8. Attenuation

We have seen that attenuation in the Elo System is achieved by means of an arbitrary constant N_o, which may be viewed as a limit for sample size in recursive averaging.  Attenuation can be achieved directly though the application of an attenuation factor f, such that 0 < f < 1.  A formula that captures the recursive nature of attenuated weighted sampling for sequential ratings would then be

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

where

[8.2]        |N_i  =  f |N_i-1 + N_i  

for events i = 1, 2, 3, . . . .  

Similar to Elo's notation, R_i is a performance rating based on N_i games.  Recursive values are preceded by a vertical line.  The constant N_o is replaced by the variable |N_i, and the new rating R_n by |R_i.  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 f and increased with addition of N_i.  Thus,

[8.5]       |N₁  =  f N₀ + N₁ ,

[8.6]       |N₂  =  f ²N₀ + f N₁ + N₂ ,

and so forth.  This is a geometric series.  Assuming N_i to be a constant N, then |N_i converges to the value

 [8.7]       lim  =  N / (1 - f).  

The process is self-correcting.  For example, if we start with some value other than N but otherwise maintain N_i at a constant N, the series still converges to the indicated value.  Interestingly, if we set f to 1-N/N_o, the weight of the old rating in Elo's blending process [7.4] , we find the limit to be N_o.

From [8.1] and [8.2] it follows that

[8.8]      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].  The one difference, aside from differences in notation, is the substitution of the variable |N_i  for the constant N_o.  From this we see that an arbitrary constant like N_o is not necessary for attenuation.  Formula [8.8] could actually be applied from the outset of the rating process, obviating the distinction between performance and established ratings.

A drawback of [8.8] is that the variable |N_i must be stored along with the current rating.  If an arbitrary constant is preferred, a better candidate than N_o would be the next to last value in the recursive sequence, which may be designated N_k.  Reverting to Elo's notation, we have

[8.9]          R_n  =  [R_oN_k + RN] / (N_k + N) .  

The advantage over Elo's blending process is that the ratio of the original sample weight to the new sample weight,

                         N_k /  N 

no longer depends on the difference between sampling weight and sample weight, as in the ratio

                     (N_o - N) / N .

In the latter, N cannot exceed N_o.