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Here's an interesting side excursion on the Elo rating system.
Say our players have ratings A, B and the probability of A beating B follows Beta(A,B). Then we observe A beating B.
We do a Bayesian update, with posterior Beta(A+1,B).
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Say our players have ratings A, B and the probability of A beating B follows Beta(A,B). Then we observe A beating B.
We do a Bayesian update, with posterior Beta(A+1,B).
1/2
What does this say about the new ratings of A and B, say A' and B'?
The expectation of Beta(A,B) = A/(A+B), so we want
A'/(A'+B') = (A+1)/(A+B+1)
subject to A'+B' = A+B.
Thus A' = (A+1)*(1-1/(A+B+1)) = A+1-(A+1)/(A+B+1).
So d = A'-A = 1-(A+1)/(A+B+1).
But...
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The expectation of Beta(A,B) = A/(A+B), so we want
A'/(A'+B') = (A+1)/(A+B+1)
subject to A'+B' = A+B.
Thus A' = (A+1)*(1-1/(A+B+1)) = A+1-(A+1)/(A+B+1).
So d = A'-A = 1-(A+1)/(A+B+1).
But...
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d = A'-A = 1-(A+1)/(A+B+1) ~ 1 - A/(A+B),
which is (observed outcome) - (expectation of observed outcome).
That's Elo's updating rule up to the scaling factor.
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which is (observed outcome) - (expectation of observed outcome).
That's Elo's updating rule up to the scaling factor.
3/3
Note I'm requiring A'+B' = A+B, because we want any points gained by player A to be taken away from player B.
