Let's just get it out of the way.

Apple to apple. Playing more hands will have higher variance than playing fewer hands. Is there any argument here?

The idea that LAG has lower variance than TAG or NIT stemmed from the idea that LAG is adjusting so well to playing environment and player tendencies that such playing style results in higher WR overall.

It's not a hard concept to grasp. Imagine a winning TAG's various ranges from losing to winning, and for the sake of simplicity, just imagine there being only two ranges: winning and losing.

If said TAG can turn 20 combinations of hands from losing range into winning range, his WR and EV of both ranges are going to go up. If he continues to do so successfully, his range is going to become progressively wider and wider to the point where he is now considered a LAG. This new "LAG" style is actually a progression of his TAG style.

However, that message is often lost because of the wide perception of the term LAG. Most people when discussing this topic often ended up with a simple notion that "playing more hands" has lower variance than "playing fewer" and that is simply untrue.

Yes, playing more hands will have higher variance. In that, your session to session results will see greater swings and dispersion.

The implication of my conclusion is that playing more hands leads your observed winrate to converge to your true winrate faster (this isn't a conclusion related to how spread out something is, but how accurate a statistical estimate is). And, in the long run, LAGs should be closer to their EV than TAGs. In the infinite run, everyone hits their EV.

That doesn't mean there is an advantage to playing more hands. It may be, and often is true, that playing more hands leads to lower EV. So who cares if it converges faster?

Moreover, in live poker, the long run never comes. Still, if you're a pro and plan to continue, you have savings, and you know your EV is equal playing TAG and LAG, you should play LAG. But, I mean, that's a lot of caveats there.

Well, in live poker, there will never be enough hands for anyone to come even remotely close to any meaningful statistical sample size. So the argument that LAG by playing more hands will be closer to realizing his true EV is not applicable.

You are right - it's not like there is a prize for actually realizing your EV. If anything, poker would cease to exist if every player realizes his/her true EV...

Not only there are a lot of caveats, there are a lot of very skewed assumptions because none of us will ever truly realize our EV.

FWIW, I would much prefer everyone to be LAG than any other style. So maybe we should just scratch this discussion and continue to promote the idea that looser playing will have lower variance =).

Just because you'll never hit your true EV doesn't mean that you aren't more likely to get closer by doing certain things.

But yeah, as I've said, there is no practical significance to any of this for virtually anyone. I totally agree.

Is there any difference how close someone is to infinity?

Another way to consider the comparison is to use yourself as an example.

Compare your current WR and variance to earlier playing days and then envision a better version of yourself that has more poker knowledge.

Finally, label this better version of yourself and label the version of yourself in the early days of playing poker.

Can you see the argument that LAG has lower variance than NIT?

The closer to infinite you are, the closer to your true EV you will be. This is true of any statistic. Some need to be closer to infinity than others before really stabilizing into a tight range of potential values.

Even statistics measuring the same thing can differ when looking at a different sample/population. Take the example I gave above. An OMC's observed winrate will converge slower than a maniac who vpips every hand and shoves every flop. The maniac is just way more likely to run even because they see so many cards run out. That's where all the noise is--in the deck. The signal is everything else.

Let me rephrase. If 1 million hands are required to have a meaningful sample size, is the sample size of player A with 100 hands going to be more meaningful than sample size of player B with 1000 hands?

Well, an OMC is going to be trending toward the same line even after reaching a statistical meaningful sample size (but OMC is probably not going to live long enough). It's like calculating win rate of someone who only plays AA. We don't need a large sample size to calculate that.

The issue is with the other side of comparison. A perceived maniac's sample size can never be large enough to know with confidence whether his result is better or worse than OMC. At end of the day, measuring maniac's sample size of 1,000 hands vs 10,000 hands are both rather meaningless if the meaningful size is 1,000,000.

There are two discussions here. What me and cannabusto are arguing is absolutely not semantics. It is important to get this straight because telling new players that Lags converge to their winrate faster is dangerous if the opposite is true (or vice versa)

What MikeStarr is arguing is semantics. It's fine to use the term "variance" in an informal sense when talking about poker. I do it all the time. It is extremely confusing to use variance informally in the context of this thread where many conversations are about statistics, because variance already has a very precise formal definition which conflicts with the way MikeStarr is using the word.

