14. The Berkin System

 

  



An interesting twist on ratio ratings involves the used of weighted ratings, an idea introduced by Berkin in 1965 [E1, 8.62].  The Berkin System was a candidate for official recognition by FIDE, but adoption of the Elo System in 1970 led to its neglect.  Berkin actually spoke of weighted points, but since his points are weighted by ratings, it amounts to the same thing.  The weighted average of opposition ratings is

[14.1]               S[RcS] / W .

Replacing the opposition average in [5.1] and reducing gives the simple ratio formula

[14.2]              R  =   S(RcS) / L ,             L > 0 .

Only losses are recorded in the denominator.  The numerator in effect does not include the ratings of winning opposition ratings, which are weighted by zero.  This loss of data is compensated in the calculation of opposition ratings, without the duplication characteristic of other systems.  The efficiency of the revised system is demonstrated as follows:    

Multiplying [14.2] by L, and expressing L as the total of lost points,

[14.3]              S[R(1-S)]  =  S [RcS] . 

Subtracting the right side,

[14.4]              S[R(1-S)] - S [RcS]  =  0 . 

The difference in totals may be regarded as a total of differences over individual pairings,

[14.5]              S [R(1-S) - RcS]  =  0  ,

which may also be expressed as a mean

 [14.6]              E[R(1-S) - RcS]  =  0  .

Again we have a mean formula to which Gauss's principle can be applied.  With zero dropping from the sum of squares,

[14.7]              S[R(1-S) - RcS] 2

is a minimum over real R where R is calculated by [14.2].  The weighted ratings may be said to be mutually predictive.

Anticipating the next section, it will be useful to derive a change formula from [14.2] for single-game events.  Scoring S against the new opposition rating Rcn gives the new rating,

[14.8]              Rn  =   [S(RcS) + RcnS ] / [L + 1 – S] .

The original rating may be given by an expression equivalent to [14.2] with the same denominator as [14.8],

[14.9]              Ro  =   [S(RcS) + Ro(1 - S )] / [L + 1 – S] .

Subtracting Ro from Rn gives

[14.10]          DR  =   [RcnS - Ro(1 - S )] / [L + 1 – S] ,        L + 1 – S  >  0 .

The denominator in this expression increases only with a loss.  It can be maintained at a constant No, as in the Elo System, though with somewhat less justification.  The resulting formula,  

[14.11]          DR  =   [RcnS - Ro(1 - S )] / No , 

is simulated in Table 10.  It is essentially the same formula cited by Elo (E1, 8.62).  See Rating by Single Games for a discussion of the table.