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BGonline.org Forums
The FIBS rating formula : why sqrt(n) ?
Posted By: Kevin Whyte In Response To: The FIBS rating formula : why sqrt(n) ? (Michael Petch)
Date: Wednesday, 14 September 2011, at 12:39 a.m.
That's a more difficult question. If the goal of adjusting the ratings is a "maximum likelihood" fit (so that your rating makes you results as plausible as possible) then one can cook up something. The change from a match makes your previous results less likely (because your rating was where it was to make them as likely as possible) but makes the new result more likely. How big a change makes sense depends on how those balance.
As an example. Suppose two players A and B both play Stick, A playing a 1-point match and B a 100 point match. Both have, based on their ratings, a 15% chance of winning their match (so B is much higher rated than A). If they both lose, how much should their ratings change? The numbers depend on the ratings model in use, but for most models the chances against a fixed stronger opponent in a single game decay exponentially with ratings difference. Let's ignore the cube, gammons, backgammons, etc. and pretend the match to 100 is just a sequence of 1-point matches.
Player A's rating says, by assumption, that he beats Stick in a 1-point match 15% of the time.
Player B's we've said predicts he beats Stick 15% in a match to 100. That means, approximately, that he should win any single 1-point match 47% of the time.
A fixed rating decrease, equivalent to winning 90% as often as before, decreases A's chances in a 1-point match to 13.5% and B's in a 1-point match to 42.3%. That makes B's chances in a 100 point match only 3% or so. Thus a change in B's rating goes much further in making the loss look likely than it does for A. That would say to maximize likelihood thet we should change B's rating by more. That fits with the intuition that a longer match should be given more weight.
How much more weight? Well, the simplest model says the changes should be proportional to the rate at which the change in rating changes the match winning chances - and that is, by the same random walk calculation, approximately proportional to sqrt(n).
There probably isn't any need to use such models though - there must be enough data from FIBS and various bots to get a good read on how match winning chances change with match length.
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