10. A Progressive System

Since a ratio scale does not confer any inherent advantage, systems such as the Dual Ratio must be viewed as methods for controlling the effects of sequential ratings, including deflation and undefined percentage expectancies.  An adaptation of the Berkin System presents another possibility for controlling these effects in the form of a progressive system, in which rating adjustments are always positive.  This possibility was hinted at in the discussion of problems arising from traditional applications.

The performance formula [8.7] of the revised Berkin System is a convenient jumping-off point for a progressive system.  The constant L_o is postulated as an arbitrary number of losses, which a player must sooner or later reach.  The formula may be written as a sum of  the event ratings leading up to the player's reaching or exceeding  L_o:

[10.1]                   R  =  _n [ (R_w + .5R_d) / L_o ] ,

where the opposition sums are variable over the n events.  There are two facts to be noted about this sum:  (1) it represents an application of formula [8.7] to the combined results, and (2) the resulting increments are always positive.  One may also write

[10.2]                   R  =  (R_w + .5R_d) / L_o 

for each event.  Once the established formula comes into effect, losses in the Berkin system cause rating decreases as a consequence of Elo's blending method. Such decreases can be avoided simply by adopting the performance rating formula [10.2] as an established formula.

The simple progressive system that emerges suffers the problem of explosive growth of ratings.  If formula [10.2] were used in the simulation below, where each player participates in 1125 games, the average rating would increase from an initial .5 to over 10,000, with the maximum at 16,892.  The average rating in the revised Berkin System, by contrast, does not increase at all and produces a maximum of approximately 1.53.  Better results obtain if the constant L_o is replaced in the progressive established formula by a variable, say L_n, which is allowed to increase as losses occur.  The average rating is then about 15, with a maximum of 39.89.  The latter version of the established formula is used in the simulation below.  Allowing L_n to increase indefinitely gives the system a sampling advantage, as noted in Averaging, which no doubt accounts for its spectacular performance.  The simulation suggests, in any case, the possibility of progressive ratings as a viable system.  The appeal of such a system to the rank-and-file player would, for course, be immeasurable.