Mod note: Carving this discussion out of the winrates thread, as it was taking over. The participants asked that it stay ITF, rather than moving to Poker Theory. Enjoy!

Variance of win rate is basically variance and there's no way a LAG (even if it's a good LAG) has a lower variance than a nit. The more hands you can play in a +EV way, the higher your win rate will be but also the higher your variance. Sure GG gets his big pairs cracked sometimes. But what about you? You're constantly in there having to push thin value and run big bluffs. How is that lower variance?

Nice graphs btw GG. I don't agree with your style of play but from what I've heard about win rates you're crushing 90% of winning regs at 1/3.

Sure, but you are a lot more likely to go on a bad run if you lose AA 7 times in a row or something like that. Its impossible to recover if you arent playing enough other hands and winning other pots.

You most likely weren't playing too loose or LAGgy in the other sample. You probably just werent playing very well because you dont have enough experience playing those hands that are tougher to play. Playing a big pair is easy, especially if you make a huge reraise preflop. You either take it down preflop or you have no difficult decisions post flop because the SPR is low. But that is 100% not the most profitable way to play those hands.

Raising hands like QJ or 87s are a lot more difficult to learn to play well because you get put in a lot more difficult spots with things like middle pair, but when you figure out how to play those well, you have so many more opportunities to make money. Assuming you play better than your opponents, the more hands you play the more money you make and the less you are at the whim of the cards.

I dont fully understand SD either as it relates to overall variance, but I do think its deceiving because it only accounts for SD based on end of session by session and not results intra-session.

MikeStarr, a nit is more likely to go on a long break even stretch because their winrate is lower, but a good LAG will have variation in their results, even they end up having less breakeven stretches because their higher overall winrate.

I believe the data shows that nittier styles have lower SD than more aggressive styles, and SD represents variance.

Not true. The more hands I play (assuming Im a good LAG and that I play better than my opponents), the more opportunities I have to make money which lowers my variance. Im winning and losing more pots than the nit but Im winning more in the pots that I win than Im losing in the pots that I lose. That means I will have a higher chance to have a winning day than a nit does. My stack will swing up and down more times than a nits stack, so intra-day variance will be higher than a nits, but at the end of the day over a decent length session, my variance will lower on average.

Now, a bad LAG, is going to have massive variance. Playing a good LAG style doesnt necessarily mean pushing thin value and making huge bluffs. I rarely make huge bluffs without a lot of equity.

Pretty sure the opposite has been proven true via online giraffes

Assuming both a good lag and nit are going to win the same general amount with big hands.
A good lag is going to win a lot of small pots, lose a few small pots
A nit is going to win a few small pot and lose many small pot

I mostly agree with this but I also think there's a lot of "variance" in people's definitions of a nit and a LAG. Some people think a LAG needs to be a maniac, constantly making huge bets trying to push people off of hands. To me that's a bad LAG and sure, their results will be all over the place. They bluff off stacks all the time but also get paid off huge when they hit a run of cards.

A good LAG does more bobbing and weaving, being aggressive often but knowing when to back off or give up. IMO, a good LAG plays more small to medium pots and not huge pots all the time.

I agree but I think you're missing much of the picture. While it's true that a LAG can get "unstuck" way easier than a nit because the LAG is involved more, the LAG is gonna be put in spots where his stack is going in way more than the nit and therefore his variance will be more. For example, if you're LAG and villains are calling you down light bet bet shove with top pair is often correct while it would generally be idiotic to do if you're a nit depending on stack sizes. Another example is a nit would back off bluffing the turn and river most of the time because of their risk-adverse style while a LAG will notice good spots to raise the turn, bluff shove the river and embrace the variance that comes with it (and of course a good LAG would be balanced or be doing it to exploit over-folding tendencies in certain villains).

Also in case we're not on the same page variance has nothing to do with win rate.

If its a good LAG, he will pick the correct times to do this and win a ton more than he loses which lowers variance....unless he picks the wrong time over and over...in which case hes probably not a good LAG.

Clearly once in a while even a good LAG will run into big hands quite often and have a bad streak but for the most part a good LAG will win more than enough more medium sized pots to make up for a few big pots he loses even within one day.

I get that, intuitively, it can seem like a tighter style would lead to less variance and thus, a less consistent winrate over shorter time periods. But looser styles lead to more decision points and hands and opportunities to apply our edge. In statistical terms, it leads to more data and more data = less variance, all else being equal.

Looser players will see greater variations in session to session results. But their winrate is easier to estimate than a tighter player's rate over an equal amount of time.

AGs of any sort, tight or loose or somewhere in between, tend to look for spots to raise turn, bluff raise river, or take any other aggressive, +EV actions. Fit or fold, passive players will see more variance as their style is more dependent on connecting with the board, which is pure luck and will vary greatly over small to medium sized samples.

So, I agree that GG's style is subject to more variance. It is likely that this causes some of the variation in his graphs. This isn't meant to discredit his style at all. It may well be the best style for his game conditions, idk. I am just speaking of the style's relationship to variance in winrates.

So just to add, variance is a term borrowed from statistics,

Var(x) = E(X^2)- E(X)^2
Var(x) = std dev(x)^2

I wrote a long explanation of this, but then realized everyone’s response will be to glaze over and say go away math nerd.

