14. A Dual Ratio System

Whether a ratio scale for rating systems represents a real advance over an interval scale is a question that we have left unanswered.  It does, however, create opportunities for statistical experiments.  An interesting example is the Dual Ratio System, which exploits the zero point of a ratio scale to restrict ratings to a specific range of values.  In addition to the usual principle of relative performance

[14.1]                  R /  ER_c  =  W / L  ,

we can define a second principle as

[14.2]                 (C - R) / (C - ER_c)  =  (1 - P) / ( 1 - P_c)

                                                          =  (1 - P) / P

                                                          =  L / W

for an arbitrary constant C larger than any rating.  A convenient value for C is 1, which has the effect of rendering all ratings as decimal values in the open interval

                          0  <  R  <  1 .

Rating formulas follow immediately from the two principles. 

[14.3]                  R  =  ER_c (W / L)                    W  < =  L   
 

[14.4]                  R  =  1 - (1 - ER_c)(L / W)        W  > =  L .

   
The condition under which each formula applies assures that ratings remain within the specified range.  The formulas are equivalent at W = L.  A simple method for calculating the corresponding "established" ratings is to adjust expected scores over the sampling constant N_o (see Methods of Calculation).  Corresponding to [14.3] is the identity

[14.5]                 R_o  =  ER_c [N_o(P_e) / N_o (1 – P_e)] .

Adjusting the identity by actual event scores gives the new established rating

[14.6]                 R_n  =  ER_c [N_o(P_e) + W] / [N_o (1 – P_e) + L] ,

where percentage expectancy is

[14.7]               P_e  =  Ro / (R_o + ER_c)

and

[14.8]               R_o  < =  ER_c  .

The latter condition implies a percentage expectancy less than half and thus corresponds to the condition attached to formula [14.3].  A change formula can be derived by subtracting [14.5] from [14.6], but the result is probably too complicated to be practical.  Corresponding to [14.4] is the established formula

[14.9]              R_n  =  1 - (1 - ER_c)[ N_o (1 – P_e) + L] /  [N_o(P_e) + W]

where percentage expectancy is

[14.10]               P_e  =  (1 - ER_c) / (2 - R_o - ER_c)

and

[14.11]               R_o  > =  ER_c  .

The two established formulas [14.6] and [14.9] are roughly equivalent where R_o = ER_c for a large sampling constant N_o.  Precise equivalence will depend on event scores.

In the test previously performed on basic rating systems (see A Test), the Dual Ratio does quite well.  Figure 5 presents a comparison of standard deviations of percentage expectancies calculated by [14.7] and [14.10] from the percentage expectancies of the predefined crosstable with s = .5.  The Dual Ratio is compared with the Elo System, along with the Progressive System of the next section.  In Figure 7, where the predefined crosstable is generated with s = .3, the comparison is even more striking.  In Figure 6 and Figure 8, the corresponding statistics for Spearman's rank-difference correlation corroborate these results.