Hello guys,

Thanks for the interest, I am definitely looking forward to any of your comments

So the first point of my research is what really interests me as a SNG player – How much can my current ROI be different from the expected ROI I would get by playing a very very big amount (theoretically infinite) of SNGs ..

In other words, say I have a x% total ROI over n SNGs, what variance can I expect ? This is a question I would like to explore and try to give some reliable answers to. So in the beginning a bit of math about ROI (the concept of ROI in SNGs is basically well known and described for example here :

http://www.investopedia.com/terms/r/returnoninvestment.asp) .. For our needs we get :

which can be rewritten in this way

where:

*BI* - buy in

*r* - rake in absolute numbers (dollars)

*%r* - rake relative to buy in ( %r = r / BI )

*paid* - number of places paid

*wi* - payout for i-th place in buyins

*pi* - probability of finish in i-th place

This formula may look a little scary, so here comes an example to make it easy : Lets take the PokerStars $3+0.4$ SNG with 10 entrants and payout structure 50%-30%-20%, and say our player wins 13% of tourneys, finishes 2nd in 10% and 3rd in 15% of tournies:

*BI* =3

*r* = 0.4

*%r* = 0.1333 (or 13.33%)

*paid* = 3

*w1 = 5 ; w2 = 3 ; w3 = 2 *

- p1=0.13, p2=0.1, p3=0.15

This leaves us with a 10.2% ROI:

Now lets look at the formula in a closer way :

- In the second line 3.866p1 means that a player will win 3.8666 net buyin with probabilty p1 (meaning if he wins the tourney)

- The last part on line 2 means he will lose 1.1333 buyins with probability 1-p1-p2-p3, meaning he will not finish in the money

It is also notable that

It is very important to realise that the ROI calculated here uses probabilities of winning and is what we may call Expected ROI. We may state that if our player played infinite amout of SNGs he would have 10.29% ROI under the assumptions of winning with probability p1, finishing 2nd with probability p2 and 3rd with probability p3.

This much for the post, I will continue tomorrow with some practical applications and assumptions