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11. A Revised Elo System
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We now come to the crux of our argument, which focuses
on the interpretation of the established rating, represented here in
abstract form as the differential [11.1]
D R
= d
D P . As Elo saw it, the established rating was essentially
a small shift along the Percentage Expectancy Curve.
The derivative d is based on the slope of the curve, and the change in rating is
calculated from the curve as the rating difference associated with the
change in percentage score.
The interpretation offered here is quite different.
The change in percentage score arises from cumulative averaging
as a correction of the estimate of long-term percentage score.
The correction is translated by the derivative
d of the performance rating with respect to
P into the rating change. We shall illustrate these different points of view by
deriving a new version of the established formula, and by implication a
new rating system. The new
system is logarithmic, like the Elo System, and uses the same constants.
The formula for percentage expectancy is also the same as the one
derived by Elo, [11.2] Pe = 1 / (1 + 10-D/400) , but not from a probability distribution. To avoid problems with averaging, we shall rate
results one game at a time.
The cumulative average for percentage score is consequently [11.3]
D P = (S – Pe)
/ No . To translate this into a rating change we take the derivative of
the performance formula [5.6] derived in our discussion of ratio
ratings, [11.4]
R' =
400 / [P(1 – P) . ln10] . The new established rating formula is therefore [11.5]
D R
= 400(S – Pe)
/ [Pe(1 – Pe)
. No
. ln10] where Pe is defined in [11.2].
The rating change resulting from a win in a pairing of various
rating differences, positive and negative, is reported in the table
below and compared with the rating change for the Elo System, interval
and ratio. The rating
changes resulting from a loss are obtained by changing
S as well as the sign of D,
giving negative versions of the reported values.
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