I'll pick some nits.

spadebidder made some valid points, although his use of language may be non-standard. You can't say that the actual results move further and further from the mean; they might just move closer to the mean at some point; we can't predict actual outcomes. However, on average, the distance from the mean grows.

This can be stated more formally. In order not to have to deal with signs or absolute values we look at the square of the distance. The expected squared distance from the mean is what statisticians (and not poker players) call variance. This can be computed for any distribution, not just normal ones. So we can write this as V=E((x-E(x))^2). The standard deviation is simply the square root of the variance. Since the variance is proportionate to the number of hands, the standard deviation is proportionate to the square root of the number of hands. Hence while its absolute effect grows, its effect on the win rate per 100 hands actually diminishes.

The other thing he said makes sense as well. funkij is right when he says that a player is more likely to perform closer to EV (meaning, closer to his expected result). However, the cards have no memory. So if I run 6BI over EV for the first two weeks I can expect to finish the month 6BI over EV. It is just as likely to gain another 3 BI, finishing 9 above, as giving back 3 BI, finishing at 3 above. So I can't expect to move closer to my EV for the month. Nor should I expect to move away from it.

The mean of what?

The mean of the total population, or if you like, the expected result.

Ok, I just wanted to clarify. For some reason, the first time I read about deviating from the mean, I took it to meant the mean of the individual player.

The individual player only has a result. But maybe I'm not really making sense right now. When I talk about "population" I didn't mean "population of players"; I was more referring to all possible outcomes. So the mean is actually the expected value.

Ok, now I'm confused. If you were referring to the population of players, I would expect you would diverge from the mean over time given that you are different from the average player. Better or worse, perhaps even slightly. But over time, your graph would represent that.

But why would you continually diverge from your own EV line? Given that the non-EV elements are luck and luck can make you run both over and under EV. Isn't that like saying a hot player will always run hot and a cold player will always run cold?

No. It just says that the longer you play, the further you will probably be from your expected result in absolute terms. If you play 100 hands you will probably be within a buy-in of your 5bb or whatever you expect to win. If you play a million hands you will probably be dozens of buy-ins off the mark, either above or below EV. Of course you could run closer to EV, but it's not very likely.

Put another way, the distribution of results looks like a bell curve centered around your expected result. When you play more hands the curve gets broader and flatter. So it's more likely to get a result further away.

Cang, the whole discussion becomes easier if we can get folks to agree that the standard way to look at future performance is to define a confidence interval.

Once that is defined, then we can talk about how the confidence interval EV change with sample size.

If folks think about the basic statistics topics discussed above while playing with udevil's poker results calculator (total winning's mode illustrates things best IMO).

NOTE: on this calculator, they show both a WR you have specified (the green line), upper and lower bounds of the confidence interval you have chosen (red parabola) and a single simulated result e.g. the results a single player might experience. Click on the "chart it" button several times and new black lines will be drawn.

Looking back on results, a single player only ever experiences a single black line. Looking into the future we do not know what the results will be but the normal (or binomial) distribution defines the probability of different random walks.

we seemed to have digressed into a basic statistics as applied to poker course. That is only slightly off topic for this thread so no harm done

harping on confidence interval some more ... most folks focus on EV (and WR) to the exclusion of stddev and confidence intervals.

It is possible for two different players to have

1. win rate 2bb/100, stddev 50bb/100
2. win rate 2bb/100, stddev 200bb/100

These are not the same! The first is much better than the second (IMO) because for any confidence interval, player 1 will see his WR dominate his variance much sooner. (Player 2 is Isildur1).

You might even prefer a WR that is 1.8bb/100, stddev 50bb/100 over WR 2bb/100, stddev 200bb/100. I.e. you may fold some slightly +EV spots to reduce your variance so that you experience smaller swings.

Obviously if standard deviation is too big, there is no "long run" for a human player -- you can't play enough hands for your WR to dominate variance. On the udevil poker results calculator, you want the bottom of your confidence interval to rise above the X-axis ($0 winnings) in your lifetime.