When someone asks "which playstyle has higher variance?" Only the formal, precise definition should be used, as the informal definition has no means to quantify itself.

There's a lot of doubt. There would be a lot less doubt if people could post some empirical or mathematical evidence. Your definition of "variance" is worthless in this discussion because it can't quantify anything. What does "smoothness" of a graph mean? We can't compare smoothness of graphs without graphs or a precise definition of smoothness.

What would it take for you to change your mind on this point?

This is the sole point we disagree on, and it's very important. Let's go back to the Wikipedia page:



"s is the sample standard deviation (i.e., the sample-based estimate of the standard deviation of the population), and
n is the size (number of observations) of the sample."

Notice it is referring to s as the "sample standard deviation," and n as the "size of the sample." It's not talking about the standard deviation and size of the distribution of means. The standard error is the standard deviation of the distribution of sample means.

If there is still doubt, also consider the derivation of the formula for SEM further down on the standard error page, and also on this page: https://en.wikipedia.org/wiki/Varian...enaymé_formula)

If you want I can run a simulation for you, as long as the parameters are easily programmable and we agree on them.

browni, I am not sure where you are going with the argument.

I do not believe there is anyone arguing the simple fact that playing more hands will have higher variance than playing fewer hands. It is as straight forward as saying that the variance of AA vs 8 hands is smaller than KK + AA vs 8 hands. It can be calculated.

Cannabusto is arguing that Lags converge to their win-rate faster (they have a lower standard error of the mean) despite having greater variance. I am arguing that greater variance necessarily equates to having a greater standard error over the same sample size.

Also, some people ITT actually are arguing that playing more hands will have lower variance.

Won't have time to address all this til tomorrow at the earliest. But note that the part you quote is still in the Population section of the that text. The Sample section starts after. The Population section is discussing the SEM and is estimating the population's standard deviation, which is equal to the SEM. It is NOT equal to the sd of any given sample. The wording in that section is super confusing.

Please note this explanation further down in the article:

"In scientific and technical literature, experimental data are often summarized either using the mean and standard deviation of the sample data or the mean with the standard error. This often leads to confusion about their interchangeability. However, the mean and standard deviation are*descriptive statistics, whereas the standard error of the mean is descriptive of the random sampling process. The standard deviation of the sample data is a description of the variation in measurements, while the standard error of the mean is a probabilistic statement about how the sample size will provide a better bound on estimates of the population mean, in light of the central limit theorem.[5]

Put simply, the*standard error*of the sample mean is an estimate of how far the sample mean is likely to be from the population mean, whereas the*standard deviation*of the sample is the degree to which individuals within the sample differ from the sample mean. If the population standard deviation is finite, the standard error of the mean of the sample will tend to zero with increasing sample size, because the estimate of the population mean will improve, while the standard deviation of the sample will tend to approximate the population standard deviation as the sample size increases."

Note that regardless of the variance within samples, the SEM will tend to 0 no matter what as sample size increases.

Yes, se will increase with variance within samples. This has nothing to do with the SEM.

Yes, Im arguing that., but we are defining variance different ways. Im pretty sure what I mean by variance is what most people mean. I have no idea what the actual definition is so feel free to ignore me.

I'll just respond to this post since this really sums up the crux of the disagreement.

Standard error of the mean is not a measure of spread or dispersion in the data points, like variance or sd is. It is a measure of the spread of sample means, and so, is measuring how close we are likely to be to the population mean. I think we all agree here.

I believe I can bust through our disagreement in this post. Yes, you're right that for any given sample of a poker player, the higher the variance/sd, the higher the SEM. The numerator in the SEM calculation is increased, which necessarily increases the SEM.

But of course every poker player's results is a separate population in statistical terms. Each will have a different distribution based on talent, tendencies, and playing style.

Despite LAGs having greater SDs, and thus greater numerators in the SEM calculation, they have greater Ns too, and so the denominator grows and shrinks the SEM being calculated despite the numerator value being higher than the average poker player's.

This is not shown in application because we measure N by hours or by hands. When actually, theory tells us the best way to measure N in our case is by measuring amount of cards seen AKA betting rounds played. Unlike most statistical applications, we know in poker that noise/luck/randomness is entirely a function of the cards. The difference in Real Bucks - Sklansky Bucks is entirely accounted for by how the cards runout. But the calculation assumes that all Ns are created equal.

