12. Cumulative Averaging and the Differential

 

  


The Elo System and its revised version postulate different uses of the derivative in arriving at the differentials known as established ratings.  In the former the derivative helps to translate percentage expectancy differences into rating differences; in the latter it provides a shortcut for cumulative averaging.  In this section we shall look at the derivative in relation to cumulative averaging, beginning with the simpler case of interval ratings.  A game-by-game rating process will be used, both for its simplicity and because averaging the competition in a ratio system is problematic.

Cumulative averaging for a percentage score is illustrated by

[12.1]         DP  =  (S - Pe) / (N  + 1) ,

where S represents the new game score.  The process is recursive because the variables Pe and N will be used in the next step with different values.  The subscript for the original percentage score indicates expectancy in some contexts.  The advantage over ordinary arithmetic averaging is that we do not have to write out the entire result at each step, but we do have to keep track of N separately.  A similar process can be applied to linear ratings.  If the new rating is

[12.2]         Rn   =  ERc + K(2P - 1) ,

then using the original rating and the new mean opposition rating, we can solve for Pe in

[12.3]        Ro  =  ERc + K(2Pe - 1) .

By subtraction

[12.4]         DR  =  2K(P - Pe) ,

and substituting for the change in percentage by [12.1]

[12.5]         DR  =  2K(S - Pe) / (N + 1) .

We get the same result by applying the constant derivative of the basic formula, 2K, to the change in percentage score [12.1].  For linear ratings, then, the cumulative average form of the established rating is the same as its differential form.

The differential form of the established rating for ratio systems is easily determined.  The derivative of the basic ratio formula [5.1] with respect to P is 

[12.6]        R'  =  ERc / (1 - P)2 .

Applying this to the change in percentage score [12.1] gives

[12.7]        D R  =  [ERc (S - Pe)] / [(1 - P)2 . (N + 1)]

as the differential.  The analogous formula for cumulative averaging is somewhat more complicated.  Given a new ratio rating

[12.8]        Rn  =  ERc [P / (1 - P)]

we can calculate expected percentage scores from the old rating and the new mean opposition rating as

[12.9]        Ro  =  ERc [Pe / (1 - Pe)] .

The rating change follows as

[12.10]        D R  =  ERc (P - Pe) / [(1 - P) . (1 - Pe)] .

Substituting for the percentage change in the numerator by [12.1],

[12.11]       D R  =  ERc (S - Pe) / [(1 - P) . (1 - Pe) . (N + 1)] .

This is the change formula for cumulative averaging in a ratio system.  We assume, for the sake of comparison with the differential, that the mean opposition rating does not change.  Since the percentage score is close to its expected value for a large sample, especially when rating by single games, we conclude that the differential provides a reasonable approximation to cumulative averaging in a ratio system.