Okay.

What theory tells us we should measure N by cards seen? Who goes around measuring variance and win-rate by card? I have literally never seen this before.

You're not arguing about anything that can be measured, or at least I'm not sure how you think it should be measured. I feel like you're just ignoring well-established math now.

Yes, I would like something more concrete, because my own mathematical models show that Lags should have greater variance and lower confidence in their win-rate than Tags. It may not be true in practice due to flaws in the models, which would be very interesting.

I'm arguing based on these models, but I'm not necessarily arguing that the models are accurate representations of the real world.

If a TAG can successfully increase the EV of fringe hands that are -EV or EV neutral, is it theoretically possible to lower the variance of the TAG by adding those hands?

If that TAG is able to do so with enough hands given his improved read and poker knowledge, and cross into the threshold of LAG, isn't it theoretically possible for the LAG to have lower variance than the TAG?

This is exactly what Ive done over the past 1800-2000 hours. Ive slowly but surely added in more and more starting hands into my raising range and from earlier positions. My original goal was to try to break even on those fringe hands which would get me more overall action on my premiums.

As I got better post flop (and you will be forced to get better if you play more hands and put yourself in more marginal situations), these fringe hands weren't just break even for me. They became profitable hands which increased my win rate dramatically but didnt raise my overall variance (measured by StnDev and also by just looking at session by session results in real time).

Then I started adding in a strategy 3 betting lite with hands like AJ / J9s / A4s / 87s / KT ect. Again, my goal was to break even on those hands by 3 betting people I knew folded a lot to 3 bets in general or when they have sizing tells that were pretty obvious. This "strategy" has also proven to be +EV overall including profit from taking it down preflop as well as any profit/loss coming post flop....and again, it gives me more action on my premiums when I 3 bet.

Ive done a few more similar things but all together they led to me passing into LAG zone. I win a lot more money without showdown now. I play a lot more small-medium pots. I win or lose $25-$150 a lot more times per day than I used to and overall, I'm winning more of these pots (and for more total money) than I'm losing. That leads to more intra-day variance in my stack size but its like the stock market. My stack bounces up and down in small amounts throughout the day but most often trends up as I win more than I lose in these small to medium pots.

Result is higher win rate and lower or maybe equal variance. It has absolutely led to less and smaller down swings which I'm sure is a result of a higher win rate.

So there's my non mathematical anecdotal evidence that variance for a LAG is lower or at least lower if you wanted to do some sort of Win Rate / Variance calculation.

This, this, a thousand times this!

The models do show this, you're right. And I acknowledge it's entirely unconventional to measure N in this way.

I am contending that poker is unique in this way. In most applications, noise is much more intangible. It makes sense to assume noise is evenly dispersed. In poker, we know all of the noise is caused by cards. So, the more cards seen, the more EV evens out(the EV does not change, just becomes stable more quickly). It's the same reason players choose to run it twice. In poker, some sets of N are richer than others, and so, converge to the mean faster.

Alright, guys, we're done here. Take this in-depth variance discussion to Poker Theory, please. The discussion has been very interesting, and y'all have handled it well, but it's eating the thread alive, and the emperics are what matter to 99.4% of the players ITF. For those of you really interested, I'm sure Poker Theory would be thrilled to host a continuation of this discussion,

If you'd like, I'll even carve out the last few days worth into a separate thread and move it there, though that will have to wait until the weekend is over if you'd like me to do it.

I'm not sure if this is better suited for Poker Theory or Probability. I think it is best suited for here, but if you disagree I'd prefer it to continue somewhere than to not continue at all. If you do move it, I hope you continue to read and post if you have something to contribute, because I believe you also have misconceptions about variance, and the topic is bound to come up countless times in the future and cause more arguments.

Empirics matter to me, too. However, not a single person has posted any empirical evidence on the topic at all. I have asked several times for you to support your argument with evidence, yet you have not come up with any.

I'm curious how you expect to handle this in the future. If someone comes in and asks questions related to this topic he's going to get barraged with conflicting answers, because neither side has convinced the other, and neither side should be content to let the other spread misinformation.

I feel like we resolved it. I was wrong about SEM actually being lower according to the calculation itself. I assumed this had to be true based on the empirical evidence I too have seen, but don't know how to conjure up. But, even assuming greater spread in results, which may be incorrect, I still contend that SEM is smaller in practice due to N being inflated by seeing more cards. This is because all of the noise in poker is a function of seeing cards. And again, this is the whole reason to run it twice and whatnot. But we can't measure N that way, and so we don't see that in the math.

I've learned a lot, we may still disagree on my conclusion since I can't show it empirically, but I think this is a good stopping point and we understand each other's points now. So feel free to continue, but I can't say I'll participate. I definitely won't here if mods don't want it.

Regardless of what the mods think, as a longtime poster on this forum i really enjoyed reading this in depth discussion. Containing lots of good reflections/arguments from both sides. Solid stuff guys.

Garick, I assume you will be moving this discussion to its own thread, and I do hope that if you do, that it will stay in LLSNL.

I think there is a misconception in the usage of noise, almost implying that any result that H loses is considered noise.

That's not noise.

Running a hand more than once is almost the antithesis of getting rid of noise. You are hoping to create more noises.

A player with 10% equity is hoping that running the hand 3 times will result in him winning at least one of them: 1/3 or 33%. In other words, he's hoping to create the "noise" that allows him to improve his result in short term variance.