And so that's where the issue lies--yes, the raw measure of SEM will be higher for LAGs. Ironically, this is the very reason they converge to EV faster--by being exposed to more and more noise/outlier events/luck sooner by virtue of playing more hands than most players, each subsequent rare event impacts the set less and less. The LAG encounters more rare events per hour than does the TAG. Over an infinite sample, the LAG gets all of the luck events that could happen to him out of the way sooner than the TAG. And so his EV is solely a function of his skill at a sooner point, whereas the TAG must get closer to infinity than the LAG. This is why 100 LAG hours are less noisy than 100 TAG hours despite being more dispersed/higher variance. Again, this doesn't mean playing every hand is good if it lessens your EV.

I believe the thought experiment I posed earlier really illustrates what I'm saying the best:

Take a maniac vpiping every hand and shoving every flop for 100 hours. Take a nit who only plays AA for 100 hours. The maniac will have much higher variance in his dataset, of course. But he is almost sure to lose. It is really hard to get all that lucky or unlucky over 100 hours of 5 card boards. The nit might only get AA a couple times in 100 hours. Who knows how it will turn out?

One hand is one hand, whether a player folds it preflop or makes a decision on every street and goes to showdown. Looser players' measured win rates are going to converge to their supposed "true" values at the same rate that tighter players are.

Player A sizes their bets so that the last of their stack goes in on the river. Player B plays shove-or-fold preflop.

Is Player A's win rate really going to converge to its supposed "true" value more quickly than Player B's, because Player A is making more decisions?

(The very idea of a true winrate is dubious, because game conditions are so variable. Is the player to your immediate left a nit, a TAG, a loose-passive fish, or an aggro-fish? Are you playing on a Friday night or a Monday morning? How long has it been since Social Security checks were mailed? What is the stock market doing? How are you feeling today? Your results are going to be sampled from different probability distributions in different circumstances.)

Actually this is a very bad example to illustrate any of your points.

Reason why a nit has small variance is because he plays such small range of hands that his results are very close to the outcome of his equity. Take for example of someone who only plays AA, against 3 other random hands, his equity is 63.8%. The mean of his results is going to be somewhere around that equity %.

Equity of AA against 3 players for a maniac who shoves every hand is the same at 63.8%. The variance of the two players will not be far apart. Difference between the two is that the maniac is also realizing his equity with 72o and all the other junks that he will have in that 100 hands, and create giant waves reflecting the volatility of varying equities.

I'll give you credit for comparing them apple to apple, but I don't think you realize that this example is actually NOT making your argument stronger.

And are we assuming that you as a player is going to play precisely the same way in the first 500 hours as you are the next?

What about players around you?

Player A is going to converge faster because they are more likely to run even by virtue of seeing more cards/betting rounds/decision points. I agree that one hand is one hand.

The true winrate, or EV, exists no matter what, it's just harder to converge to for some variables compared to others. You're just naming a bunch of variables that affect it directly and indirectly. That doesn't mean it doesn't exist.

Are we basically arguing whether playing 10 hands vs 1 hand is going to have more meaningful sample size even though we can all agree that we cannot actually achieve meaningful sample size?

But it doesn't exist because of those variables mentioned.

Because if it does exist, the game would be solved.

I think a pretty intuitive example that shows that it in fact should converge faster is a roulette comparison.
Person A, our "TAG" only bets once every 20 spins. The other spins he doesn't bet. (This is like folding.)
Person B, our "LAG" bets every 3rd spin. He only "folds" some of the time.

Which of these two will reach long term expectation in the shortest number of spins?

Knowability and existence are way different things. The true winrate always exists even though it is essentially always unknowable.

There is no bright line of meaningfulness. The closer we get to infinity, the more meaningful the results become. We don't need to be anywhere near infinity to see this effect. We have seen plenty of meaningful results itt.

10 pages going back and forth arguing that a guy traveling 60 mph (LAG) will arrive at his destination faster than a guy going 30 mph (TAG).

completely separate issue from the fact that one is driving in a straight line and the other is driving all over the road like a maniac while blindfolded.