variance of a strategy as a decision tree. Subtrees can be looked at by themselves and do have both a mean and a variance.

"Holy wall of text, Batman!"

Different people have different ideas about what it means to be Tag vs. Lag so I won't be using those labels in this post.

We can think of the game of poker as a list of payouts x_i which are arrived at with some frequency, f_i. For example, consider a simple game with a pot of 100 on the river. We bet pot on the river with our entire rang and our opponent calls half the time. We win when called 2/3 of the time and lose when called 1/3 of the time. If you have ever studied game theory you should know the EV of the bettor is pot and the EV of the caller is zero. Let's check.

We bet pot on the river and the opponent folds:
(x₁, f₁) = (100, 1/2)
We bet pot on the river, the opponent calls and we lose:
(x₂, f₂) = (-100, 1/6)
We bet pot on the river, the opponent calls and we win:
(x₃, f₃) = (200, 1/3)

The expected value is the sum of the payouts times their frequencies:
EV = 100*1/2+-100*1/6+200*1/3 = 100, as expected.
The variance is the sum of the squared deviations of the payouts from the mean, multiplied by their frequencies:
Variance = 1/2*(100-100)^2+1/6*(-100-100)^2+1/3*(200-100)^2 = 10000

More generally:





Now, in order to show that forking a payout node can not decrease variance.

First, we hold these conditions to be true.

1. The sum of the frequencies of the new payouts equal the frequency of the original payout.



2. The expected value of the new payouts equals the expected value of the original payout.



3. The rest of the payout/frequency pairs do not change.

Now, compare the variance of the old strategy with the variance of the new strategy. The terms for the payout/frequency pairs that don't change cancel and we are left to prove that:



Expand the squared term on each side and use the distributive property to get individual terms:



Use equations 1 and 2 to substitute and eliminate mu:



In equation 2, square both sides, divide both sides by f_t and substitute with equation 1 to get:



Substitute and multiply both sides by the denominator:



Expand both sides with the distributive property:



Re-write stipulating j > i. Terms where j = i cancel.



Move all terms to one side and factor. Now we have a sum of squares on the right side, which must be greater than or equal to zero.



Notice also that the equation can only be equal to zero if all of the payouts are equal to the original payout by equations 1, 2 and the final result.

On the other point, here is the formula for the approximation of standard error, where s is sample standard deviation and n is the number of hands:




Standard error is proportional to the standard deviation. An increase in standard deviation due to a change in strategy will increase the standard error over the same sample size.

Tldr: Deciding not to fold 0 EV hands increases variance.

I can't find the ones that actually show the numbers in a quick search, but here's a post on variance by a professional poker database investigator (or as he called it "leak finder") coach during the boom.

His post illuminates what I meant by increasing the number of decisions per hand. They get in more spots and have more opportunities to apply their skill edge. This allows the statistic (our winrate in this case) to converge at a faster rate.

Still, I was wrong about session to session variance. It appears nits do have both higher variances/standard deviations as well as standard errors.

Here's another post that 'splains it a bit more:

I'm still very unclear on it, as I don't understand the nit vs lag variance concept very well at all.

All I can do is refer to my results.

I think it would be agreed that I likely play a very nitty style.

I've played 4328 hours at 1/3 NL.

I've never been on a 1000bb downswing.

I've only been on a 500bb+ downswing 6 times. I've only been on a 800bb+ downswing 2 times.

Over my last 1020 hours of my most nitty Super Nit style, here's a breakdown of my results:

140 sessions over 1020 hours (averaging 7.3 hours per session), 66.4% session winrate, 6.06 bb/hr, SD = 45.06 bb/hr

+200bbs: 9
-200bbs: 1

+300bbs: 6
-300bbs: 1

+400bbs: 0
-400bbs: 0

Sessions that were less than +/- 200bbs: 123 (i.e. 88%).

