5. Ratio Ratings

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

                     =   ER_c (P / 1 - P) ,        P < 1 ,

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

with the same variables as the interval formula. Taking the reciprocal of this relation,

[5.2]         ER_c / R  =  P_c / P ,

and from this we conclude by Gauss's principle that

                 S[(R_c / R)  -  (P_c / P)] ² ,        R > 0 ,  P > 0 ,

is a minimum over real values of R for the calculated value.  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.

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. In the realm of statistics, their primary advantage is that probabilities can be expressed over a full range of rating ratios. In any case, Elo went on to develop a ratio version of his rating system which was eventually implemented by the USCF.

Elo's development of his ratio system was very different from that of his interval system, though he aimed in the end to show the near equivalence of the two systems.  As we shall see, his ratio system follows, roughly at least, from the basic ratio principle of [5.1]. We begin with the performance rating formula, which is oddly missing from the 1978 treatise but may be inferred from Formula (46):

[5.3]         P(D)  =  1 / (1 + 10^-D/2C) .

Solving for D gives

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

where C is the class interval of 200 rating points.  A performance formula would thus be

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

This formula follows directly from [5.1], assuming a geometric mean for the opposition ratings, by taking the logarithm of each side to a small base b.  There is no special notation for ratings as logarithms.  The base may be calculated from

[5.6]         log_b R  =  400 log₁₀ R .

By the rule of logs

[5.7]         log_b R  =  log₁₀ R / log₁₀ b ,

and setting equals to equal,

[5.8]         log₁₀ R / log₁₀ b  =  400 log₁₀ R .

Consequently,

[5.9]         log₁₀ b  =  1/400

and

[5.10]         b  =  10^1/400

or about 1.00577.  Formula [5.4] as a logarithm of [5.1] assumes a geometric mean for the opposition ratings of [5.1], but the mean proposed is arithmetic to take advantage of least-squares approximation.  This is a potential source of error in the Elo System.  Possible improvements are discussed at Rating by Single Games.  It is also worth noting that taking the logarithm of [5.1] does not produce a linear formula equivalent to [4.1] since relative performance in the result would be

                  log_b(P) - log_b(P_c) .

Consequently, the efficiency of [5.4] cannot be demonstrated from the theory developed for interval ratings.  Another ratio system is the Berkin System, discussed under that heading.