5. Ratio Ratings

Having discovered the secret of relative performance in typical rating systems, we now ask whether it might just as well be represented by a ratio of percentage scores as by a difference.  The corresponding principle for ratio systems is that rating ratios reflect percentage score ratios.  To avoid the complexities of harmonic averaging, we can express this by reciprocals as 

[5.1]        E(R_c / R)  =  P_c / P 

                                          =  (1 - P) / P .

The basic ratio formula then follows as  

[5.2]        R   =   ER_c (P / P_c) ,            P_c > 0 ,

                     =   ER_c (W / L) ,            L > 0 ,

with the same variables as in the interval formula.  It also follows from Gauss's principle that

                S[(R_c / R)  -  (P_c / P)] ²

is a minimum over real values of R for the calculated value.  The rating ratios may thus be said to predict relative performance, though in a somewhat weaker sense than in the case of  interval ratings.  Here relative performance cannot be defined for individual games because of the possibility of division by zero.  Using a geometric mean for the average opposition rating leads to Elo's treatment.

Elo was much impressed with the possibilities of a ratio scale, though perhaps for the wrong reasons.  In the realm of physical measurements, ratio scales represent a considerable advance over interval scales, but it is by no means certain that this holds in the realm of statistics. In any case, Professor Elo went on to develop a ratio version of his rating system that was eventually implemented by the USCF.  His version of the basic ratio formula is oddly missing form the 1978 treatise but can be inferred from the context.  Solving his formula (46) [E1, 8.43] for D gives

[5.3]         D(P)  =  2C ^. log₁₀(P / P_c) ,

where C is the class interval of 200 rating points.  The basic formula would thus be

[5.4]         R   =   ER_c + 400 log₁₀ (P / P_c) ,          P > 0,  P_c > 0 .

This formula follows directly from [5.2], assuming a geometric mean, by taking the logarithm of each side to a very small base b, approximately 1.005773.  It is worth noting that this does not produce a linear formula equivalent to [4.1] since relative performance would be

                  log_b(P) - log_b(P_c) .

Consequently, the efficiency of [5.4] cannot be demonstrated from the theory developed here for interval ratings.  Perhaps an argument can be made for its efficiency based on the ratio formula from which it derives.  For large ratings, say above 1000, and for a small ranges of values, say within the class interval of 200,  the geometric average generally tracks within a point or two of the arithmetic mean.  Discrepancies are therefore likely to be subtle.  

An interesting twist on ratio ratings involves the used of weighted ratings, an idea introduced by Berkin in 1965 [E1, 8.62].  A weighted rating is associated with the point score against it.  (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

[5.5]             S[R_cS] / W .

Replacing the average in [5.2] gives the simple ratio formula

[5.6]              R  =   S(R_cS) / L ,             L > 0 .

Moving L to the left side, an equation of means follows as

[5.7]              E[R(1-S)]  =  E[R_cS] .  

If the terms of the respective means are arranged in pairs corresponding to individual games, the means can be combined to give

[5.7]              E[R(1-S) - R_cS]  =  0  .

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

                      S[R(1-S) - R_cS] ²

is a minimum over real R where R is calculated by [5.6].  The weighted ratings may be said to be mutually predictive.