Author: Uri Blass
Date: 13:44:46 02/18/01
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On February 18, 2001 at 16:14:34, Peter Fendrich wrote: > >In another message Bruce Moreland raised the question >about how to interpret a match result. >With a match result between two players, he wanted to >express things like: >Player1 is better than Player2 with a certain significance >level. > >The following algorithms will try to give a general method >to deal with and evaluate match results. The Bruce example >will show up in the end of this message. > >This is not found in the standard statistic course and I >haven't seen any description of this so I wanted to give it >a try because it should be rather straight forward. I have >to admit however that it gave me some headache before the >obvious solution appeared. > >My intention is to create a program based on these ideas, >maybe adjusted after your comments, and place that program >here or on any other well known site so we don't have to >struggle all this over and over again whenever some new >results drop in. :) > >If you don't want to read all this "crap" you can instead >get my little program and test it. In the end of this message >there is some more info. >Just send me an email and I'll return the source code and an -exe file. > >Be patient because the program isn't written yet but very soon!!! > >Some basic definitions and formulas >=================================== >The real problem here is when we have small number of games. >When we have a lot of games he normal distribution could be used. >We will use the Trinomial distribution which is the only one >giving the exact probabilities when the outcome has three distinct >outcomes. > >N = number of games >W = number of wins (for Player1) >D = number of draws >L = number of losses (for Player1) >Pw = probability of a win (for Player1) >Pd = probability of a draw >Pl = probability of a loss (for Player1) >hence N = W+D+L and Pw+Pd+Pl = 1.0 > >P = probability of a certain match result with exactly W, L and D >cumP = probability of a certain match result or better (for Player1) > This function is often used but we will avoid it here. >With the help of the Trinomial distribution we can get the P if we >know the values of both Pw and Pd: >P(N,W,D,Pw,Pd) = (N! / W!D!L!) Pw^W * Pd^D * Pl^L where L=N-W-D > >One problem, of course, is that we don't know the real values of >Pw, Pd and Pl. > >The Question to answer >====================== >Given a match result W, D and L between Player1 and Player2. > >H0: Player1 better than Player2 or in other words > Pw+Pd/2 > 0.5 >P(H0) is the probability that H0 is true. (I don't care about rejecting >H0 because it's not necessary) > >Let's start with the assumption that we don't have any prior knowledge >of the distribution of Pw and Pd. That means that they are evenly >distributed within the area formed by Pw and Pd each as an axis between >0 to 1 and where Pw+Pd <= 1. The assumption is not correct because usually we know something. We also know that the probability of white to win is not the same as the probability of black to win. Uri
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