[Delphin]

I have been thinking about this a lot lately. One problem is that every player really has at least 10 different VPIP's for a full ring game (one for each position). There aren't just 1326 possible events. There are at least 1326*10 possible events (receiving each possible hand in each possible position).

And even this doesn't factor in the action prior to the player's turn. Obviously my VPIP on the button is much different if there there is a raise, reraise, 3 bet, and cap, than if it is folded to me. The space of all possible preflop situations is quite large.

When you have a 100 hand sample, you only have 10 samples in each position at best. If the player was running hotter or colder in one or two positions than normal (which is overwhelmingly likely), then you have a large potential for over/underestimating the player's true VPIP. If there is a maniac at the table raising every hand, or if the player is seated to the left of several LPP players this will have a large effect on the observed VPIP.

There is a general consensus that VPIP converges very quickly. I think that I believe this relative to other stats (BB/100 in particular). I think that an estimate obtained over 50/100/200 hands aren't nearly as reliable as most people think.

I would love to see a detailed mathematical treatment of this problem. I can easily do the calculation for one position assuming that other players previous action has no effect. Once I try combining multiple positions or accounting for other players actions I get lost.