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Winning probability doesn't depend on match length?!

Posted By: Bob Koca
Date: Monday, 5 June 2023, at 9:20 p.m.

In Response To: Winning probability doesn't depend on match length?! (Timothy Chow)

Here are a few thoughts:

i) The equation from the Larry Kaufmann article is more than a prediction. It is at the heart of the Elo system itself and is used as the predictor of win chance. http://www.fibs.com/ratings.html#explained

ii) One might hear that ratings have a good feature of being self correcting. On average if one is underrated (overrated) the person would win (lose) more than they are predicted to which adjusts the rating in the correct direction. There are several assumptions though. One is that knowing P(A defeats B) and P(B defeats C) allows one to calculate P(A defeats C). For example in a 1 point match if A is .640065 to defeat B and B is .640065 to defeat C then for things to work is should be the case that A defeats C with probability .749747. Those are predicted win chances for a 500 and 1000 point Elo difference. Suppose though that the win chances for A vs B and B vs C actually mean a larger (lower) win chance for A vs C. Players who play matches against those 1000 lower than themselves would be overrated outperforming (underperforming) the predicted win chance.

iii) The factor of square root of N is basically a guess. I think it might have some justification from a weighted coin flipping first to N successes model but that might match with actual backgammon.

iv) It seems that apples and oranges are being compared to some degree. Kaufmann's formula is based on Elo garnered from many matches and is a proxy for one's overall ability. The new formula is based on PR for just an individual match.

v) Chris Yep pointed out to me that the data was from 19000 matches from Terje Peterson. If they are mostly self reported a bias could be introduced. For example a player who plays a 2 might report it but wouldn't report it if a 10 was played. If a 6 was played might report it only the match was won. I don't know if this is important but there are more clean sources that could have been used. For example matches from dailygammon.

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