9. A Dual Ratio System 

The zero point on a ratio scale can be exploited in a dual formula system which confines ratings to a specific range, the open interval from 0 to 1.  The formulas are

 [8.2a]                  R  =  ER_c (W / L)                    W  < =  L   
and
 [9.1]                    R  =  1 - (1 - ER_c)(L / W)        W  > =  L .

The latter is derived from the relation

[9.1a]                   (1 - R) / (1 - ER_c)  =  (1 - P) / ( 1 - P_c) 

[9.1b]                                                 =  (1 - P) / P.

ER_c is an arithmetic mean.  There is no need, as has been pointed out, for the mean to be geometric.  In the Berkin System division by zero can be avoided by the use of weighted averages.  Here it is avoided simply by maintaining W/L as a constant over game instances.

Established formulas are developed as described in Averaging.  The derivatives are taken at P = .50, where R_o = R_c, to simplify calculations.

[9.2]                    R  =  4R_o(W - NP_e) / N_o                   R_o < = .5 ,

where

[9.3]                     P_e  =  R_o / (R_o + R_c) ,

and

[9.4]                    R  =  4(1 - R_o) (W - NP_e) / N_o            R_o > = .5 ,

where

[9.5]                    P_e  =  (1 - R_c) / (2 - R_o - R_c).

The condition under which each set of formulas applies, i.e. the player's rating in relation to the midpoint of the range, assures consistent rating changes from event to event and between players of similar strength.

The restriction of ratings to the range from 0 to 1 provides a frame of reference that is lacking  in other systems.  It can be argued further that a true ratio system, such as the Dual Ratio or the Berkin System, provides a control for deflation by virtue of its zero point.  Deflation is the tendency of ratings to drift lower in response to changes in the rating pool, a quirk in the sequential rating process to which Elo devoted considerable attention. In an interval system there is nothing to prevent the rating scale from shifting to the negative indefinitely.