13. The Berkin System
The Berkin System is a novel rating system. It is based on the idea 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. We have taken some liberties in presenting the system, but the essential idea remains intact. 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
[13.1] S[RcS] / W .
Replacing the opposition average in [5.1] and reducing gives the simple ratio formula
[13.2] R = S(RcS) / L , L > 0 .
Only losses are recorded in the denominator, with draws counting as half-point losses. The numerator in effect does not include the ratings of winning opposition ratings, which are weighted by zero, but it does include the ratings of drawing opponents, weighted by the usual half point. The absence of data for losses is compensated in the calculation of opposition ratings, without the duplication characteristic of other systems. As a consequence [13.2] can be calculated simultaneously for a competing field, and the Berkin is the only recognized ratio system that can claim that advantage. A case can furthermore be made for the system as an improvement over ordinary ratio systems. For a given percentage score P, if the opposition ratings are all the same, a Berkin rating yields the same result as a standard ratio rating. Consider now a group of opponents with ratings that vary, and let us suppose that the percentage score against these opponents is P, such that 0 < P < 1. The standard ratio formula yields a single rating regardless of the individual outcomes, while the Berkin formula yields a variable rating depending on the scores against individual opponents. The latter will be higher than the standard result if the wins are against higher ratings, lower if the wins are against the lower ratings. The Berkin rating, in short, extracts more information from precisely the same data, and the conclusion seems inescapable that it is a better statistic.
The efficiency of the system is demonstrated as follows: Multiplying [13.2] by L, and expressing L as the total of lost points,
[13.3] S[R(1 – S)] = S [RcS] .
Subtracting the right side,
[13.4] S[R(1 – S)] – S [RcS] = 0 .
The difference in totals may be regarded as a total of differences over individual pairings,
[13.5] S [R(1 – S) – RcS] = 0 ,
which may also be expressed as a mean
[13.6] 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,
[13.7] S[R(1 – S) – RcS] 2
is a minimum over real R where R is calculated by [13.2]. The weighted ratings may be said to be mutually predictive.
Avoiding the difficulties that arise from opposition averaging, we derive a change formula from [13.2] for single-game events. Scoring S against the new opposition rating Rcn gives the new rating,
[13.8] Rn = [S(RcS) + RcnS ] / [L + 1 – S] .
The original rating may be given by an expression equivalent to [13.2] with the same denominator as [13.8],
[13.9] Ro = [S(RcS) + Ro(1 – S )] / [L + 1 – S] .
Subtracting Ro from Rn gives
[13.10] DR = [RcnS – Ro(1 – S )] / [L + 1 – S] , L + 1 – S > 0 .
The denominator in this expression increases only with a loss. L can be maintained at a constant value from one game to the next when it is large enough to avoid significant error, giving
[13.11] DR = [RcnS – Ro(1 – S )] / [No + 1 – S] ,
which is a reasonable change formula. Using Elo's blending process, we can simply write
[13.12] DR = [RcnS – Ro(1 – S )] / No ,
and for multiple results we can substitute W for S, and L for (1 - S). Now with slightly different notation we have Elo's formula (56), representing the established rating formula of the Berkin System [E1, 8.62].
[13.13] DR = (RcW – RL) / No ,