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

Posted By: Timothy Chow
Date: Thursday, 15 September 2011, at 2:37 p.m.

In Response To: The FIBS rating formula : why sqrt(n) ? Because it works! (Rich Munitz)

Interesting experiment, Rich. However, what it suggests to me is that it might not matter much whether one uses sqrt(n) or n or some other power of n, provided one jiggles the numerical fudge factors appropriately and restricts oneself to a small range of match lengths.

If one really wanted to test whether sqrt(n) is better than (say) n, then one would need to compare both formulas against your experimental results, to see which one gives the better fit. Given your model, the mathematics of large deviations predicts that "n" is more accurate than "sqrt(n)," at least for large n. (But I'm not sure if the difference would be noticeable for the small values of n that arise in practice.)

The main limitation of your experiment may be the elimination of the cube from your model. Others have already pointed out that n = 1 and n = 2 are basically equivalent in real backgammon, but obviously only because there's a cube in play. Trying to model the effect of the cube, though, looks pretty tricky.

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