Author: Uri Blass

Date: 04:04:45 12/24/02

Go up one level in this thread

On December 23, 2002 at 12:14:57, Peter Fendrich wrote: >On December 23, 2002 at 09:10:59, Rémi Coulom wrote: > >>On December 23, 2002 at 06:31:57, Peter Fendrich wrote: >> >>> >>>Remi, >>>I don't get the same results as you: >>>(all the sample values of Pw, Pd, Trin and cumTrin are the same) >>> Mine Yours >>>4 6 0 : 0.2745363 0.2745533 >>>4 6 1 : 0.274543 0.2747703 >>>4 6 10: 0.2740947 0.2773552 >>> >>>I'm not sure what's happening here but as you say, the Monte Carlo method >>>doesn't give exact values. >>>I thought that the program could be reliable to 3 decimals but maybe not... >>>However, if I'm right about draws, the prob would slowly move towards 0.5 when >>>the number of draws increases. I continued up to 500 draws and got the >>>following: >>>4 6 100: 0.2758096 >>>4 6 200: 0.2820387 >>>4 6 300: 0.2906555 >>>4 6 400: 0.3076096 >>>4 6 500: 0.3273436 >> >>That is because the Monte Carlo method is inaccurate: think about the x^n >>function, x varying between 0 and 1: when n grows large, the function has an >>extremely thin peak, that is very difficult to integrate accurately with a Monte >>Carlo method. >> >>I am totally certain about this, because I first started by implementing a >>program that calculated the big trinomial integrals. I wrote a small polynom >>library based on the GNU multiprecision library so that the _exact_ value was >>found. That's how I noticed that the result does not depend on draws, and went >>further in the calculations. > >Yes, that's quite possible. What's funny is that I repeatably get these kind of >results for other combinations of win/lose. I have a sligth hunch about what >might happen. Let's come back to that later. > >>> >>>The probability for A neatly grows with more draws. >>>OTOH I can't argue against your formulas. Give me some more time. >>>Could you please elaborate the first formula in section "3 Draws do nout Count". >>>How do explain 1 - p0 - p0.5 = 1 - u + up0.5 - p0.5? >>>I'm a bit interested in the term up0.5 that seems to be superfluous. >> >>u is defined by p0 = u * (1 - p0.5). If you replace p0 by u(1 - p0.5) in >>1 - p0 - p0.5, the you get 1 - u + up0.5 - p0.5 > >Ok, u is a variable and nothing else... >Then it's prefectly clear that your math is allright. The connection to the >rating system is not as close as I, from the beginning, took for granted. It's >not the probability that is changing it is the interval describing the bell >curve, surronding 0.5 that is shrinking when the number of draws increases. I >had to get a short discussion with another person in order to get really >convinced about this strange fact. >That gave rise to a new question. >Maybe he will post something himself otherwise I will come back with an better >explanantion of what's folling ... > >The new question was about the problem formulation. The issue is not right >formulated from the beginning ("is A *better* than B" given a certain match >result). Draws contains some information and when we have a question formulated >in a way that makes draws not to be included, we will not get full quality of >the conclusion. The area of interpreting match results is interesting and not as >evident one might think. >I'll return later on... If you want to give an interval for the difference in rating that you want to be sure in 95% that the real difference in rating is in the interval then the number of draws is relevant. Uri

- Re: Proving something is better
**Peter Fendrich***05:59:19 12/27/02*

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