My last session out was a perfect example of this. In an 11 hour session, I got stacks in and was called exactly twice: $200 stacks with KK vs AK preflop, and $40 stacks with 88 vs AK preflop, losing them both. I lost $95 (32bb) overall on the session.

Maybe I'm not understanding what variance is, or maybe I'm not understanding exactly what a super nitty style is. But it seems to me I'm playing very nitty and yet having very low variance results, no?

Interestingly enough, those lone -200bb+ and -300bb+ sessions listed above came back-to-back and helped lead to a 800bb+ downswing; is this the variance everyone is talking about here? Or am I missing something?

GcluelessNLnoobG

mpethybridge's posts seem to ring true to me and I think your data agrees. From your posts I'd argue that you're raising range is nitty but once your include your infamous overlimps you are VPIP more than the 8% range referenced here. The extra hands you're playing allow you to make up for the big losses when your KK and QQ get cracked but it's not enough to make the session break even. The argument is that as you move to TAG / LAG if you can maintain the same winrate those big hands will become less and less session crushing. Also, big hand doesn't mean the whole stack goes in as in your example. Just that it's a big starting hand and a big pot.

^^^^

Yeah, that's definitely possible. I mean, in EP I'm folding 66/A9s (super nitty!)... but in LP I'm seeing a cheap flop if possible with 74o (super loose?).

GcluelessvariancenoobG

Dont forget that the min buy in online back then was 20BBs (maybe it still is, I dont know). It doesn't take a huge number of hours to figure out how crazy the variance is when jamming you're just 20BBs over and over again.

Your conclusion doesn't follow from the premises. Mpethy is actually talking about Lags having a higher win-rate, which leads to less intense downswings. He doesn't make a claim about Lags having a lower standard deviation. I have already proven above that vpipping more hands while holding the rest of the strategy constant results in higher variance. Can you point out a mistake in the proof? I hope it is not difficult to follow. I have never taken a class or studied how to do formal proofs, I just go about it in a way that makes sense to me.

Evidence of this has yet to be seen in the thread. I'd be very interested in actual empirical evidence. None of the logical arguments for this claim hold up very well.

----------

This comes up so often and causes so much confusion that there really should be a sticky about the difference between variance and swings. It turns out mpethy is not talking about variance in the statistical sense at all.

So mpethy even says himself he's not talking about the technical definition of variance. He's talking about swings.

The bolded sounds a lot like the conditions for my proof above. A Lag has less bad swings than an ubernit not because he plays more hands, allowing his win-rate to converge more quickly, but because he is finding more profitable spots, leading to a much greater win-rate.

If it wasn't clear before, it is very clear here that mpethy is not talking about variance = σ^2. Increasing win-rate does NOT necessarily decrease variance. It DOES decrease n-nought, holding standard deviation constant.

I am not sure if n-nought is a well known term, so let me explain. It represents the length of expected breakeven stretches. I will quickly show how it's derived and defined. This formula is for the bottom end of the confidence interval. Let us set B to zero and solve for t in terms of standard deviation and win-rate.

B: change in bankroll
n: sample size
σ: standard deviation
z: z-score
μ: win-rate









It is clear that breakeven stretches become shorter with an increase in win-rate, and become longer with an increase in standard deviation. It is possible for a Lag to have a lower n-nought than a nit despite having a higher standard deviation due to a higher win-rate. In fact, I would not be surprised if an analysis of online players revealed a winning "Lag" style to be less swingy than a winning "nit" style. However I highly doubt that a Lag has a lower σ than a Tag or a nit.

For a quick example, let's say a nit has a win-rate of $5/h with a SD of $100/h. His one standard deviation n-nought is (100/5)^2 = 400 hands, which means he is approximately 15% likely to breakeven or worse over his next 400 hands.

Now let's say a Lag has a win-rate of $20/h with a SD of $200/h. His one standard deviation n-nought is (200/20)^2 = 100 hands.

So, mpethy did not disagree with anything I said. He's talking about a completely different concept. Let me repeat one more time: N-nought (length of breakeven stretches/downswings) is NOT the same thing as variance.

