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The FIBS rating formula : why sqrt(n) ? Because it works!

Posted By: Rich Munitz
Date: Thursday, 15 September 2011, at 3:42 a.m.

In Response To: The FIBS rating formula : why sqrt(n) ? (Fabrice Liardet)

I've been rather busy lately, but this one caught my eye and looked interesting. While I cannot justify the formula, I was able to convince myself that it can serve as a reasonable approximation for a predictor of winning chances based on skill difference.

First of all, note that the formula not only makes use of SQRT(match length), but also involves decay of the underdog's winning chances in an inverse exponential of SQRT(L). It doesn't actually matter that the power used is 10 since any base can be used with a suitable constant multiplier. But it seems just as reasonable to question the inverse exponential as questioning the SQRT in the formula.

What I did as a test was to see if I could model skill-based match winning chances over a range of match lengths without the Elo formula and then compare my model to what the formula produces and see if I could get the two different methods to line up. The model I chose was one in which each game is worth exactly 1 point and the first player to reach the match length wins. This makes winning chances of the underdog in a match of length L computable via the cumulative binomial distribution function, given a chosen probability of the stronger player winning (i.e. the favorite wins L-1 or fewer games in L*2-1 trials). I arbitrarily chose a 100 Elo difference because it produces an easy 100/2000 = .05 factor in the formula. I then played with the single game win% value to see how close I could get the match win% via binomial distribution to the Elo prediction. I looked at all match lengths between 3 and 29 points (even and odd). I found that the average difference between the two methods over this range was minimized with a single game win% value of 52.555%. At these settings, over this range of match lengths, the two formulas differed by an average of only 0.06% mwc. Pretty darn close.

What this tells me is that with properly chosen constants (no small challenge, and no claims that the chosen constants are good ones), the FIBS Elo formula can serve as a rather good predictor of match winning chances in backgammon over the range of common match lengths.

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