Thanks for that. Sometimes I think my posts aren’t appreciated in this subforum, I get salty and it leaks into my posts as arrogant/condescending or otherwise negative.

This is just wrong. I'm sorry, but I don't know how else to put it.

Noise is unexplained variability in winrates, or whatever we may be measuring. In poker, the unexplained variability is a result of the cards dealt.

The best way to work out unwanted noise is to increase N. That is what happens when you run it multiple times. Rather than being all in with AK v QQ once, you're all in with it twice. And everyone gets twice as close to the long run.

I've found your posts to be insightful and valuable.

No.

https://forumserver.twoplustwo.com/1...twice-1667202/

Again, I will stress that the goal to run it twice IS to create noise, not to reduce it.

You're not telling me why, first of all. Secondly, post #4 in that thread says what I'm saying. I don't understand how the thread speaks to there being more noise whatsoever. I can only conclude you misunderstand what is meant by noise.

Please don't refer to multiple runouts as noisy. It makes things messier, more complex, etc. But it lessens noise in the statistical sense by causing outcomes that are more in line with the equities more often.

First of all, what does noise mean to you in this context?

Applying the concept of noise in a situation where Hero has 10% equity is to increase its likelihood of winning in “short term,” aka variance. Assume H is to run the river 10x with 10% equity and wins the first river card, any subsequent river card resulting in H winning can be considered noise because it distorted H’s actual EV.

So when you ask to run it twice, you are not actually hoping to realize your equity, you are hoping to run above it. I thought that link explained pretty well that running it twice doesn’t change your actual EV.

Running it twice is actually a pretty bad example.

However, yes, if you can run it a million times and have 51% equity, it would certainly benefit you.

On the flip side, 49% might prefer to live in the noise and run it far fewer times.

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!

Here's an example of that, and it's self-perpetuating. When you post stuff like this it makes people less likely to value your posts and more likely to get mad.

This is simply not true. I did post evidence of it, and you responded to that evidence. In fact, you agreed with the results but said: This is the empirics that matters to most posters. You guys enjoy your technical math/semantics discussion about what the definition of variance is, but the important info for most players is understanding what styles are swingier. That's also what most players mean when they talk about variance, whether that is mathematical definition or not.

Let's just establish terminology before moving forward if we hope to have a fruitful discussion.

Statistical noise:

In other words, if we look at a large sample plotted out using scatter chart, data that are outliers and unexplainable can be considered statistical noises.

In this context of running it multiple times, in which I referenced in this discussion:

If running it twice has no effect in your EV and the valid sample size of a hand scenario is most certainly in the thousands, what exactly is the purpose of running it more than once?

For someone with a significantly worse hand? It is the hope to have much better result than actual EV. So the worse hand that you have, the more likely that any positive result is an outlier in a large sample. It is not hard to see that you are indeed hoping to create noise - not in a sense that you are actually creating it, but the result is the noise.

Are we sticking with LAG is less swingier than TAG and NIT?

Like I said, I am fine with that belief for personal gain, but there is a ton of caveats to go with that.

Good read. The things I miss not following that thread...

I do have a stats background. browni's mathematical analysis is correct and generalizable; you cannot decrease variance by substituting a certain outcome (e.g. folding) with a distribution over multiple outcomes (calling or raising). It seems like taking tons of small gambles should largely cancel out and result in lower overall variance, but that's simply not how it works. Total variance is the sum of the variances of each individual gamble; you can't reduce it by adding more gambles. (Running it twice decreases variance because you're not adding a second gamble, you're splitting a gamble for X into two gambles for 0.5X.)

That said, preflop is not the whole story, as a few people have pointed out. There can be other aspects of a good LAG's game that decrease variance and explain the empirical results. Betting/raising to steal a pot is a gamble, but so is checking, so is barreling an overpair on a wet board, etc. Estimating the effects of flop & turn decisions on variance is extremely complicated. The decision tree is huge, with many outcomes whose probabilities are subjective and Villain-dependent.

Can you elaborate this further?

Follow-up question: is running it twice beneficial to higher equity or lower equity?

Let us please not turn this in to a benefits of RIT discussion. RIT has no effect on EV, and is therefore not more advantageous to either the higher or the lower equity. This has been well proven a zillion times.

What RIT does have an effect on is variance, so we can discuss its effect there. I'm not sure how to elaborate on Jay S's point, though. It seems pretty clear. Because you have two smaller gambles, rather than one larger gamble, you will average closer to your expected EV.

Here's a toy example: You are AI for $100 with 50% equity. Your EV is $50, but you will never end up with $50. You will always end up with $100 or zero.

Now you RIT. Half the time you split and end up with exactly your EV of $50. Only a quarter of the time do you end up with $100 and only a quarter of the time with nothing.

Your EV hasn't changed at all, but your observed results are usually closer to it than they would have been if you just ran it once.

Actually it does in the sense for someone with significantly lower EV. Basically you get an extra chance to "bink" it.

But you know that if the scenario runs out 1 million times, there is no such thing as binking it.

It's basically a coin flip scenario. We had already talked about this. If we do not know the EV of a scenario, data sample of running it 100x could be as misleading as running it 10x.

I also do not understand how RIT lowers variance, hence I asked. If my EV remains the same whether it's running it once or twice, it's no different than I am running it once in two different hands with the same equity. Or in the coin flip analogy, I will basically take another coin flip. So how is it lowering my variance?