Vlad Dumitrescu wrote:
On Jan 6, 2008 11:00 PM, Don Dailey <[EMAIL PROTECTED]> wrote:
The idea of a non one dimension rating model is interesting. If you
decide to pursue this I can give you the CGOS data in a compact format,
1 line per result.
Hi all,
I'm not sure I get the whole picture regarding multi-dimensional
ratings. How can you compare two players with a 2-dimensional rating?
You can't, so how would one use this rating? In my book, a rating's
goal is to make things comparable...
best regards,
Vlad
The idea is that players would not be ranked on a linear scale, but we
would have a formula to estimate the probability of winning between any
pair of players.
For instance, if player A has rating (A1, A2, A3) and player B has
rating (B1, B2, B3)
Delta = ((A1-B1)^3 + (A2-B2)^3 + (A3-B3)^3) / ((A1-B1)² + (A2-B2)² +
(A3-B3)²)
P(A beats B) = 1 / (1 + exp(-Delta))
if A1 = A2 = A3 and B1 = B2 = B3, this is the usual Bradley-Terry model.
But with 3 dimensions, it is possible to get a cycle for instance with:
A=(1, -1, 0)
B=(-1, 0, 1)
C=(0, 1, -1)
With these ratings and the formula above, P(A beats B) > 0.5, P(B beats
C) > 0.5, and P(C beats A) > 0.5.
It is exactly the same principle as the basic Bradley-Terry model. The
very big difficulty is finding the maximum-a-posteriori of the ratings
from the observation of game results. There is no easy optimization
algorithm like for the one-dimensional model. The probability
distribution has many local optima, so it is tricky.
Rémi
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