Q- is the $EV line really applicable to SnG or purely just cash games?? You could win an all-in by hitting your draw and still not cash in the SnG thus not gaining any actual $'s....or does it using an ICM technique????

At FunkyJ's request, I am posting an exchange we just had in the chat thread regarding the various ways in which i have been running good and bad in my last 120k hands:

This is a super interesting set of questions, and I think the answers are pretty interesting, too.

During most of the 120k hands in which I have run break even, I have been above all in EV for big stretches of it. BUT, what has clearly been going on is that I have been getting set up with an unusually high frequency, and sucking out some. Just as a for instance, I had one 2000 hand session in which I got dealt KK14 times--in 7 of them it folded around preflop, and in the other 7 I got all in preflop against AA all 7 times. It was bizarre.

At the moment, and for the last 30k hands, I have been exactly at all in EV.

I have been running bad in the following ways:

I have gotten set over setted twice as often as I have set over setted someone.

I'm seeing a lot of monotone flops when I flop a set.

I'm hitting the flop with AK about 1 time in 5.

I'm flopping top pair when I 3 bet an ace high hand 3% of the time.

In the 120k hands, when I have a straight and get all in on the flop v. a set, i have a 0% W$SD. When i get a set in against a straight, I have a 0% W$SD. Total 11 losses with sets to a straight all in before river and 3 losses with straights to sets that boated after we got all in).

Here is the worst of it, though, because it is the most dangerous thing to have happen to me: I have started (within 500 hands) 21 of the last 29 sessions stuck at least 2.5 buy ins. This plays to my weakness of getting desperate to get unstuck, and I usually go through a period where I make a few bad plays trying to win some hands. Interestingly, I usually get unstuck; I have won 13 of the 19 sessions where I started out stuck a couple buy-ins. Those are the sessions in which I regain control of myself.

You can't run bad for this long (since early December) and not have it affect your game.

The major leak that has crept into my game is that I am making a huge number of stupid river calls in spots where I am obviously coolered/sucked out on. My river call efficiency has plummeted from a long-term average of 1.9 to 1.16 in the 120k hand sample. I have posted about this emotional play before; it is "jesus christ, they can't have sucked out again," when, of course, that is exactly what happened. It has been the major leak in my game since I started playing.

The vast majority of the dip is just emotional play--tilt.

Another leak that has crept into my game is that my c-bet stat in 3 bet pots has come down dramatically. I am passing on obvious spots because I am playing emotionally and reacting to the fact that I missed the flop AGAIN, rather than saying, "yeah, OK, I have A5 but this K high flop is a good board to c-bet.

The last leak that has crept into my game is that I am making "desperation" bluffs. I'll get sucked out on a few times, get a little tilty, and then make a stupid bluff the only justification for which is "god, i don't want t take another hit on my red line." This leak is especially pronounced from the blinds in hands where I wanted to improve and didn't, and I donk the river.

All in all, I'd say that I should have been winning a bit more than I have. When I said that I was ~break even in the last 120k hands, I was rounding down. I've been winning at about .3ptbb/100. I probably should have been winning at about 1.3ptbb. That's a rough guess, but I suspect it's pretty close.

The only thing I am proud of about the last 120k hands is that between sessions, I have been really clear-eyed about my results. I KNOW I am playing emotionally, and each session I tell myself to not do it, and at the beginning of sessions, I do a good job. But then I get stuck a few stacks, and I start to lose it and make desperation plays trying to get even.

I forgot about this thread, maybe someone here can answer a question for me concerning AIEV.

It seems that everyone runs below AIEV. (I understand that partly this is b/c people who run +AIEV don't post about it.) How does HEM account for the rake in their calculations? As an example, if HEM doesn't properly consider rake the following could happen.

10 times we get it in with 90% equity. We win 9 and lose 1 so we should be AIEV neutral. But if HEM deducts the rake from our winning pots we'll actually be 27bbs below AIEV. Assuming full stacks, 9 times we win 197bbs (200bbs less 3bbs rake), 1 time we lose 100bbs.

