8. Attenuation

Elo's blending method is a improvement on older systems that relied on performance samples of fixed size.  The Harkness System took a rating average of the last four events [E1, 8.53], which allowed the unfortunate possibility of a winning result leading to a drop in rating.  There are, however, improvements on Elo's improvement to be considered.  Elo's method maintains a constant sampling weight at the expense of the original sample.  Thus, in the blending method previously presented as

[7.4]         R_n  =  [R_o(N_o – N) + RN] / N_o  ,

the sample weight of the original rating,

                 (N_o - N) / N_o ,

clearly depends on the weight of the new sample.  The size of the new sample must never exceed the size of the original.  Somewhat better results obtain if sampling weight is allowed a temporary expansion at each step in the recursive process.  The formula previously presented as

[7.3]         R_n  =  (R_oN_o + RN) / (N_o + N) ,

serves to describe this process, except that here the sampling weight reverts to its constant value after each recursive step.  This method of combining ratings more closely resembles the ordinary averaging process.  The original sample weight is essentially independent of the new sample weight.  

The primary benefit of a constant sampling weight, as we have seen, is attenuation of sample weights.  In the Elo System this attenuation does not occur until the advent of established ratings, but logically there is no reason why it should not occur from the outset of the rating process.  Attenuation can be achieved more directly though the application of an attenuating factor a, such that 0 < a < 1.  At this point it is best to abandon Elo's notation in favor of one that is more mathematically descriptive.  Our concern is to calculate a sequence of rating results based on cumulative averaging,

               |R₁, |R₂, |R₃, . . . .

The vertical line before each variable indicates cumulative results.  The sequence is calculated using a corresponding sequence of variable sampling weights,

              |N₁, |N₂, |N₃, . . . ,

as well as a corresponding sequence of performance ratings for events i = 1, 2, 3, . . . .

A formula that captures the cumulative nature of attenuated weighted sampling for sequential ratings would be

[8.1]        |R_i  =  (a|N_i-1^. |R_i-1 + N_i^. R_i) / |N_i ,

  where

[8.2]        |N_i  =  a|N_i-1 + N_i  .

Analogous to Elo's notation, R_i is a performance rating based on N_i games.  The variable |N_i corresponds to Elo's constant N_o, and |R_i to the new rating R_n.  The process begins with the original performance rating, which is to say,

[8.3]         |R₀  =  R₀

and

[8.4]       |N₀  =  N₀ .

At each stage in the rating process, the sampling weight is decreased with multiplication by a and increased with addition of N_i.  Thus,

[8.5]       |N₁  =  aN₀ + N₁ ,

[8.6]       |N₂  =  a²N₀ + aN₁ + N₂ ,

and so forth, which is a geometric series.  If N_i is assumed to be a constant N, then

 [8.7]       lim |N_i  =  N / (1 - a)

as i goes to infinity.  Interestingly, if we set a to

                (N_o - N) / N_o ,

that is, the weight of the old rating in Elo's blending process [7.4], we find the limit to be N_o.

A change formula analogous to [7.6] is found by using the identity

[8.8]      |R_i-1  =  (a|N_i-1^. |R_i-1 + N_i^. |R_i-1) / |N_i .

It becomes clear that this formula is an identity by using the substitution

[8.9]      N_i  =  |N_i  -  a|N_{i-1 ,}

which follows from [8.2].  The expanded equation quickly reduces to an identity. 

Subtracting the identity [8.8] from [8.1],

[8.10]      DR  =   |R_i - |R_i-1  =  (R_i - |R_i-1) (N_i / |N_i ) ,

which is essentially a more subscripted version of [7.6], although here |N_i is a variable rather than a constant.

We find, then, that a sampling constant is not necessary for the implementation of rating attenuation.  Accurate attenuation by a factor of  a can be maintained by Formula [8.10] from the very outset of the cumulative rating process.  The downside is that sampling weight must be updated and stored at each step in the cumulative process, along with the cumulative rating itself.