Basically the larger your losses and wins the larger your variance.
Who has larger wins and losses? The nit or the lag, obviously the lag, so the lag has larger variance.

I can provide proofs and explain graphically, but does anyone really want to delve into calculus based statistics here? I don’t think so.

I was going to add this as well, variance should be mathematically related to standard deviation.

My 2c is the higher variance player is the one who routinely gets in marginal spots for lots of money. Nits can do it just as often as lags, it depends a lot on the odds you lay people (in GG's case he prides himself on giving dummies really bad odds) or in the lag's case how good at hand reading he is (and thus able to put himself in super +ev spots with thin value bets on the river).

At any rate say what you want about GG but he's a true savage grinder.

Sample variance (variance between observations) increases. The variance of the sample mean (our winrate) decreases though. This is what matters when it comes to measuring our winrate.

If you look at a graph of 2k LAG hours, it will be swingier than the TAG graph of 2k hours. But the winrate on the LAG sample will have a lower standard error.

To sum, you will be closer to achieving your true winrate playing LAG despite the increase in session to session variance.

Ok that make sense. I was thinking variance in the bankroll not variance of the sample mean.

I have not ever seen it more succinctly explained then this. wp

Thank you, I appreciate the kind words.

That's what I said.... in a much dumber way.

I kind of roll my eyes at all the "benefits of LAG" talk.

Once you get to $10/20 and higher there really aren't winning LAGs anymore. Someone playing close to 40/30 is going to get owned by most regs these days. Note: opening 87s in MP is not LAG.

It's not a benefit. It just is what it is. I mean, it's not like you should LAG it up just because of that. Just exploit your games as they should be exploited. That said, I don't think even 2% of this forum has or will play 10/20+ NL. That's like nosebleeds in 2019.

I think your post naturally highlights why tight players experience more long-term variance. At 10/20, you must play a tight game, and so you do. You play less hands. You are more at the mercy of the shuffler and seat-choice-roulette when it comes to getting it in good because you need fresh ground peanutty hands and edges are smaller as the pool is more skilled.

Standard deviation * Standard deviation = Variance

While admittedly subjective, preflop lag stats would certainly be lower than 40/30...especially for fr.

But even more important would be non-showdown winnings. There are players that I would consider somewhat taggy with their preflop hand selection but batsh*t crazy postflop. This is where some of the bigger edges still lie in live poker 2019, imo. And I consider them lag. (I realize loose is in the acronym , but loose can apply to flop/turn calls too)

The best player I have ever played with is a near hyper lag in deep games. He reminds me of durrrr on HSP. As I’ve mentioned in chat many times, the second best played a very unique passive game. So ymmv.

In general though I would agree that lag is way over-idolized on these forums. It’s pretty funny. I remember catching streams of players that talk up a big aggression game and then realizing they were tighter than me.

Unfortunately I don't think he's correct. I'd be happy to be proven wrong, which could happen since I'm a high school dropout who has never taken a stats course.

This is true because samples can be biased. Think about what would happen to our SD if we tried to quit every session at EV. Of course the population SD remains the same, but our estimate of it changes.

Show the graphs.

This doesn't make sense. Every time we fold is a data point, too, so VPIPing more doesn't give us more data points to help our win-rate converge faster. Folding is zero variance at the decision point because the result equals the EV. Holding EV and all other nodes constant, replacing zero variance nodes in a strategy's decision tree (hands a Tag folds) with positive variance nodes (hands a Lag doesn't fold) can't decrease the overall variance of the strategy. I think this can be proven with calculus. If it is important to the discussion I can try to do a proof.

This explanation makes no sense to me at all. Doesn't an increase in sample variance necessarily mean an increase in the variance of the sample mean? The standard error of the mean is estimated by the sample standard deviation divided by the square root of the sample size. Therefore the standard error is directly proportional to the sample standard deviation holding the sample size constant.

Folding is not zero variance. If our overall win rate is (say) 10 big blinds per hundred hands, we should win 0.1 bb/hand on average. That's our mean. If we fold, we win 0 bb. The contribution to our variance from a hand we fold will be (0.1 bb - 0 bb)^2 = 0.01 bb^2

This is why I carefully said that folding is zero variance at the decision point. The expected change in stack by folding is zero from the decision point. The result of folding is also zero. I'm not talking about the variance of the entire strategy there, just for the decision in isolation.

Think of our entire strategy as a decision tree. In reference to the root of the tree, folding is neither zero variance nor zero EV. In reference to the current decision node folding is zero variance and zero EV because we have no knowledge of money previously contributed to the pot up to that point. If we take these locally zero variance nodes and fork them into multiple locally non-zero variance nodes, I believe it can be proven that the variance of the entire decision tree necessarily increases.

I may not know correct terminology because I do not have a formal education in statistics, which may make communication difficult. However, perhaps I can post some math tomorrow which may be more easily communicated.

Folding is a decision too, yes. But if you fold pre, you cap your decisions at 1 for the hand. If you vpip, you may make several more decisions in the hand.

You're right about variance in general. But that's not what I'm saying. I'm arguing that loose players and aggro players have a lower standard error to their winrate. Playing a high variance style actually helps you get to the long run quicker.

https://stats.stackexchange.com/ques...e-measurements