I don't know about HEM in particular, but generally, AIEV calculations use the AI pot size after rake is taken out for their AI calculation.

Yes, it would be a silly mistake to compare actual results (reported after rake is accounted for) and AIEV that calculates based on pre-rake pots size.

Yeah, it's selection bias. Since starting over in November I'm running about 13 BI above EV, 100k hands.

Thanks a lot for this explainations

This EV simulator (currently malfunctioning, but worth checking later when it is working again) is awesome for getting an idea how big total poker variance is.

bump to prevent COTW from being archived

Fantastic post, very informative!

I reviewed the first half of this thread. There is still one point I don't get - isn't a big reason why all-in EV as calculated pointless that it doesn't include the profit you make from bluffing?

For example, let's say you're playing heads up against a player and you think it's profitable to four or five bet shove any pair and AK/KQ, because he will only call shoves with AK/JJ+.

So the times he calls, you could compare your money earned to your all-in ev to see how you're running (if you filtered for just those hands that saw a flop and were all in preflop), and which would be a loss (since any hands he calls with are ahead of your shoving range); but overall, your profit line will include the times you fold out better hands than yours or hands that actually had proper equity to call! And so the profit line would run above all-in ev, no, if shoving light is part of your strategy?

Actually I want to show you that the AIEV line is not a horrible measure of your luck. Actually it shows it to some extent but not exactly.

First, there is no thing called luck in poker in a long run. so if your sample is big enough you are getting what you expected to get, otherwise you sample is not big enough.

I'll tell my point using 3 scenarios:

1. You go all-in preflop holding AA vs KK. in this case your AIEV (yellow) line shows your expected amount to win and your green line shows what you actually won. So your yellow line has a realistic picture of the scenario and considers completely the factor of luck.

2. you check shove on the flop on your opponent. Now, if he folds you get all the equity in that pot. your yellow line will rise to the amount you have won in that pot. your green line also rises the same amount. So your AIEV line doesn't show your fold equity here and it is the same as your green line.
However, if your opponent calls your AIEV line is calculating exactly what you are expected to gain (same as scenario 1).

So overally, although your AIEV line doesn't calculate exactly what is happening, it is still a better approximation of your variance and expected value than the green line in this scenario.

3. your opponent chases his flush draw to the river and the result doesn't really matter. in this case the green and yellow line change exactly the same amount. So in this case your AIEV line doesn't contribute at all.

Conclusion:

Your yellow line is more likely to show what is really expected than your green line. So if Variance has effect on your actual result (green line) 100 %, then your yellow line reduces the effect of the variance to some extent. It shows a more realistic view of what is happening. the bigger your sample is it is more likely that these lines converge each other.


If you are running above your AIEV line you can't say certainly that you are running good. But it is more probable that you are running good and vice versa. the bigger the sample the more probable that you are running good (or vice versa)



A note about the original post:

On a long run all the things that the original poster sent are neutral. the amount of AA vs KK hands. the times you've been dealt cold cards. the % of times you're against a nit (actually in this case you can choose who your tables and seats and this has a big effect on your results. you don't want to play with better players than you). In the scenario 3 on a long run the amount of times that your opponent hits his flush is a defined %.


In the long run your green line has to converge your yellow line. on a shorter run however, your yellow line is a better approximation.

interesting. do you mean to say that for a large enough sample:

• ... the graph of my actual winnings (vs hands played) will eventually cross the graph of my expected winnings at least once, or
• ... at the end of the sample, my actual winnings = expected winnings?

in either case, your logic is unassailable.

Perchance you use the martingale system to beat the casinos?

EDIT: I wonder if Arashh is really just Cang trolling me

Yes I think they have to converge with a small error which is negligible compared to the number of hands played. Why not ?

EDIT : actually converge is not a good word. they move in the same direction and if there's some differences at some points it is because of luck factor which the yellow line can assess with some error.

Arashhh is not me, and I disagree with him. Your lines will not converge. The winrate does, though. Note however that "long run" and "convergence" involve the notion of infinity, so it can take arbitrarily long