8. Some Basic Systems

Performance rating formulas follow immediately from [3.1] and [4.1] as


 [8.1]                    R  =  ER_c + K(2P - 1) 

[8.1a]                        =  ER_c + K(W - L) / N  

and

 [8.2]                    R  =  ER_c (W / L)                    L > 0 .

Performance formulas are used for an initial sample of N_o games, which becomes sampling weight in the context of established ratings.  It has a value of 25, 32 or 50 in the USCF version of the Elo System, depending on rating levels, and a value of 80 in the FIDE version.

The established version of the interval formula has 2K as the constant derivative with respect to P in formula [8.1].  Thus,

 [8.3]                  R  =  2K(W - NP_e) / N_o ,
where
 [8.4]                  P_e  =  (R_o - ER_c) / (2K)  + .5 .

P_e is restricted to the range of a percentage score in accordance with its definition from the basic formula .  It may be substituted into [8.3], giving 

 [8.3a]              R  =  [ K(W - L) +  R_c - NR_o ] / N_o ,

where the absolute difference | R_c - NR_o | should not exceed KN.

A simple established formula based on [8.2] was proposed to FIDE by Berkin in 1965 [E1].

[8.5]                 R  =  (ER_cW  -  R_oL) /  N_o ,

The development presented by Elo involves one or two questionable manipulations. Contrary to his main objection, however, a geometric average for ER_c is not required.  Again the distinction must be made between direct measurements based on a ratio scale and rating statistics which make use of ratios.  The important thing in a ratio rating system is to avoid division by zero.  By cross multiplying in the basic ratio relation,

[8.2a]                 R(L)  =  ER_c(W) ,

which suggests the relation

[8.6]                    E[R(1-S)]  =  E[R_cS] 

though it is not equivalent.  The latter involves a weighted opposition mean in which the ratings of opponents are weighted according to the scores against them, which may be written

                         [R_w + .5R_d] / W , 

where R_w designates ratings won against, and R_d ratings drawn against.  Substituting for the unweighted mean in [8.2] yields the new performance formula 

[8.7]                  R  =   [R_w + .5R_d] / L ,          L > 0 ,

which follows also from [8.6].  The established formula becomes

[8.8]                  R  =  (R_w + .5R_d  -  R_oL) /  L_o ,

which is similar to the original Berkin formula [8.5].  References to the Berkin System henceforth  will indicate these revisions, using formulas [8.7] and [8.8].  Here L_o is an arbitrary number of losses. The Berkin performance formula is used until a player loses L_o times.  Consequently, the number of games in the initial sample varies from player to player.  The new established formula is seen to follow from [8.7] by an adaptation of Elo's blending method,

[8.9]              R  =  (R - R_o) (L / L_o) .

The resulting formula [8.8] conveniently deals with events in which a player wins every game.  Expressing this result as a separate performance by [8.7] would involve division by zero.

It can be shown that the effect of the new Berkin formulas is to minimize the difference

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

which becomes clear from [8.6] when terms are arranged in pairs, as

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

The minimization then follows from Gauss's principle, with zero dropping from the expression.