VP$IP Convergence/Sample Size

[fimbulwinter]

We've all heard from the winrate sample size nazi's, but what about VP$IP? It must converge much more quickly than any other stat, but what is an accurate sample size for getting say within 5% of true VP$IP.

I'd guess 500-1K hands. you?

fim

[Pog0]

just like winrate, this has got to be game selection specific.

[PinkSteel]

I might be missing something about your question, but VP$IP is fairly easy to calculate analytically. I have a spreadsheet that does it for me.

You can calculate/look up the probability of being dealt each opening hand.
You'll play each differently depending on your position relative to button.
Some hands, in a given position, you may play only a percentage of the time -- raise or fold -- so you'd have to declare what that percentage is.

Pretty easy calculation to roll up -- I calculate my long-term VP$IP at about 16% (full ring).

I haven't gone this far, but one could certainly tweak the numbers based on game selection; e.g., LAG to your left, table full of rocks, etc.

[Ghazban]

[ QUOTE ]
I might be missing something about your question, but VP$IP is fairly easy to calculate analytically. I have a spreadsheet that does it for me.

You can calculate/look up the probability of being dealt each opening hand.
You'll play each differently depending on your position relative to button.
Some hands, in a given position, you may play only a percentage of the time -- raise or fold -- so you'd have to declare what that percentage is.

Pretty easy calculation to roll up -- I calculate my long-term VP$IP at about 16% (full ring).

I haven't gone this far, but one could certainly tweak the numbers based on game selection; e.g., LAG to your left, table full of rocks, etc.

[/ QUOTE ]

This sort of calculation shouldn't be accurate. As hands are playable or not based on position, they are also playable or not based on other players at the table. For example, in limit games with a lot of limpers and little preflop raising, you can limp more hands early because of the lowered chance of a raise behind you. In no-limit, if a deepstacked player who can't lay down aces postflop makes a small raise, calling with any two should be profitable (assuming you also have a deep stack) because of the immense implied odds you are being laid.

Anyway, to get back to the OP's question, I have no idea at what rate VPIP converges. I vaguely remember seeing a thread that dealt with that issue in the small stakes (limit) forum a few months ago so maybe you can find it. The probability forum might get you better answers, too, as there's a lot of pure math and statistical discussion there (not just probability).

[PinkSteel]

I agree that the analytical calculation isn't "accurate". We're dealing with an infinity of variables as soon as we introduce any measure of opponent behavior into the mix.

But if the exercise is to get to a reasonable walking-around number, I'm arguing that you can probably get to a very solid number with a quick spreadsheet. I certainly believe that my VP$IP estimate is +/-5%.

I'll also argue that pursuing the same number with historical data could take a lot of data and easily lead to a worse prediction.

After all, if you want to get to a VP$IP that factors in specific situations such as you suggested -- playing any two against someone you know can't lay down Aces -- then (1) there will be a great many such specific situations, and (2) many of them will be rare events.

That's a very reasonable way for an advanced player to view VP$IP -- such a player's view of circumstances is infinitely varied, unlike the beginner who only sees a few dimensions to the game.

But arriving at a stable long term average rate for a many-dimensional probability distribution, consisting of a large number of completely nonlinear, relatively rare events will necessarily take many, many iterations.

I would think for VP$IP, the way you're suggesting the level of nuance to the analysis, you would have to be well into 7 figures of trials. At any rate, as soon as you talk about very large numbers of hands, you run into the practical matter of the player refining his game, changing stakes and thus opponent profiles, and such other fundamental shifts that will render all prior hands moot, since they were played in a different environment.

I think you can get a very good number very quickly by crunching a few numbers. With a huge hand history, after a great deal of time and analysis, I think you could very likely get yourself to a less reliable number.

[PairTheBoard]

[ QUOTE ]
We've all heard from the winrate sample size nazi's, but what about VP$IP? It must converge much more quickly than any other stat, but what is an accurate sample size for getting say within 5% of true VP$IP.

I'd guess 500-1K hands. you?

fim

[/ QUOTE ]

I think you're right that it should converge a lot faster than win rate. If your data is collected from tables that are more or less "typical" then I think a decent aproximation for the sd can be had by treating it like repeated Bernoulli trials with your unknown vpip being p, the probabilty of success for the trial. If p were known then you would see a standard deviation for the number of hands you enter in N trials to be SQRT(Np(1-p)). Say p=20% for example and N=1000 hands dealt to you. You would expect to put money in preflop 200 times with a sd of about 12.6. Thus +- 2 sd's would be a range from 175-225 or 17.5%-22.5%.

That may not be an acceptable level of accuracy. To get it down to a range of say, 19%-21% you would need to collect data from N large enough so that

2Sqrt(.2*.8*N)/N = .01

or N = 6400 hands dealt to you.

This should give you a decent quick and dirty idea of where you're at with the question. Since p is unknown the statistics are a little different and of course the real situation is not exactly modeled by homogeneous repeated Bernoulli trials. But it would be good enough to satisfy my curiosity and as you can see this is far fewer hands than you would need to get this accurate a measure of your long term win rate - as you guessed.

PairTheBoard

[TomBrooks]

[ QUOTE ]
what is an accurate sample size for getting say within 5% of true VP$IP. I'd guess 500-1K hands. you?

[/ QUOTE ]
I'd guess 1k hands would have you within 10% of your long term VPIP 90% of the time. 5% error is only 1 VPIP point if your VPIP is 20. I think it might take more than 1k hands to get that accurate. Just a seat of the pants guess.

[mosta]

(aren't you the guy who condescendingly tells people to do simple math themselves? irony??)