Since I'm sitting on this mathematical problem for a while I decided to ask the community around here (I'm usually more of a "reader").
In this situation I'm trying to predict Villain's VPIP. This is an example and I will leave all other factors that can help to get an idea of Villain's VPIP (such as stack-size, position, stakes, etc.) out of the question. This is a purely theoretical situation.
Now, it could be that I'm missing something but I consider the following two facts to be true:
- If I have 0 hands on Villain I will look entirely to the population to estimate Villain's VPIP. If I know that most people have a VPIP around, let's say, 25%, I consider it most likely that Villain's VPIP is also around 25% (or around the population's average) and will play accordingly.
- If I have 1000 hands on Villain I will look entirely to the data I have on this particular Villain. So, if this statistic says Villain's VPIP is 45%. I don't care about the population's average of 25% and will play as if Villain is playing 45% of his hands.
Now, since there are situations between these two extremes, I want to know the math.
For example: if I have four hands on Villain, and Villain's VPIP shows up as 100%. I know that it's more likely that Villain just -by chance- got four playable hands in a row than that Villain really plays 100% of his hands. But, I think (but I could be totally wrong) that this small sample of only four hands says tells me -in combination with my knowledge about the population- at least a bit more about Villain. Intuitively I would estimate Villain has a higher chance of being a 40-VPIP kind of player than I would estimate before those four hands.
But, how to do the math? How to weigh a big and accurate sample over a population and a small (but bigger-getting) sample on a Villain together to form my most accurate prediction about Villain?
Visualizing what I mean in a table (numbers are completely made up, but should give an idea of what I mean):
To not regard a Villain's VPIP as 100 after opening just two hands is for sure a good idea. But what would be a good approach? Or is there none and do I -statistically- have to regard the chance Villain plays the next hand as 100%?
One approach I took was to calculate the probability of a particular event (let's say, opening two hands in a row) to happen with every possible VPIP-statistic. So, for example. If Villain is to actually have a VPIP of 25%, but plays the first two hands I play with him/her, the chance of this occuring is ~6%. If Villain was to open two hands out of three hands, the chance if this occuring would be ~14%. Now, after calculating this for every possible VPIP stat (rounded to 0 decimals) I could make a weighted sum and estimate Villain's VPIP more accurately. However, this doesn't take into account the fact that there are simply
more people with a VPIP of 25% so this makes it (at least intuitively)
extra likely.
The following image explains my approach a bit more:
The image consists of five tables with each two columns. The situation is that Villain opened 40% of the hands but each table represents a different sample size. From left to right: 10-25-50-100-250.
The table should be read as for example: if Villain's
real VPIP is 39%, then there is a 25,03% chance this player plays 4 hands of the next 10 hands in a row.
Thanks a lot!
Last edited by wtbhhz; 09-11-2017 at 10:11 AM.