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win probability for a PR difference and match length

Posted By: Rick Janowski
Date: Friday, 21 September 2012, at 3:58 a.m.

In Response To: win probability for a PR difference and match length (Jason Lee)

Take the example of a 25-point match with an Elo rating difference of 400 - This was set in the Elo system to give the underdog odds of 1 in 10, ie a probability of 1/11 (or 9.091%).

Let us now consider a 200 rating difference in the same length match. The underdog odds will be 1 in 10^0.5 (ie, 1 in 3.162, or a probability of 0.2403). So if A is 200 rating points higher than B and C is 200 points higher than B, then the odds for A beating C are 1 in (3.162 x 3.162) = 1 in 10 clearly.

Lets look at combining a 100 rating difference with a 300 rating difference again in the same length match. The 100 rating difference gives underdog odds of 1 in 10^0.25 (ie, 1 in 1.778, or a probability of 0.3599). The 300 rating difference gives underdog odds of 1 in 10^0.75 (ie, 1 in 5.623, or a probability of 0.151). Combining the odds as before the odds of A beating C are 1 in (5.623 x 1.778) = 1 in 10.

If you have extremeGammon, you can check these probabilities against the rating differences in a 25 point match.

I appreciate that this logistic curve may be somewhat counter-intuitive but I believe I have applied the rational developed by the originators correctly. Morover, as I mentioned in my earlier post, an alternative "more rigorous" approach using standard deviations and the normal distribution will give very similar answers.

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