11. Methods of Calculation

 

  


Percentage expectancy in the Elo System is associated with a probability distribution function..  Rating changes are calculated with the derivative of this function.  If it is true, however, that probabilities are to be viewed as long-term percentage scores, then every percentage score is associated with a probability, and rating changes may be calculated by ordinary averaging.  We begin with a percentage score, such as might result from any chess event:

[11.1]          Po  =  Wo / No ,

which in chess has the special meaning of Wo points scored (including draws) in No games.  The subscripts indicate original values since we are considering the process by which percentage score changes.  For the next event, in which W points are scored out of N games,

[11.2]          Pn  =  (Wo + W) / (No + N) , 

which is the new percentage score by cumulative averaging.  Cumulative averaging can be applied recursively, which avoids having to write out the entire average at each step.  The original percentage score [11.1] as an estimate of long-term percentage score was the expected score, which may be written as the equivalency

[11.3]         Pe  =  (Wo + NPo) / (No  + N) .

The change in percentage score now follows as

[11.4]         DP  =  Pn - Pe  =  (W - NPo) / (No  + N) .

This can be converted to a rating change by using the constant derivative of the basic linear formula with respect to P, which is 2K:

[11.5]          DR  =  2K(W - NPo) / (No  + N) .

A change formula can also be derived from the basic linear formula [4.1].  The new rating, based on a new percentage score, is

 [11.6]         Rn   =  ERc + K(2Pn - 1) .

The original rating may be written using a percentage expectancy for the same mean opposition rating

[11.7]         Ro  =  ERc + K(2Pe - 1) .

Subtracting Ro from Rn ,

[11.8]           DR  =  2K(Pn - Pe) .

We can substitute into [11.8] using the percentage difference [11.4], giving the same result as in [11.5].  There is no need to use a derivative.   

Similar observations apply to ratio ratings.  The derivative of the basic ratio formula [5.1] with respect to P is 

[11.9]        R'  =  (R + ERc)2 / Rc ,

which can be used to convert [11.4] into a change formula for ratio ratings:

[11.10]      DR  =  R' (W - NPo) / (No  + N) .  

The derivative is not necessary for deriving ratio change formulas, as will be seen in a subsequent topic (Rating by Single Games).  The formula developed there is based on single-game events:

[16.10]       DR  =  [S(Ro + Rc) – Ro] . [Ro + Rc] / [(1 – S)(Ro + Rc) + NoRc] .

Since it does not use the differential form of [11.10] it is likely to be more accurate, but it is interesting to compare the two formulas.  First, applying [11.10] to single-game events,

[11.11]     DR  =  R' (S - Po) / (No  + 1) ,

We can write the write the expression S-Po in terms of percentage expectancy as

                [S(Ro + Rc) - Ro] / (Ro + Rc) .

Substituting this expression back into [11.11], along with the derivative for a single game by [11.9], and reducing,

[11.12]      DR  =  [(Ro + Rc) / Rc] . [S(Ro + Rc) - Ro] / (No + 1) .

Multiplying through and rearranging terms,

[11.13]     DR  =  [S(Ro + Rc) - Ro] . [Ro + Rc] / [Rc  + No Rc] ,

which is proportional to [16.10], the result obtained without a derivative.