No, I cannot rebut your proof. Way too dumb.

But, I mean, it's right here: "So, someone might argue that the lag is more susceptible to variance because he's in more and thinner spots. But the difference is that a lag is in spots where luck is not the determinative factor in the outcome, but skill is"

That's what I said. LAGs get in more spots, and thus, ply their skill advantage more often. Maybe I have the semantics wrong regarding statistical variance. But that's my entire point in a nutshell.

Here is some grist for the mill of the "whose variance is bigger, LAG or nit?" debate:

Imagine a player who folds everything. Everyone else knows this, so our hero's big blind is always raised, and hero folds. It's a nine-handed game.

Hero loses 1.5 bb every nine hands, for a mean loss rate of -1/6 bb/hand.

What is hero's variance?

The deviation from hero's mean is -5/6 in the big blind, -2/6 in the small blind, and +1/6 in the seven other positions. The squared sum is 25/36 + 4/36 + 7*(1/36) = 36/36 = 1, so the mean square deviation, i.e. the variance is (36/36)/9 = 0.111 bb^2/hand. 11.11 bb^/100, or (at 30 hands/hour) 3.33 bb^2/hour.

By comparison, my own variance at 2-3-5 NLHE is 5,081 bb^2/hr.

Now imagine that our hero loosens up a bit, so that they play pocket aces (and only pocket aces). This is going to raise our hero's win rate, and it is going to dramatically raise their variance, but I doubt it is going to raise it a thousand-fold.

I think there is little doubt that adding more hands to our hero's range will continue to increase their variance.

It seems quite likely to me that a player who plays a nitty preflop range is going to have a lower variance than a player who plays a wider range.

Okay, define "variance." Do you means swings, or σ^2? This is not just semantics.

Did you track those hands? PT4 can calculate the statistic.

I highly suspect that all of the "shortstacking is higher variance!" folks are falling for the same variance vs. swings confusion as the Tag. vs. Lag folks.

I dont know what the real definition of variance is in the mathematical sense, but I think what we mean when we use the word in the poker world is... results that "vary" all over the place. Results that are non consistent or regular.

I dont have any of my data anymore so this is from memory but I remember it pretty well. I played about 2 hours very early each morning before work when the Euro players were playing.

First month I short stacked 5/10 online I made about $14,000
First 2 weeks of the next month I was down $6000
Next 2 weeks of month 2, I won $7000.

That's only 120 hours and I only did it for about 250 hours total so its obviously not a large sample but the results continued like that sample I listed. It was all over the map.

Playing my normal style of poker never has "swings" like that. My day to day or week to week results never "vary" anywhere near that much.

I was 4 tabling most of the time and I was all in almost every hand I played so it was a lot of all ins and a bigger sample than it looks like.

I think that what I mean when I say "variance" is the mathematical one: 1/N * sum ((x[k] - <x>)^2)

I'll try to clear up my point of view one more time.

Winrate is a mean. Being that our statistic is a mean, we should look at the standard error to see how much our sample mean (observed winrate) may differ from our population mean (true winrate). Note that this has nothing to do with the observed variance of the sample.

Now, imagine you have a 10 billion hour sample. Regardless of our winrate, our standard error should be extremely close to zero. This makes sense since we should definitely be right on the nose of our true winrate if we played 10 billion hours. Likewise, if you have a 20 hour sample, your standard error will likely be relatively large because the sample size is so tiny.

So, swings, the sample's variance value, etc. has nothing to do with my point. My point is that loose players need less hours for their observed winrate to track to their true winrate. The interval estimates of their winrates will be narrower because the standard error associated with it is lower. This makes intuitive sense since their is less luck involved for loose players than tight players over equal sample sizes.

This does not mean loose is better in any way. I can't imagine it being practically useful. Maybe if you believe to have a similar winrate playing tight or loose, you would decide to play loose because you're more likely to hit your true winrate over time. But that scenario seems far fetched.